A phase-time-path-difference approach for online wave direction and wave number estimation from measured ship motions in zero and forward speed using a single inertial measurement unit

This study investigates the potential capability of a relatively new and unexplored signal-based approach for shipboard wave estimation. The approach uses the phase-time-path-differences (PTPDs) from an array of shipboard sensors to uniquely resolve the wave propagation direction and wave number. We derive a kinematic PTPD model accounting for forward vessel speed and assess its theoretical foundation to model the sensor delays on a rigid body. The forward-speed PTPD model is structurally equivalent to the zero-speed model considered in previous works, thus retaining the same observability results provided by a noncollinear array of a minimum of three sensors. Moreover, based on the outlined theory and PTPD model, we propose a methodology to estimate the main wave propagation direction and wave number online by employing a fast Fourier transform (FFT), an unscented Kalman filter (UKF), and a rigid-body measurement transformation based on a single inertial measurement unit (IMU). Provided that the vessel in question can be considered a rigid body, a single IMU is sufficient to obtain the desired wave quantities instead of three IMUs, as initially proposed in our previous work. Additionally, our methodology incorporates a novel frequency threshold to avoid distorted wave components caused by the effect of vessel filtering. The performance of our PTPD method is evaluated on data collected from a wave tank and full-scale experiments involving a vessel with zero and non-zero forward speed. The results show very good agreement with the reference wave values reported from a commercial wave radar and wave buoys operating in proximity to the vessel, indicating that our proposed method is competitive with existing wave measurement technology in terms of accuracy and online performance while being cheap, easy to install, flexible, and robust against environmental influences.