Least Squares Estimation of Sailing Yacht Dynamics from Full-Scale Sailing Data

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1 THE st CHESAPEAKE SAILING YACHT SYMPOSIUM ANNAPOLIS, MARYLAND, MARCH 3 Least Squares Estimation of Sailing Yacht Dynamics from Full-Scale Sailing Data Katrina Legursky, University of Kansas, Lawrence, KS ABSTRACT Linear dynamic models are extremely useful in autonomous vehicle controller development because they are straightforward to estimate from real data and they enable advanced control system design. To date an in depth investigation of the possibility of using a linear dynamic system model to represent sailing yacht dynamics has not been made. A non-linear simulation of a m racer/cruiser racer and full-scale data collected aboard a 3 ft day-sailer are used in conjunction with the one-shot least squares estimation algorithm to evaluate this possibility. Using the non-linear simulation data it is found that it is possible to use a linear model to represent sailing yacht. Full-scale data collected while sailing upwind on a port tack is used to estimate different linear models. NOTATION R A x Effective angle of attack on rudder (rad) Parameter vector Apparent wind angle (rad) Effective wind angle (rad) Leeway angle (rad) Sail deflection angle (rad) Rudder deflection angle (rad) Average deviation of the states Downwash angle on rudder from keel (rad) Pitch angle (rad) Eigenvalues Density of air (kg/m 3 ) Density of water w (kg/m3 ) Roll / heel angle (rad) Yaw / heading angle (rad) C Rudder roll coefficient K R C Rudder yaw coefficient N R C Rudder surge coefficient X R C Rudder sway coefficient Y R D K Draft of yacht f Aerodynamic force vector aero Acceleration due to gravity (m/s ) Metacentric height of the yacht (m) Identity matrix Ixx, Ixz, I Rigid body mass moment of inertia (kg-m ) zz K Total rolling moment (N-m) Total roll aerodynamic coefficient Waterline length (m) WL Rigid body mass of yacht (kg) N Total yawing moment (N-m) Total yaw aerodynamic coefficient Total roll angular velocity (rad/s) Total pitch angular velocity (rad/s) Total yaw angular velocity (rad/s) S Wetted surface area of sails (m ) A Control vector Total surge velocity (m/s) Total sway velocity (m/s) V Non-dimensional sway velocity Total boat velocity (m/s) Apparent wind velocity (m/s) Effective wind velocity (m/s) State vector

2 X S Longitudinal location of rudder c.o.e. (m) Total surge force (N) Total upright resistance of yacht (N) Total surge aerodynamic force coefficient Output vector Total sway force (N) Total sway aerodynamic force coefficient INTRODUCTION Autonomous sailboats have great potential to serve as platforms for long term oceanic observing (Cruz and Alves, ). A yacht under sail is a beautifully complex dynamic system. Building an autonomous controller for such a vessel is a challenge because of that very complexity. Where a seasoned sailor gets his insight into the operation of his sailboat from a wealth of past experience and sensory inputs, a computer has only the model given to it and the information available from its sensors to make the same decisions. Translating this wealth of experience to a computer controller which is capable of autonomous operation is a formidable problem. In the last several years there has been a growing interest in designing autonomous sailboats for competitions such as the World Robotic Sailing Championship, SailBot, and the Microtransat Challenge. The focus of competitors in these events has been the design of a complete autonomous sailing system, including hull and rig design (Miller et al., 9; Rynne and von Ellenrieder, ), sensors (Neal et al., 9), onboard processing (Alves et al., ), navigation, and control (Stelzer et al., 7). s and control systems for existing boats have been similar to a standard autopilot driving a heading and choosing a sail angle based on wind angle. Only recently have physicsbased models been presented for future use on autonomous sailing platforms (Petres et al., ; Xiao and Jouffroy, ). System identification is a powerful tool that allows engineers to use real data to build models which can later be used for control system design. These techniques have been widely explored by the aerospace industry. They have qualities, develop controllers for autonomous flight systems, improve physics-based simulation models, and to compare wind tunnel vs. flight test measurements (Jategaonkar, ; Morelli and Klein, ; Tischler and Remple, ). These ideas have great potential for increasing the fidelity of physics-based sailing yacht models through full-scale sea trial data, but have yet to be explored. The base upon which all computer-driven autonomous control systems are built is the mathematical model of the system dynamics. The most simple form of model to use for advanced control system design is a linearized system model. Even though full physics based models of aircraft are highly non-linear, linear model have been very successfully used for autonomous flight control design. These linear representations of aircraft dynamics are largely responsible for the success of autonomous flight systems. This paper presents the status of a project whose overall goal is to use system identification techniques to estimate a dynamic sailing yacht model from full-scale data. Since linear models are straightforward to estimate and have such potential for control system development, they are a natural first choice in model structure. A non-linear model of a sailing yacht is implemented in simulation to generate data for estimation. This data is used to explore the feasibility of applying the one-shot least squares (LS) technique to the full-scale data that has already been collected aboard a 3ft day-sailer. Ten linear models are estimated from full-scale data recorded while sailing upwind on a port tack. NON-LINEAR SAILING YACHT MODEL For sailing yachts, it is possible to describe the dynamic yacht system with a set of physics-based differential equations of motion derived from Newtonian physics. This set of differential equations represents the model, and contains the information necessary for prediction and understanding of the physical system behavior. For a surface vessel symmetric about the xz plane, the four degree of freedom rigid body equations of motion expressed in a body coordinate system fixed to the center of gravity can be fully expressed as (Fossen, ; Stengel, ): [] [] [3] [] where U and V are the velocities along the surge and sway axes, and P and R are the angular velocities about the roll and yaw axes, all in the body fixed coordinate system. X and Y represent the total external forces acting along the surge and sway axes; K, and N represent the total external moments applied around the roll and yaw axes. m is the rigid body mass of the yacht, and,, xx xz zz I I I are the rigid body mass moments of inertia of the yacht. For a sailing yacht, the forces and moments are generated by the sails, hull, keel, and rudder. The effects of added mass and buoyancy/righting moment are in this case considered to be external forces. The external for model is defined as follows (Legursky, b; Masuyama et al., 99): []

3 H Rud S [] K K K K mggm sin [7] H Rud S [] [7] H Rud S [] where X represents total upright hull drag which is a function of forward velocity, the subscript H denotes hydrodynamic forces and moments, Rud denotes forces and moments generated by the rudder, and S denotes aerodynamic forces and moments generated by the sail plan. g is acceleration due to gravity, and GM metacentric height. The hydrodynamic force model is based on non-linear non-dimensional hydrodynamic coefficients (Masuyama et al., 99): Figure : The Wind Triangle [] VV H H WL K VR VVVV V V H 3 H WL K VV VVV P R [9] [] [9] Where C X R, C Y R, C K R, and C N R are non-dimensional coefficients, R is the physical rudder deflection from centerline, and R is the effective angle of attack on the rudder as defined by: [] K N H H Where: KV K K V V V VV V K V K V 3 VVV q L D K P H WL K P NV N N V VV V N V N V 3 VVV q L D N R H WL K R [] [] [3] [] [] V is a non-dimensional boat velocity, is leeway, H is the dynamic pressure, is the density of water, w V is the B total boat velocity, WL is the waterline length, D K is the total draft of the yacht, and are nondimensional hydrodynamic derivatives of the hull and keel. The rudder force model is: Where is the angle of inflow from the downwash generated by the keel, and x R is the longitudinal distance of the center of effort of the rudder to the c.g. of the boat. The aerodynamic force model includes the main sail deflection angle as a control input to the system. It is important to include the sails as a control to the system from a safety perspective, especially if the model will be used on an autonomous boat. This aerodynamic model allows for the sails to be in less than perfect trim. It is a very simplified version of the complex aerodynamics of the sails, but what is important from the modeling perspective here is to capture the largest control effect of the sails, which is accomplished through mainsail deflection. Of course, the sails have several secondary controls but the effect of these is small and difficult to measure, and thus is ignored by this model. The aerodynamic model is expressed in the body coordinate system rather than the airflow reference frame which would entail lift and drag coefficients (Legursky, a). To eliminate the dependence of aerodynamic parameters on heel angle, effective angle theory is used to calculate an effective wind speed and angle (Hansen, ): [] []

4 The aerodynamic force and moment vector may be expressed as: [3] [] The non-dimensional coefficients, and, represent the total aerodynamic force and moment coefficients. S A is the reference sail area, and a is the air density. The total sail angle b is considered a control input to the system, and is measured w.r.t. the centerline of the yacht (See Figure ). It may be expressed as the sum of the optimal sail angle,, plus the deviation of the angle from b the optimum, b: [] The optimal sail angle is considered to be a function of the effective wind angle, so for every effective wind angle there is a known sail deflection. The total sail force and moment coefficients are defined as: [] [7] [] [9] The terms represent the sail coefficient at the optimal sail deflection. The terms represent the change in force or moment coefficient w.r.t. b: b [3] This model is implemented in MATLAB and simulated using the ode3 function with pre-defined control inputs. The data generated by the simulation is recorded and used for estimation. LINEAR SAILING YACHT MODEL The linear model is not meant to fully capture the nonlinear behavior of the sailing yacht, it is meant to predict the near future motion of the yacht given a set of initial conditions. Thus, a linear model simulated in open loop is only valid for a few seconds, after a few seconds its behavior is expected to diverge from what the non-linear model would predict, as well as from what the full-scale yacht experiences. The benefit of the linear model is for control purposes, knowing the near future is enough for advanced robust control and filtering strategies to be implemented. A linear time-invariant state space model is used to represent the sailing yacht as a dynamic system in a trim condition (initial condition). This trim condition may be thought of as a point of constant true wind speed and direction on a typical yacht polar diagram. From the true wind speed and direction, the equilibrium sailing condition of the yacht may be found as is done with a velocity prediction program (VPP). The linear model is defined in terms of perturbed motion variables: [3] [3] [33] [3] [3] [3] [37] The capitals denote total variables, which would mean the actual total measured motion variables of the yacht. The small letters denote the initial condition about which the system has been linearized, so these are the motion variables which represent the steady state motion of the yacht at the chosen wind speed and direction (as would be determined by a VPP). The deltas represent the perturbation away from that initial condition. A primary assumption in model linearization is that all perturbations away from the initial condition are small. Small is relative, and what it means for the sailing yacht system has yet to be determined. The linear model is cast in state space form: Where the state vector will be defined as: The control vector will be defined as: [3] [39] [] Linear time invariant state space models are characterized by the eigenvalues of the system A matrix. of motion of the system. These may be first or second order. First order modes have only a real part, and are said to be stable when they are negative. Second order modes have a real and imaginary part, and are said to be stable when the real part of the eigenvalue is negative.

5 ONE-SHOT LS FORMULATION The LS algorithm may be solved in one step for linear systems, thus, computationally it goes very fast and can be applied to several large sets of data and yield estimated models very quickly. One of the assumptions of the one shot least squares algorithm is that measurements of the independent variables, in this case the states, is perfect. This, of course, is never the case with real measured data, however the algorithm remains very useful as long as the data is of high-quality. Pre-processing the data to remove noise and biases is commonly done to prepare data for use with a least squares estimator. The full-scale data used for this analysis has been pre-processed. The unknown parameters to be estimated of the model described by Eqs. [3] to [] are the numbers in the A and B matrices. One-shot LS allows the numbers in these matrices to be calculated in one step given a set of measurements: [] The unknown parameter matrix is formed by combining the A and B matrices into a single partitioned matrix, and may be solved for using the formula in Eq. [3] where N is the number of measurements (Jategaonkar, ). T N T u u N u u X X X X k m m k m m T [] [3] Computation of the unknown parameter vector is very fast, especially using an analysis tool like MATLAB that can do matrix multiplication and inversion very quickly. This technique, then, is well suited to a preliminary investigation of estimation techniques to develop a linear model from full-scale data. FULL-SCALE TEST PLATFORM AND DATA The test vehicle is a Precision 3ft day-sailer, which has been equipped to perform system identification. All tests were performed on Clinton Lake in Lawrence, KS. Data is collected from a variety of sensors (Legursky, b) at Hz by a LabView Virtual Instrument. The data includes GPS position, heading, and velocity, hull speed, Euler angles, rate of change of Euler angles, linear accelerations, apparent wind speed and velocity, rudder deflection angle, and boom (mainsail) deflection angle. Data has been collected on several days in differing weather conditions. The data from each day includes segments where the boat is sailed at constant apparent wind angles on both port and starboard tacks ranging from close hauled (true wind ~ ) to a beam reach (true wind ~ 9 ). During such a segment inputs in the form of singlets and doublets are given to the system through the rudder and main sail angle. The Precision 3 is a sloop rig and is not equipped to measure the set of the jib. Thus, during all constant A time segments, the jib is set and not varied. Since the nonlinear model that has been formulated varies with mainsail angle, it is assumed that the effect of the jib is included in the overall sail force and moment coefficients. The secondary sail controls are never adjusted or utilized, even in between days, in order provide the most consistent data possible. ESTIMATION OF SAILING YACHT DYNAMICS FROM SIMULATION AND FULL-SCALE DATA Data generated by the non-linear simulation was used to estimate a linear model using the one-shot LS method. The non-linear simulation data is used because it closely resembles full-scale data sets and does not contain the noise of real world data. The total non-linear data set used with the algorithm consists of six maneuver segments, two rudder singlets, two rudder doublets, and two sail inputs. Each segment represents seconds of simulation. With a sampling rate of Hz, this results in total data points used for the estimation. Two of these maneuvering segments, a rudder singlet and sail input, are shown in Figure. The dashed line in the center marks where one data segment ends and another begins. The linear model estimated from this data is given in Eq. [], its initial conditions are given by Eq. [], and the eigenvalues given by Eq. []: R B [] [] [] model is shown in Figure which shows the estimated linear model fits the non-linear data well. To quantify this fit, we will look at two things. The first is the average deviation of the estimated model from the measured data over the first five seconds of simulation. Only the first five

6 seconds seconds system average of the linear simulation are informative, after five the accumulated errors inherent in the linear model. The deviation of each of the states for the model represented in Figure and Eqs [-] is reported as: [7] Measured Non-Linear Response - Estimated Linear Response Marks Start of New Data Set and Simula Zero (Initial Condition) 3 Figure : Comparison of non-linear simulation data and the response of the linear model estimated from that data. Especially in the first seconds of simulation, the linear model fits the non-linear data well Figure 3: Sample full-scale data estimation set -

7 Table : Error analysis of selected models estimated from the full-scale data. The first five columns are the average deviation of each of the states, and the last column is the total standard deviation for the model. (m/s) (m/s) p (rad/s) (rad/s) (rad) The second is the total standard deviation of the model, summed over all five states. This is calculated according to Eq.. Ns model in, i i n [] Where x is the deviation of the model from the data at in, each point, and is the mean deviation of each of the states, as reported in Eq.[7], and Ns is the number of samples. This results in a single number by which models may be compared to one another. The total standard deviation for the linear model response estimated from the non-linear simulation data shown in Figure is.. As shown in EQ 7, the deviation in is. m/s, which is a very small percentage of the total surge velocity, 3.7 m/s. The deviation of the sway velocity is much higher,.33 m/s, where the total sway velocity is -. m/s. However, as may be seen by the fit in Figure, the rate of change of the sway velocity is very well matched, and the deviation of the linear model appears to contain a definitive fixed negative bias. Overall, this model as shown in Figure is a very good fit. Therefore, the total standard deviation of this model fit will be used as a benchmark for comparison of the models estimated from the imperfect full-scale data. After this preliminary result, the one-shot LS method was used to estimate the linear model from full-scale data. The full-scale data represents data collected while sailing in the upwind condition on a port tack. The rudder singlets, worth of sailing data. Each of these maneuvers and associated responses were recorded in intervals of approximately 3 seconds. The initial conditions for each maneuver are calculated by averaging the first seconds before a maneuver begins. A set of estimation data contains one of each of these three maneuvers, resulting in data points per set of estimation data. A sample set of a fullscale data set used for estimation is shown in Figure 3. The vertical dashed lines again represent where one maneuver segment ends and another begins. A total of estimation data sets were formed this way, resulting in estimated linear models. The estimated models were quite different, as shown by their eigenvalues plotted in the complex plane in Figure. The response of each of the ten models is visually inspected and four models were selected for further analysis. This analysis consists, as with the analysis of the simulation data, of an evaluation of the model fit during the first five seconds of an open loop simulation of the estimated linear model. In this respect, the full-scale data is much more difficult to analyze because the times at which the maneuvers begin vary greatly and must be manually selected in order to capture as much of the maneuver in the five second open loop simulation. This is done for the four selected models. Three five-second simulations are done for each of the four models, representing the beginning of each of the three maneuvers that comprised the original data set. The error analysis is done over all three simulations for each model. The results are shown in Table. The first five columns represent the average deviation in each of the states, and the last column represents the total standard deviation of the model fit. The fit of is shown in Figure. It is very useful to evaluate the accuracy of an estimated model by comparing it to a set of data that was not used in the estimation procedure. Figure shows where is used to simulate the response of a different full-scale measurement segment. The average deviations are shown in Eq. [9], and the total model standard deviation is.3, which is % smaller than.33 deviation when it is compared to the data used to estimate. [9]

8 Figure :Roots of the ten estimated models estimated from full-scale data. This plot shows that the roots are widely scattered among the models, indicating very different A matrices Measured Full Scale Data Estimated Linear Response data Figure : Evaluation of predictive capability of which was estimated from full-scale data. The blue lines represent the full-scale data used to estimate, and the orange line is the estimated model response to the measured inputs. The dashed lines indicate where one maneuver segment ends and another begins. Only the first seconds of each maneuver segment are shown, as the linear model simulation performance will degrade rapidly after the first seconds. The model fit is not as close as that estimated from the non-linear simulation data in Figure due to noise in the measurements.

9 Discussion of full-scale analysis results There are two possible reasons the full-scale estimation did not yield a very consistent model in terms of the modes exhibited by the A matrix. The first is obvious, the data is not perfect, rate measurements are particularly noisy, which the one-shot least squares algorithm is not well equipped to deal with. The full-scale data sets used to estimate the ten models were relatively small, containing only three maneuvers each. While on the water behavior is different depending upon whether a turn (singlet or doublet input with the rudder) is started to windward or to leeward first. Data was collected with turns starting in both directions, but has not yet been sorted and collated that way. Thus, the full-scale data used for the estimation did not specify which direction turns started. Segregating the data sets into two groups may reduce the variability within the model. Increasing the size of a data set used for estimation will also increase the fidelity of the model. This will be shown using non-linear simulation data. The non-linear simulation was used to generate data sets each containing two rudder singlets, two rudder doublets, a maneuver where the sail is let out, and a maneuver where the sail is let out and brought back to its original position. Twelve new data sets with six maneuvers each were used to estimate twelve new models. The resulting roots are plotted in Figure, which shows the roots are much more closely grouped together than those in Figure 7, and the difference between the models estimated at the different true wind speeds is also more pronounced. From this it can be concluded that having all six maneuvers would reduce the total variability in the estimated models. The other reason for variability among the ten estimated full-scale models involves the assumption of a linear time invariant system. A linear time invariant (LTI) system will exhibit constant modes and the A matrix will remain constant. This assumption is commonly used for aircraft where the flight condition, for example cruise, defined by the state initial conditions, is well known and predictable. Speed and attitude are to remain constant. This is not the case for the yacht, which will have high variability in the roll degree of freedom, and its speed, while predictable, is governed by true wind velocity. It is possible that a linear time varying (LTV) model may be required for the sailing yacht. An LTV system will not exhibit constant modes because the A matrix varies along a trajectory. The cause of variance in the A matrix for a sailing yacht would be the effects varying true wind speeds and directions introduce to the dynamic behavior of the yacht. The full-scale data has been collected on days with significantly varying true wind speeds, and it may be this is what is causing the inconsistency in the estimated models. More data than currently has been taken at varying true wind speeds would be necessary determine whether this has been the cause of model variability. In the meantime, the non-linear simulation may give more insight into whether the assumption of the LTI system is valid. A brief analysis of a more extensive set of non-linear simulation data is presented here. Twelve simulation data sets were created with varying input sequences similar to that of the full-scale data, meaning each set contains one rudder singlet, one rudder doublet, and a sail input. Six of these data sets were in the upwind condition with a true wind speed of m/s, and the next six were also in the upwind condition but with a true wind speed of m/s. As is shown by the plot of the roots in the complex plane Figure 7 and Figure, the roots of the different estimated models in each case are grouped together for models at the same wind speed, but they shift for the models estimated at the different wind speed of m/s. This shifting of roots illustrates the differences in the A matrix at the two different wind speeds. Besides just the roots, it is important to look at the differences in the predictive capability of the model. Two of the twelve models represented by their roots in Figures 7 and are simulated in open loop and their response compared to the non-linear response. One of the models is the true wind speed of m/s. the other is m/s. The time history is shown in Figure 9. All three models match well up to about seconds, then begin to diverge. This divergence is caused more by the errors accumulating from time propagation than the differences between models. The fact that initially the models do match well indicates that there is not that much difference between them. The two linear models could be used interchangeably, and for this range it is appropriate to assume an LTI system and the difference in roots as shown by Figures 7 and are not having a significant effect on the overall behavior of the system. This conclusion is reinforced by Figure, which showed that a model estimated from one particular data set was able to similarly predict a different set of data. CONCLUSIONS AND FUTURE WORK The overall goal of this research is to arrive at a physics based mathematical model to represent sailing yacht dynamics for control design purposes. As has been shown, the linear model is straightforward to estimate using oneshot LS. As expected, models estimated from noisy fullscale measurements exhibit larger standard deviations than models simulation data. The fullscale data as it is arranged currently, it is not yielding a consistent linear model in terms of modes, however the predictive capability is fairly consistent across models. Increasing the number of data points used for estimation by increasing the number of maneuvers included from three to six has the potential to alleviate some of this inconsistency and also improve the predictive capability of the model. Preliminary investigation using non-linear simulation data indicates that the assumption of an LTI system is valid for a range of true wind speeds from to m/s. To strengthen this conclusion, further work encompassing a larger range of true wind speeds and directions needs to be done.

10 Measured Full Scale Data - Estimated response of linear model NOT estimated from this data data3 Figure :Evaluation of the predictive capability of which was estimated from full-scale data. The blue lines represent the full-scale data NOT used in the estimation of, and the orange line is the estimated model response to the measured inputs. The dashed lines indicate where one maneuver segment ends and another begins. Only the first seconds of each maneuver segment are shown, as the linear model simulation performance will degrade rapidly after the first seconds. The model fit is similar to that of Fig re.. True Wind m/s True Wind m/s Real Figure 7: Roots of linear models estimated from non-linear simulation data with 3 maneuvers each, just as the full scale data True Wind m/s True Wind m/s - - Real Figure : Roots of linear models estimated from non-linear simulation data with maneuvers each. The roots are in similar places as those in Figure 7, but much more tightly grouped. There is a significant shift in the location of the roots on the real axis for models in the same trim condition but with different wind speeds, indicating an LTV system may be necessary for representing the sailing yacht.

11 Non-linear model response for VT= m/s - - Linear response of model estimated w ith VT= m/s Linear response of model estimated w ith VT=m/s Figure 9: Comparison of linear models estimated to the non-linear simulation data. All three models match well up to about seconds. This shows that the two different linear models may be used interchangeably. To date, the analysis has been completed with a small portion, about %, of the total full-scale data available to the author. At this point in time, the complete set of fullscale data has not been sorted and pre-processed for system identification. Utilizing a larger portion of the full-scale data will no doubt shed further light on model discrepancies. With more confidence in the assumption of the LTI system and more data, other system identification strategies for use with noisy measurements may be explored. Future research will pursue estimation of both linear and non-linear models from the full-scale data. ACKNOWLEDGEMENTS This work would not have been possible without the generous support of the Madison and Lila Self Graduate Fellowship, which enabled to author to develop a unique research topic blending aerospace and naval engineering. The inspiration for the project originated with the donation of the test vehicle by Ted Kuwana. Advice and technical expertise has been generously provided by Dr. Rick Hale, Dr. Shahriar Keshmiri, and Wes Legursky. The research is made possible by the equipment and funds provided by the University of Kansas Aerospace Engineering department. REFERENCES Alves, J., Ramos, T., Cruz, N.,. A reconfigurable computing system for an autonomous sailboat, International Robotic Sailing Breitenbrunn, Austria. Conference, Cruz, N.A., Alves, J.C.,. Autonomous sailboats: An emerging technology for ocean sampling and surveillance, IEEE/MTS OCEANS., Quebec City, QC. Fossen, T.I., Handbook of Marine Craft Hydrodynamics and Motion Control. John Wiley & Sons.. Hansen, H., Enhanced Wind Tunnel Techniques and Aerodynamic Force s for Yacht Sails, Mechanical Engineering. University of Auckland, Auckland.. Jategaonkar, R.V., Flight Vehicle System Identification: A Time Domain Methodology. AIAA, Reston, VA.. Legursky, K., A Modified, Simulation, and Tests of a Full-Scale Sailing Yacht, IEEE/MTS OCEANS, Hampton Roads, VA. a. Legursky, K., System Identification and the ing of Sailing Yachts, SNAME Annual Meeting. Providence, RI. b. Masuyama, Y., Fukasawa, T., Sasagawa, H., Tacking simulation of sailing yachts numerical integration of equations of motion and application of neural network technique, th CSYS, Annapolis, MD, pp

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