System Identification and the Modeling of Sailing Yachts
University of Kansas
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This research represents an exploration of sailing yacht dynamics with full-scale sailing motion data, physics-based models, and system identification techniques. The goal is to provide a method of obtaining and validating suitable physics-based dynamics models for use in control system design on autonomous sailing platforms, which have the capacity to serve as mobile, long range, high endurance autonomous ocean sensing platforms. The primary contributions of this study to the state-of-the-art are the formulation of a five degree-of-freedom (DOF) linear multi-input multi-output (MIMO) state space model of sailing yacht dynamics, the process for identification of this model from full-scale data, a description of the maneuvers performed during on-water tests, and an analysis method to validate estimated models. The techniques and results described herein can be directly applied to and tested on existing autonomous sailing platforms. A full-scale experiment on a 23ft monohull sailing yacht is developed to collect motion data for physics-based model identification. Measurements include 3 axes of accelerations, velocities, angular rates, and attitude angles in addition to apparent wind speed and direction. The sailing yacht herein is treated as a dynamic system with two control inputs, the rudder angle, dR, and the mainsail angle, dB, which are also measured. Over 20 hours of full scale sailing motion data is collected, representing three sail configurations corresponding to a range of wind speeds: the Full Main and Genoa (abbrev. Genoa) for lower wind speeds, the Full Main and Jib (abbrev. Jib) for mid-range wind speeds, and the Reefed Main and Jib (abbrev. Reef) for the highest wind speeds. The data also covers true wind angles from upwind through a beam reach. A physics-based non-linear model to describe sailing yacht motion is outlined, including descriptions of methods to model the aerodynamics and hydrodynamics of a sailing yacht in surge, sway, roll, and yaw. Existing aerodynamic models for sailing yachts are unsuitable for control system design as they do not include a physical description of the sails' dynamic effect on the system. A new aerodynamic model is developed and validated using the full-scale sailing data which includes sail deflection as a control input to the system. The Maximum Likelihood Estimation (MLE) algorithm is used with non-linear simulation data to successfully estimate a set of hydrodynamic derivatives for a sailing yacht. As there exists a large quantify of control algorithms which may be applied to systems described by a linear model, the non-linear model is simplified to a 5DOF MIMO state space model with a state vector including surge velocity, sway velocity, roll rate, yaw rate, and roll angle: , and a control vector: . Over 100 singlet and doublet maneuvers specifically designed to identify linear model dynamic responses are included in the full-scale data. The one-shot least squares (OSLS) technique offered a simple and fast means to estimate many linear models from this large sub-set of the full-scale data. As no sailing yacht linear dynamic model exists, especially for the test yacht, the only way to evaluate the fidelity of estimated models is to evaluate their predictive capability. This is accomplished through two separate criteria, Theil inequality coefficient UT and R2 > 0.75 0.75, which are shown to provide sufficient quality models to enable control system design. Each linear model is estimated from only 3 maneuvers, one rudder singlet, one rudder doublet, and one sail singlet, and validated with a similar set of independently collected maneuvers. In total, 102 linear models are estimated in the Jib configuration, 17 linear models in the Genoa configuration, and 1 linear model in the Reef configuration. The dynamic modes of the models estimated from the full-scale data are investigated using the eigenvectors and eigenvalues of the linear state space model A matrix. First, the estimated models are characterized by the number of first and second order modes observed for each given model, and are referred to herein as Type A or Type B models. Type A models exhibit two second order modes, and one first order mode, whereas the Type B models exhibit one second order mode and three first order modes. The modes are then separated by natural frequency. A subset of models from the Jib configuration which exhibit an ,R2 > 0.88 0.88 are analyzed via eigenvector modal analysis. It is shown that all sailing yacht models will contain a second order mode (referred to herein as Mode 1A.S or 4B.S) which is dependent upon trimmed roll angle. For the test yacht it is concluded that for this mode when the trimmed roll angle is , roll rate and roll angle are the dominant motion variables, and for surge velocity and yaw rate dominate. This second order mode is dynamically stable for . It transitions from stability in the higher values of to instability in the region defined by . These conclusions align with other work which has also found roll angle to be a driving factor in the dynamic behavior of a tall-ship (Johnson, Miles, Lasher, & Womack, 2009). It is also shown that all linear models also contain a first order mode, (referred to herein as Mode 3A.F or 1B.F), which lies very close to the origin of the complex plane indicating a long time constant. Measured models have indicated this mode can be stable or unstable. The eigenvector analysis reveals that the mode is stable if the surge contribution is 20%. The small set of maneuvers necessary for model identification, quick OSLS estimation method, and detailed modal analysis of estimated models outlined in this work are immediately applicable to existing autonomous mono-hull sailing yachts, and could readily be adapted for use with other wind-powered vessel configurations such as wing-sails, catamarans, and tri-marans.
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