Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 6547312, 8 pages

https://doi.org/10.1155/2017/6547312

## Heave Motion Measurement by Adaptive Filter Based on Longuet-Higgins Wave Model

School of Instrumentation Science & Opto-Electronics Engineering, Beihang University (BUAA), 37 XueYuan Road, Haidian District, Beijing 100191, China

Correspondence should be addressed to Jiazhen Lu

Received 13 September 2016; Accepted 7 February 2017; Published 28 February 2017

Academic Editor: Alberto Borboni

Copyright © 2017 Jiazhen Lu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A method is proposed to obtain heave motion information based on the Longuet-Higgins wave model. The Longuet-Higgins wave model which is closer to the sea wave is introduced. Based on it, random process of the noise is analyzed and the highpass filter is designed to reduce errors. Then it is the key point in this article that an adaptive algorithm is put forward because of the complexity of the waves. The algorithm adjusts the cutoff frequency to reduce the amplitude attenuation of the filter by analyzing the wave. For the same reason the comprehensive parameter of the phase compensation can be also obtained by the algorithm. Simulation measurement results show that under the rough sea situation the maximum value of absolute error is 0.4942 m according to the normal method, the method is 0.1170 m, and the average error ratio of the rough sea test reduces to 3.89% from 12.54%, which demonstrates that the adaptive filter is more effective in measuring heave movement. A variety of simulation cases show that the adaptive filter can also improve the precision of the heave motion under different sea situations.

#### 1. Introduction

Heave motion is generally used in the research of floating offshore engineering. For example, the introduction of heave information is beneficial for ship replenishment, landing of shipboard aircraft, and the design of heave compensation of offshore oil drilling platform. Moreover, the measurement of heave motion is used to provide a reference for the safety of vessels sailing. For example, “sheng lu” ship capsized due to the factor of the storm in 1991.

How to get heave information has been mentioned in many studies. Basically there are two main types of filtering methods for heave motion. One is the Kalman filter. For example, El-Hawary estimated heave component with Kalman filtering which established a handled value [1]. Triantafyllou et al. used Kalman filtering techniques to estimate the ship motion [2]. Aminzadeh and El-Hawary used the Generalized Kalman to solve the problem of heave compensation in underwater applications [3]. Fossen and Perez introduced the ship motion-control models and used Kalman filter to eliminate the oscillatory [4]. But the method has two main drawbacks. One drawback is that the model of the heave dynamics cannot be equivalent to the actual model. The other drawback is that Kalman filtering requires GPS or other external reference information. However, the result of GPS height is not accurate enough to compute heave motion value. In this paper according to the requirment of navigation system design, a long time pure inertial navigation is desired, so Kalman filter is not applicable in this case.

The other type is the highpass filter. For example, in the 1990s, an adaptive filter algorithm for heave motion was developed by Godhavn and was implanted in the seatex MRU [5]. Yang et al. did active heave compensation experiments to validate the validity of active heave compensation system in 2008 [6]. Simulations and the tests with MTi unit were done by Yang et al. and the results have proven that the adaptive highpass filter was effective in 2009 [7]. Wei and Feng used highpass filter to eliminate Schuler errors to get heave information in 2012 [8]. Hu and Tao put forward the weighted-frequency Fourier linear combiner algorithm for phase compensation in 2013 [9]. However, little attention has been focused on the sea wave model which decides the cutoff frequency of the filter. It is often assumed that the wave model is a simple sine function. In fact, ship motion in waves is a complex random process as a result of sea tides, currents, and wind effects.

In order to get accurate heave motion which is based on strapdown inertial navigation system (SINS) composed of inertial sensors with high accuracy, it is necessary to select a sea model which can better reflect the actual situation to analysis. Thus, this paper chooses the Longuet-Higgins wave model which is closer to the actual situation and analyzes the random process of the narrowband noise. On the basis of this analysis, an algorithm is proposed to adjust the cutoff frequency to reduce the amplitude error caused by the filter. In addition, the same algorithm is used to change all pass system parameters to compensate phase error.

The structure of this paper is given as follows. In Section 2 the Longuet-Higgins wave model and the accelerometer error model are introduced to analyze the noise. In Section 3, the method is presented and analyzed. Moreover, the steps of heave measurement are shown. Simulation measurement results are given in Section 4. Finally, the article ends with concluding remarks in Section 5.

#### 2. Review and Analysis

Referring to previous literatures, the sea model and the inertial unit error model [10–12] are set up. In addition, the correlated noise is analyzed based on the model.

##### 2.1. Mathematical Model

###### 2.1.1. Longuet-Higgins Wave Model

In order to describe the characteristics of the actual wave, Longuet-Higgins wave model is established on the basis of the measurement data [13]. According to the assumption of the model, the wave motion is a random process. The fixed point fluctuation of the wave is composed of some different phases and different amplitude sine wave.

The wave surface displacement is expressed as In the equation above, and are the amplitude and frequency of the th wave component, respectively, represents time, and is the random phase uniformly distributed in the . Furthermore, the sea wave only has a dominant component under most condition.

###### 2.1.2. Inertial Unit Error Model

The accelerometer error model is given as follows [14]:where is the accelerometer error, is the white Gaussian noise, is a constant bias, is linear error coefficient, , are, respectively, second-order and third-order nonlinear error coefficient, and , are cross-axis coupling coefficient, respectively. , , and are, respectively, the specific force of input axis, output axis, and pendulum axis.

The relationship of two models is given as follows:where is the measured value of accelerometers; is coordinate transformation matrix of SINS.

On the other hand, the gyroscope error contains the constant error and random drift. Affected by inertial unit errors, the coordinate transformation matrix is not accurate. When SINS works for a long time, the 6-degree-of-freedom navigation results are in oscillation because of incorrect . According to the error analysis of SINS, under the undamped case the results include three kinds of different cycles of persistent oscillation [15]. They are, respectively, Schuler oscillation period , earth oscillation period , and Foucault oscillation period .where is the earth radius; is self-rotation angle velocity of the earth; is latitude; is gravitational acceleration. Foucault oscillation period decreases with increased latitude. In addition, the period of earth oscillation and the period of Foucault oscillation are greater than the period of the heave motion, so this paper mainly considers the effects of Schuler oscillation for heave motion measurement.

###### 2.1.3. Highpass Filter

Schuler oscillation whose period is 84.8 min in the process of SINS is caused by the constant bias of inertial sensors with high accuracy. The existence of Schuler oscillation makes heave information divergent. The frequency of Schuler oscillation is less than the frequency of heave motion. The highpass filter is designed to eliminate Schuler oscillation and reduce noise [16].

In this paper a fourth-order highpass filter is given by the following equation:where is the damping factor, is the Laplace variable, and is the cutoff frequency of the highpass filter which can be tuned.

##### 2.2. Analysis of Narrowband Noise

Due to the influence of random phase of the wave, the high order nonlinear error part sensed by the accelerometer can be equivalent to a narrowband noise after two integrations [17]. The white Gaussian noise is also in the form of narrowband noise in the heave information because of the output with a range of sampling frequency. The random process of narrowband noise in the wave model is discussed as follows.

Without considering the impact of other errors which are the constant bias and linear errors of the accelerometer, narrowband noise in the random phase wave model is analyzed. The model can be expressed as follows:where is the sine wave with the random phasewhere is the known amplitude, is the random phase uniformly distributed in the , and is the known angular frequency.

In (5), narrowband noise is a narrowband stationary Gaussian random process. The mean is zero and the variance is . It is given as follows:

According to (6) and (7), (5) is rewritten asIn the equation above,Under the condition of a given signal phase and the time of , with the value range of and being , , the probability density function of the envelope is given bywhere is the zero-order Bessel function and is the number of Bessel functions.

Assuming that , , (10) can be expressed asIn the actual situation, the useful signal of the accelerometer is much larger than the noise; therefore, With and , the mean and variance of are given by

However, some of frequency of the noise is the same as frequency of heave motion. So the important thing is to select the appropriate cutoff frequency of the filter to minimize errors.

#### 3. Design and Error Analysis of Adaptive Algorithm

According to Section 2, it is important to design the filter to get the heave information. While there are the phase errors and the amplitude attenuation in the filter, it is necessary to compensate them.

##### 3.1. Adaptive Phase Compensation of Filter

The peak to peak error which is induced by the filter is presented as follows:From (14), the peak to peak error cannot be ignored. Phase compensation is realized to reduce the peak to peak error by using first-order all pass system in this paper [18, 19]. It is given by

The amplitude response of the system is not affected by the frequency so that all frequency components of the input signal can be passed without amplitude attenuation. The phase error caused by the filter can be compensated by choosing the appropriate value.

Then the plane is mapped to the plane through the expression , soWith (15) and (16), we can getIn (17), is the sampling period.

By multiplying the highpass filter and the all pass system , the expression is given as follows: is the frequency of the heave motion. By making the phase zero, we can determine the parameter value. But the heave motion is composed of several different phase waves. So with comprehensive consideration, an adaptive algorithm is proposed to select the parameter to minimize the phase error.where is the comprehensive parameter; is the amplitude of the th wave component of the sea, and is the phase compensation parameter of the th wave.

##### 3.2. Adaptive Selection of The Cutoff Frequency

The amplitude attenuation of the filter is given by

The magnitude of errors is closely related to the cutoff frequency and the input frequency of the filter. Furthermore, according to analysis of narrowband noise, the variance of is , so the narrowband noise error is given as follows:

The magnitude of the errors and the noise are given as follows:According to the change of the frequency , selecting the optimal cutoff frequency minimizes the magnitude of errors. The method is given by

Only with one frequency of the signal, a single optimal cutoff frequency can be obtained by (24). However, the heave motion is composed of some sine waves. Therefore, on this basis, an adaptive algorithm is proposed to find the comprehensive optimal cutoff frequency. It is given bywhere is the comprehensive optimal cutoff frequency and represents the single optimal cutoff frequency of the th wave component of the sea wave.

##### 3.3. Error Analysis

The method can estimate heave motion, but there still exist errors at heave information. It is necessary to analyze range of errors. SoIn the above equation, is the frequency of the th wave component. They are determined by the sea wave. Even though the sea wave motion is changing, the range of error can be obtained by (26).

##### 3.4. The Steps of Heave Measure

The steps of heave measure are shown in Figure 1. Firstly, the displacement of altitude channel is calculated by SINS. Secondly, the frequency and the amplitude of each wave component are obtained by FFT spectrum analysis. Next, the comprehensive parameter and the comprehensive optimal cutoff frequency are, respectively, calculated by the adaptive algorithm. Finally, the heave motion information can be obtained by the adaptive filter and phase compensation.