Discrete Wavelet Transform-based Approach of Real-time Wave Filtering for Dynamic Positioning of Marine Vessels

. Te ongoing development of deep-sea resources has contributed toward the widespread use of dynamic positioning (DP) systems that can operate in arbitrary sea areas without the limitation of operating water depth compared to mooring systems. Te second-order slowly varying forces can be compensated by DP feedback control. Wave fltering techniques are crucial for designing the DP controller, which can improve the positioning performance and service of multiple actuators. Te objective of wave fltering is to separate the motion induced by the frst-order high-frequency oscillatory waves from the motion caused by the second-order low-frequency (LF) slowly varying disturbances. Tis paper proposes a novel real-time wave fltering approach for DP systems on the basis of the discrete wavelet transform without relying on the previous knowledge of dynamics of the vessel. Meanwhile, to suppress the end efects of wavelet fltering and achieve real-time application, symmetric boundary extension and the sliding window method are investigated in order to capture the slow time-varying LF motion of DP vessels. Te proposed real-time wavelet fltering approach can signifcantly improve the quality of the separated components by selecting the optimal decomposition level, wavelet function parameters, and threshold denoising method. Te real-time and fltering performance of the proposed approach is verifed using a scaled model of an ofshore supply vessel as well as full-scale experiments on the EXPLOERE I scientifc investigation vessel under the action of winds, currents, and waves. Te experimental results confrm that the proposed wave fltering algorithm is efcient and that the LF component waveform can be efectively recovered after wavelet fltering.


Introduction
Dynamic positioning (DP) systems are defned as a set of components for maintaining a foating vessel in a specifc position or along a predefned track by means of sufcient thrusters and propellers to ensure that the position and heading of the vessel are unafected by the disturbances to which it is subjected. Compared to the traditional vessel positioning operation mode using mooring systems, DP systems can perform well in deep and shallow water, and its costs do not increase signifcantly with the water depth. Moreover, DP systems can arrive at the designated sea area more rapidly and efectively and perform the required operation immediately. Meanwhile, the higher redundancy for ensuring the safety of DP systems has resulted in higher positioning accuracy and increased payload compared to mooring systems. At present, DP systems are used not only in the oil industry but also in various marine applications such as drilling, pipe laying, and supply.
For surface marine vessels with DP systems, only the horizontal plan position and heading are controlled. Te environmental disturbances mainly include winds, waves, and ocean currents. In a DP system, only the low-frequency (LF) estimation of the vessel motion must be counteracted by the thrust system; the rapid and purely oscillatory wavefrequency (WF) motion (frst-order wave-induced load) should not enter the feedback control system because the WF components in the applied thrust may endanger the propulsion system by causing excessive wear, resulting in shorter propulsion unit service intervals and service life.
When designing DP systems for vessels, a key issue to consider is wave fltering. In most cases, the position and heading measurements are corrupted with sensor noise.
In the frst DP systems invented for marine vessels in the 1960s, the conventional PID controller with a low-pass flter was employed to remove the frst-order wave-induced motion [1]. However, low-pass flters introduce phase lag, and the fltered data are processed backward using a lowpass flter to reduce the phase lag [2]. Subsequently, the performance of wave fltering was improved using a Kalman flter [3][4][5][6]. Te DP model proposed in [4] has the advantages of self-adaptive noise fltering, optimum combination of diferent measures of the position and heading of the vessel, and efective estimates of the environmental forces. However, this approach is based on a linear mathematical model of the dynamic system, which is only valid for small heading changes (around 10°); hence, the lineation process of the state-space model must be carried out at diferent heading angles between 0°and 360°. To ensure efective DP control for diferent operation conditions, the matrix parameters must be tuned with the change in the heading angle. Tus, to enhance the robustness of wave fltering for diferent heading angles, the extended Kalman flter (EKF) was developed [7,8], and it remains in use in existing DP commercial applications, such as the Kongsberg DP system. Compared with the KF and EKF, H∞ fltering has the advantage of robustness against the unmodeled dynamics. A state-space H∞ flter was proposed in [9]. Recently, many studies were reviewed in [10]. Te H∞ flter has been applied to DP, and it has a similar potential compared with EKF [11]. In the late 20th century, with the increasing application of observer techniques, a nonlinear observer with wave fltering and estimation was designed using passivity methods [12]. In [13], this observer was extended to adaptive wave fltering. In particular, second-order wave transfer function approximations have been used extensively [14][15][16][17][18]. A new linear design model was employed for DP vessels with wave fltering by suggesting that the WF motions should be modeled in a hydrodynamic frame. Meanwhile, a second-order wave transfer function approximation and a modifed model for the WF components of motion were proposed; the gains and design of the observer were based on the WF model [19,20], and three diferent environmental conditions were considered. Tere is a need for an observer that achieves wave fltering and separates the LF and WF position and heading. However, all these observer designs assume that the DP system operates in a stable sea state, or a priori wave model parameters have been established by designing separate observers from calm to high seas [21]. Te aforementioned assumption is not realistic, as the sea state is constantly changing, and the mass gain-scheduled parameters must be tuned to satisfy diferent sea states. Terefore, for the ideal observer design, it is difcult to automatically adjust the reconstruction in accordance with the varying sea state.
To mitigate the aforementioned difculties, an adaptive fltering approach using empirical mode decomposition (EMD) was adopted for wave fltering of a DP vessel, and experimental results were presented in [22]. Te obvious advantage of the EMD flter approach is that it is modelindependent, as it requires no information of the system dynamics. Although the EMD method can remove the WF motion components, useful LF motion components may be lost because of the overlapping of diferent mode functions. Te signals from a full-scale oil tank were analyzed using an ofine fltering algorithm, and online EMD of wave fltering was proposed [22]. In [23], the performance of the online EMD flter with a sliding window was evaluated via DP numerical simulation, and it was proposed that the time window and sliding window lengths should be reduced for real-time DP applications.
At present, few studies on wavelet fltering have been conducted in the feld of vessel motion signal processing [24,25]. Wavelet fltering has two main advantages. First, it is possible to flter short signals. Second, there is no signifcant signal delay caused by the fltration process. In a previous study, we verifed that the wavelet fltering method can flter the measured noise to improve the precision of ship motion prediction under irregular waves [24]. Wavelet threshold denoising of signals has also been adopted in the identifcation of the maneuvering model on the basis of preprocessing data corrupted by high-frequency measured noise in calm water [25].
In view of the aforementioned advantages of the wavelet fltering algorithm, this paper frst proposes a new wave fltering approach for DP vessels based on the discrete wavelet transform (DWT), which is model-independent and only needs information of the wave characteristics in advance compared to the traditional commercial DP system using EKF. In the actual operation process of the DP system under diferent sea states, the acquisition of the wave characteristics is much easier compared to dynamic modeling, and the modeling difculties caused by diferent vessel types under hypothetical conditions are avoided. Next, to suppress the end efects of wavelet fltering and achieve realtime application, symmetric boundary extension and the sliding window method are investigated in order to capture the slow time-varying LF motion of DP vessels. Another advantage of the proposed method is that it is hardly afected by phase lag and amplitude distortion owing to the use of near-symmetric symlet wavelets. Finally, the efectiveness and feasibility of the proposed wave fltering approach are verifed using a scaled model of an ofshore supply vessel as well as experiments on a full-scale scientifc investigation vessel under the action of winds, currents, and waves. Te experimental results confrm that the wave fltering algorithm is suitable for analyzing nonstationary and nonlinear signals when the position and heading of the DP vessel contain high-frequency components.
Te remainder of this paper is organized as follows. Section 2 introduces the mathematical problem considered in this study, including the wavelet transform, decomposition, and threshold fltering steps. Section 3 describes the scaled and full-scale experimental set-up as well as the experimental environment conditions. Section 4 presents, discusses, and analyzes the experimental results of the scaled and full-scale vessels. Finally, Section 5 concludes the paper.

Wavelet Transform.
A wavelet is essentially a small wave that grows and decays over a limited period. In the continuous wavelet transform (CWT), a given signal of fnite energy is projected to a continuous family of frequency bands or subspaces of various scales in L 2 (R). Te wavelet transform is visualized in the continuous domain as follows: where f(t) ∈ L 2 (R) is an arbitrary function to be transformed, and ψ a, b(t) is the conjugate function of ψ a, b(t).
For instance, the signal may be represented in each subspace of a scale for all a. For each a, the subspace is generated by the translation and rescaling of a single function ψ(t) ∈ L 2 (R) called the mother wavelet which is given as follows: where a, b ∈ R and a ≠ 0; a denotes the scaling parameter and b denotes the translation parameter. Further, a represents the time and frequency resolutions of the scaled mother wavelet, while b represents its translation along the time axis. Te continuous scale is computationally intractable; thus, DWT comes into play. DWT minimizes the transformation for discrete values of a and b while guaranteeing invertibility [26]. To use this transform for sampling data, the scaling and shifting parameters are discretized on a sampling grid as follows: Tus, we get Notably, the shifting parameter b is a function of the scale a. Tis represents a crucial advantage of DWT, which corresponds to the sampling of a and b such that the consecutive discrete values of a and b as well as the sampling intervals difer by a factor of two. More formally, a 0 � 2 and b 0 � 1. Te discrete wavelet is used as shown in the equation below, which constitutes a family of orthonormal basis functions: In general, the wavelet coefcients can be derived accordingly as follows: Further, f(t) can be reconstructed using the discrete wavelet coefcients as follows: When the input function f(t) and the wavelet parameters a and b are represented in a discrete form, the transformation is referred to as the DWT of the signal [27].

Wavelet
Decomposition. DWT extracts the time and frequency localized coefcients from an input signal or function, which constitute its wavelet decomposition. Mallet proposed a fast algorithm to compute an input signal by using the wavelet multiresolution algorithm [27]. Tis algorithm is based on orthogonal multiresolution analysis.
According to the scalability and inclusiveness of multiresolution analysis, wavelet decompositions are conducted from wavelet basis functions that consist of a scaling function ϕ(t) and a wavelet function ψ(t). Tese wavelet decompositions decompose a signal into a series of low-and high-frequency sub-bands. An orthonormal basis is formed by the scaling and wavelet functions in the Hilbert space (H) [28,29]: where j 0 denotes the initial decomposition level, and c j, k and d j, k denote the approximation and detail coefcients at scale j and location k, respectively.

Treshold Processing of Coefcients.
A typical threshold processing function consists of hard and soft value processing functions [30]. For hard thresholding, it can be defned as follows: For soft thresholding, it can be defned as follows: In general, g is a linear function, and σ denotes the threshold value.

Wavelet Filtering.
During wavelet decomposition, the signal is decomposed into a series of low-and high-frequency sub-bands via scaling and wavelet functions.

Mathematical Problems in Engineering
Successive lower-frequency band-pass flters (wavelets) are used to orthogonally decompose the signal, and the lowest band is analyzed using the scaling function. Te coefcients in this domain are removed such that only the local modulus maxima are retained. Te signal is then reconstructed with the new coefcients to produce a fltering version of the original signal.
Realizing the signal flter using the wavelet transform involves three basic steps as follows: Step 1: Wavelet transform for decomposing the data in the trend part, applied to the original time series data to obtain the same number of coefcients as the size of the data.
Step 2: Determination of the optimal decomposition level.
Step 3: Treshold fltering method of approximation and detail coefcients. Te new coefcients are used to regenerate the signal using the inverse DWT (IDWT).
Decomposition and reconstruction can also be performed using Mallet's pyramid algorithm [27]. Figure 1 shows an example of the overall process of decomposition and reconstruction by the wavelet transform.
In Figure 1, c j,k represents the coarse approximation (LF trend), and d j,k represents the detailed information (HF noise). Te diference between the frst-level approximation coefcients c 1,k and the original series x yields the detail coefcients of the frst-level d 1,k . To obtain c 2,k , c 1,k is approximated by a set of basis functions. Te coefcients that pass through the low-pass flter are called approximation coefcients, and the coefcients that pass through the highpass flter are called detail coefcients.

Symmetric Boundary Extension.
DWT is implemented using a two-band perfect reconstruction flter bank. In many applications, such as signal fltering [31,32], compactly supported wavelets are required to realize perfect signal reconstruction. However, a real orthogonal symmetric wavelet basis, such as the Haar wavelet, is not continuous and the flter is of the order 1 only, which is not adequate for practical applications. Te design of the ideal orthogonal and symmetric wavelet flter is challenging. For applications of signal processing in DP experiments, we can select orthogonal and near-symmetric wavelets to have a minimal phase response rather than an exact linear phase. Terefore, an orthogonal and near-symmetric wavelet of the symlet wavelet family is selected in this study. Te efect of the optimal decomposition level and wavelet family will be discussed in Section 4.
In practical engineering applications, iterative decomposition and reconstruction are required when using the mallet algorithm. Terefore, the end efects of the fltered signal are inevitable because the input sequence of the signal has a fnite length [33]. To suppress the end efects of wavelet fltering for real-time application, symmetric boundary extension (SBE) is applied to wavelet fltering in this study. Te extension procedure is as follows. Assuming that x(n)(n � 0, 1, . . . , N− 1) is the N point vector of the input samples, for the sequence x(n), the input signal is extended by adding N/2 (N-1/2) values when N is even (odd) at each end, and then, the new sequences x(n) can be constructed by SBE as follows:

Sliding Window Method.
To realize the real-time application of wavelet fltering, a sliding window method [34] is adopted to capture the slow time-varying LF motion of DP vessels. Te development of the sliding window is shown in Figure 2. Te sliding window is fxed, and the realtime fltered result is obtained using the newly measured and historical data.

Scaled Experimental Model.
To validate the proposed method based on wavelet fltering, we used a 1/32 scaled model of an ofshore supply vessel, as shown in Figure 3, appended with a skeg and a bilge keel. Te details of both the scaled model and the full-scale vessel are presented in Table 1. Te vessel is equipped with three azimuth thrusters (two located aft and one in the fore) and one tunnel thruster (in the fore). Experiments were conducted under winds, currents, and waves in the maneuvering and seakeeping basin of the China Ship Scientifc Research Center (CSSRC). Te basin is 69 m long and 46 m wide, and the water depth is 4 m. Te current and wind load coefcients were obtained through tests in the wind tunnel. All the loads were simulated (the wind velocity was defned as 13 kn, and the current velocity was defned as 1.5 kn), and disturbances in terms of irregular waves with a signifcant wave height of 4.0 m and a characteristic period of 8.3 s were applied (corresponding to full-scale ones). Te ITTC spectrum was employed in the model test to simulate the wave environment. Waves were generated by a wave generator with a system of 188 rocker faps. Wind and current were generated by a large number of wind fans and a water pump system, respectively.
Te ITTC spectrum is defned as follows: where H 1/3 is the signifcant wave height, T 01 is the characteristic period, and ω is the circular wave frequency. Te irregular seas were adjusted such that the spectral density distribution of the generated sea is compared well with the required theoretical energy distribution before the experiment. Te requirements for the wave spectrum were set as follows: the signifcant wave height (H 1/3 ) and the spectral characteristic period (T 01 ) were within 5% of the theoretical values. Te measured and target wave spectra are shown in Figure 4.

Full-Scale Experimental Vessel.
To verify the ability of wavelet fltering for the full-scale vessel, full-scale experiments were conducted on EXPLOERE I, a scientifc investigation vessel, with the DP system (the DP control system was developed by CSSRC and approved by the China Classifcation Society), as shown in Figure 5. Te details of EXPLOERE I are presented in Table 2. Te vessel was ftted with two main propellers and four tunnel thrusters (two at the stern and two in the bow).
Te CSSRC DP control system uses data from gyrocompasses, diferential Global Positioning System (DGPS), mechanical reference systems (taut wire), and the BeiDou satellite position reference system, which allow the DP control system to position the vessel.
Te operating condition was sea state 4, the signifcant wave height and main wave period were measured to be 2.0 m and 6.0 s, respectively, by a wave radar, and the mean wind speed was measured to be 8-11 m/s by a wind sensor. Te surface current speed was not measured in this experiment.

Scaled-Model Experiments.
Prior to the experimental verifcation of wavelet fltering in DP, experiments on a scaled model were conducted with EKF. Te three degrees of freedom of the horizontal plan (sway, surge, and heading) were analyzed corresponding to the wave direction of 180°(coming from the vessel bow) by using wavelet fltering. Te motion of the scaled model was measured using an optical measurement system.
To test the efectiveness of the proposed method, we employed the stand deviation (SD) σ x and the signal-tonoise ratio (SNR) in order to evaluate the reconstruction accuracy of real signals: where x(i) denotes the measured signal and x(i) denotes the fltered signal.
Te wavelet function, sliding window length, decomposition level, and thresholding method directly afect the fltering performance. In this study, we tested a range of window lengths and near-symmetric symlet wavelets (sym2, sym4, sym6, and sym8), as shown in Table 3. Te optimal decomposition level was 4, and the detailed signals of levels 1-3 were processed using the hard thresholding method. Te SD and SNR do not change when the window size is ≥2 6 for diferent symlet wavelets. To validate the efectiveness of wavelet fltering, EKF has been used in [7]. Te SD and SNR of EKF are 0.0335 and 8.6018, respectively, which is closer to the fltering performance of the symN wavelet (N � 4, 6, and 8), as shown in Table 3. Considering that higher-order N of vanishing moments will make the fltering signal smoother while consuming more time for fltering, the sym6 wavelet was selected for wave fltering in this model experiment.
Furthermore, the relevant source code for wave fltering was developed using the LabVIEW platform on a personal computer with an Intel Core i7-4790 processor. To validate the program for real-time fltering, the time consumed can be determined using the LabVIEW Tick Count function. By fltering 1500 samples, diferent sliding window lengths with the sym6 wavelet were detected, and the average and maximum time consumed were obtained, as shown in Table 4. It can be concluded that when the window length is 2 6 , the execution time is around 0.4 ms, and the time consumed is around 15 times greater than that when the window length is 2 10 . Te sample time is 200 ms in the model experiment; thus, real-time fltering can be realized using the sym6 wavelet.
As mentioned above, the smaller the window size, the shorter is the time consumed. However, the efectiveness of wave fltering must also be guaranteed. From the results presented in Tables 3 and 4, we comprehensively considered the computational complexity and fltering performance for diferent wavelet functions, and we selected the sym6 wavelet with a window length of 2 6 for appropriate wavelet fltering in the model experiments.
Furthermore, the reasons for selecting the optimal decomposition level of 4 and processing the detailed signals of levels 1-3 by the hard thresholding method will be analyzed.
As shown in Figure 4, the WF components corresponding to the wave spectrum are nearly in the interval [0. 5 3]. Te high-pass and low-pass flters are complementary and have a cut-of frequency of f s /2 at each level in the wavelet optimal decomposition. When the level of decomposition is 4, the theoretical approximation coefcients of the fourth level are in [0 0.3125] (the sample frequency f s is 5 Hz in this DP experiment).
Te decomposed approximation signals and detailed signals for surge motion are separately given with level 1-4 decomposition using the sym6 wavelet, as shown in Figure 6. Further, the decomposed signals are subjected to fast Fourier transformation (FFT) and amplitude (AM) spectrum analysis, as shown in Figure 7. As can be seen, the corresponding high cut-of frequency of the detailed signals is around 0.3 in the fourth level. Terefore, the motion of the WF components can flter out the detailed signals of levels 1-3 by the hard thresholding method. Figure 8 shows the measured and fltered results of surge motion; the time evolution of the WF and Raw data x (t) Mathematical Problems in Engineering LF components of the motion and their estimates are presented by wavelet fltering with SBE, wavelet fltering without SBE, and EKF. As can be seen, the second-order LF motion induced by slowly varying disturbances is efectively separated from the frst-order WF oscillatory motion using wavelet fltering with SBE. For a clearer comparison, Figure 8(b) shows a magnifed view of Figure 8(a). As can be seen, wavelet fltering provides a smoother estimation of the LF part of the motion compared with that obtained using EKF; the end efects of the fltered signal are restrained with SBE compared with that without SBE; the phase lag and amplitude distortion ... Figure 2: Schematic diagram of the sliding window.        To further verify the efectiveness of real-time wave fltering with the sym6 wavelet in diferent wave directions, DP station keeping experiments were conducted in the wave direction of 135°on the basis of the aforementioned selections of the wavelet function, window length, decomposition level, and hard thresholding method. Te sway, surge motion, and heading were analyzed by real-time wavelet fltering, as shown in Figures 9-11. As can be seen, the same efective fltering performance is verifed compared to EKF.

Full-Scale Experiment.
To verify the feasibility of wavelet fltering in a full-scale vessel, frst, the ofine data were analyzed by conducting a full-scale experiment on the EXPLOERE I scientifc investigation vessel on the basis of the wavelet fltering with SBE and the sliding window method. Compared to the scaled-model experiment, the wave spectral density distribution was not accurately measured in the full-scale experiment. Hence, the decomposition level and thresholding method were selected by amplitude spectrum analysis of the ofine data.
Te measured signal frequency was distributed on the whole frequency coordinate axis, and the amplitude spectrum distribution was in the high-frequency components, as shown in Figure 12. Clearly, the amplitude spectrum distribution with frequencies greater than 0.5 Hz was reduced to 0. Trough a comparison of the amplitude spectra of the measured signal and the fltering signal, we found that the signal energy after fltering was clearly reduced to 0 in the high-frequency components. Hence, the decomposition level of 4 and the detailed signals of levels 1-3 were processed by the hard thresholding method and selected in the fullscale experiments.
Te same sym6 wavelet and the sliding window length of 2 6 were used as a reference for real-time fltering in the fullscale experiments. Ten, the high-and low-frequency components after threshold fltering were combined for signal reconstruction. Te fnal fltering signal was obtained through reconstruction, as shown in Figure 13. Tus, the LF signal waveform was efectively recovered after wavelet fltering.
According to the aforementioned results, we could further verify the efectiveness of wavelet fltering. Online data fltering was conducted in the full-scale experiments on the basis of real-time wavelet fltering with SBE and the sliding window method. Te experimental results are shown in Figure 14. For the real-time fltering results, the amplitude spectrum analysis is shown in Figure 15. As can be seen, the WF motion component was efectively fltered out and the smoother LF motion waveform was efectively recovered after fltering.

Comparison of Filtering
Performance. Furthermore, we quantitatively compared the fltering performance of wavelet fltering and EKF. Te fltering approaches were compared and the cost function E is used as follows:  Mathematical Problems in Engineering 13 where ω c is a cut-of frequency of WF component, ω s is circular frequency corresponding to sample frequency f s , s m (ω) is energy spectral density function of the measured signal, and s WF (ω) is energy spectral density function of the measured signal of WF motion. E represents the energy with the frequency band ω c ≤ω≤ω s ; the smaller the value E, the better the fltering performance. Te energy spectral density functions of the model and fullscale experimental results are shown in Figures 16-18. According to the results, we can see that the low-frequency motion components are efectively separated, and wave-frequency motion components can be fltered out by wavelet fltering.
Te fltering methods are compared using E(ω c ) for the aforementioned scaled model and full-scale vessel, and three horizontal degrees of freedom (X or surge, Y or sway, and heading) were analyzed (ω c is 0.625 π·rad/s), and horizontal three degrees of freedom motion (x or north, y or east, and heading) are analyzed. Te results of E are shown in Table 5. As can be seen from Table 5, compared to the EKF approach, the energy of wave-frequency components is lower using wavelet fltering, and the wavelet fltering performs better for all cases in the model and full-scale experiments.
As mentioned above, compared with the EKF with ship dynamics, the wavelet fltering returns better results of position and heading fltering. However, the most important point we presented in the work is that the wavelet fltering approach is model-independent, and the low-frequency and wave-frequency motion model is not established compared to the EKF,  especially a damped oscillation wave-frequency model is not accurate in EKF, and the characteristics of waves still need to be considered to obtain the better fltering performance.
E WF-wavelet is the WF motion using wavelet fltering; E WF-EKF is the WF motion using EKF; and E WF-m is the WF components of measured motion.

Conclusions
A real-time wave fltering approach for DP systems was proposed on the basis of the wavelet transform, SBE, and the sliding window method. We analyzed the performance of wavelet fltering using DWT for separating the high-frequency oscillatory motion from the second-order low-frequency slowly varying motion by using a scaled model of an ofshore supply vessel and conducting full-scale experiments on the EXPLOERE I scientifc investigation vessel. Specifcally, the noise component was in the high-frequency part of the wavelet coefcient, the hard thresholding method was preferred, and a decomposition level of 4 was used for wavelet fltering. Te scaled model and full-scale experimental results demonstrated that the proposed approach can flter out the high frequencies of a raw signal by retaining only the low frequencies, and no signifcant signal delay is caused by the selection of a near-symmetric wavelet function. In addition, diferent sliding window sizes were analyzed, and the end efects with and without SBE were compared. By comprehensively considering the computational complexity and fltering performance for diferent wavelet families, we selected the sym6 wavelet and a window length of 26 for appropriate wavelet fltering in the scaled model and full-scale experiments. Te proposed real-time wavelet fltering method was shown to be suitable for nonstationary and nonlinear signals of the position and heading of DP vessels, containing high-frequency components. Moreover, it was found to signifcantly improve the quality of the separated components without relying on the previous knowledge of dynamics of the vessel.
Although the current performance of real-time wavelet fltering seems acceptable in DP experiments on the scaled model and full-scale vessel, some problems remain owing to the diferent characteristics of the wave spectrum in diferent sea operation areas for a full-scale vessel. Tese problems must be addressed in future studies. Tus, further  investigation is required to fully develop the adaptive wavelet fltering approach in diferent sea states. Moreover, considering that the motion periods will be diferent for different types of vessels under waves, to realize a wide range of engineering applications for DP vessels, the wavelet fltering approach should be further validated using diferent types of vessels.

Data Availability
Te data used to support the fndings of this study are available from the corresponding author upon request.

Conflicts of Interest
Te authors declare that they have no conficts of interest.