Evaluation of wave-frequency motions extraction from dynamic positioning measurements using the empirical mode decomposition Paula B. Garcia-Rosa Department of Electric Power Engineering Norwegian University of Science and Technology Trondheim, Norway p.b.garcia-rosa@ieee.org Astrid H. Brodtkorb Department of Marine Technology Norwegian University of Science and Technology Trondheim, Norway astrid.h.brodtkorb@ntnu.no Asgeir J. Sørensen Department of Marine Technology Norwegian University of Science and Technology Trondheim, Norway asgeir.sorensen@ntnu.no Marta Molinas Department of Engineering Cybernetics Norwegian University of Science and Technology Trondheim, Norway marta.molinas@ntnu.no Abstract—For dynamic positioning operations, high-frequency wave induced motions cause excessive control action, and conse- quently additional power consumption and wear of actuators in the propulsion system. Thus, such operations require the control of only low-frequency motions, which is achieved by proper filtering of high-frequency motions. This study investigates the use of the empirical mode decomposition (EMD) method for wave filtering purposes. EMD is a data-driven method that decomposes an oscillatory waveform into a number of modes from the highest to the lowest frequency. The decomposition process in the standard EMD algorithm relies on repetitive iterations through the entire data span, which is impractical for wave filtering in real-time applications. Thus, an online EMD algorithm is also considered. The online decomposition process features a time lag, and measurements of the ship motions have to be taken at a point ahead of the center of gravity so that high-frequency motions are estimated in advance. In this study, the performance of both standard and online EMD algorithms, in terms of wave filtering and control efforts, is evaluated through a comparison with a nonlinear passive observer (NPO). Furthermore, the time lag of the online EMD is also of interest, as it indicates the required prediction time window. Simulation results with a simple maneuver of a vessel in moderate, and calm seas, show that the control action with wave filtering from the online EMD can be up to 40% lower than with wave filtering from NPO. Index Terms—wave filtering, dynamic positioning, empirical mode decomposition I. I NTRODUCTION In most ship applications, the oscillatory motion due to first- order wave forces and moments should not enter the control loop, as it causes unnecessary tear and wear on the machinery and thrusters, and consequently, increases fuel consumption. Such an oscillatory motion is caused by waves in the frequency range [1]: 0.05 <f 0 < 0.2 (Hz) . (1) The ship is also subject to wave drift forces that are caused by second-order wave disturbances and induce nonzero slowly- varying motions. However, second-order wave drift forces can be counteracted by the motion-control loop [1], [2]. In order to avoid high-frequency wave induced motions, which cause excessive control action, proper filtering of state variables must be performed by using wave filtering tech- niques. In this study, the focus is on wave filtering for dynamic positioning (DP) systems. In this framework, a number of studies have developed wave filtering techniques based on conventional filter design, and state estimation methods, see, e.g., [2]–[7]. Techniques based on state estimation consist of using a wave-induced motion model and an observer to separate the ship position and heading into wave-frequency (WF) motions (i.e., the high-frequency wave induced motions) and low-frequency (LF) motions [2], [5]. A non-model based approach is considered here, where a scheme based on the empirical mode decomposition (EMD) method is proposed to extract the first-order oscillatory mo- tions from the ship position and heading measurements. EMD is a data-driven method with an adaptive basis that relies on the local characteristic time-scale of an oscillatory waveform. The method decomposes a waveform into a number of in- trinsic mode functions (IMFs) from the highest to the lowest frequency modes, and a residue, which can be either the mean trend or a constant [8]. An IMF is defined as a symmetric signal with respect to the local zero mean, and with numbers of zero crossings and extrema that differ at most by one. Such a signal satisfies the necessary conditions for a physically meaningful interpretation of instantaneous frequency obtained from the Hilbert transform [8]. The aim of this paper is to evaluate the performance of the EMD method for wave filtering purposes in DP systems.