Application of wavelet denoising in aircraft strapdown inertial navigation initial alignment
Mohammad Elias Yang Yunchun Zhang Ren
Beihang University NavTechnology Co., Ltd. Beihang University
Electrical Engineering and Automation Major North Ring Center Electrical Engineering and Automation Major
China, Beijing, 1001915 China, Beijing, 100029 China, Beijing, 1001915
[email protected] [email protected] [email protected]
Abstract: Initial attitude error is one of the most important error sources in high-precision inertial navigation. Accurately estimating the initial attitude angle is the guarantee of accurately determining the position, velocity and attitude of the mobile platform. For military aircraft, high initial alignment accuracy and short time are required. The required inertial signals (accelerometer specific force and gyroscope angular rate) are masked in high frequency measurement noise. When the aircraft is started, the vibration of the engine and other disturbances will cause high frequency noise. Kalman filters have been widely used in the initial fine alignment of strapdown inertial navigation systems (SINS). This paper proposes a multi-resolution wavelet de-noising (MRWD) method for removing high-frequency noise from inertial sensor measurement signals prior to initial fine alignment. The method proposed in this paper is tested and verified using the actual data collected when the carrier engine is powered on. The results show that this method can effectively improve the accuracy of initial alignment.
Key words: SINS (strapdown inertial navigation system) initial alignment wavelet de-noising Kalman filter
Inertial Measurement Unit (IMU) consists of 3 accelerometers and 3 gyroscopes
1 Introduction
Today, inertial navigation systems are used in a variety of commercial and military aircraft, ships, military tanks, submarines, missiles of all sizes, and space vehicle boosters. For aircraft navigation, since the coordinate system is oriented relative to the Earth's ellipsoid, the local-level geographic coordinate system can be used as the calculated coordinate system. The initial alignment can be done approximately by computing the elements in the rotation matrix from the body to the navigation coordinate system [1]. Since initial alignment is critical for high-accuracy inertial navigation, extensive research on this topic is required [2][3][4]. An estimation of fine alignment for inertial navigation has been done [5] and it has been shown how it affects navigation accuracy.
An inertial navigation system (INS) is a self-contained system consisting of inertial sensors that measure the acceleration and rotation of a moving object. The sensor ensemble is an inertial measurement unit (IMU) consisting of three accelerometers and three gyroscopes mounted on quadrature triples. These measurements are made against an inertial frame of reference. The position, velocity and attitude of the main carrier are obtained by integrating the acceleration and angular velocity in a well-defined coordinate system. The implementation of this concept requires computing the transformation between the navigation coordinate system and the measurement coordinate system defined by the IMU-sensitive axes. The relationship between the b- and n-frames is achieved by continuously updating the transformation (or pose) matrix. In order to limit errors in the derived parameters, it is crucial to determine high-precision initial values for such matrices [6].
Since the SINS are independent, a self-calibration method can be used to compute the rotation matrix elements of the navigation-level SINS, as described in Section 3. The process of calculating the initial value of the rotation matrix is called SINS initial alignment. Alignment is accomplished in two steps, coarse alignment (CA) and fine alignment (FA). The purpose of the coarse alignment is to determine an approximation of the attitude angles (roll, pitch and heading) between the b- and n-frames. Fine alignment uses iterative techniques to improve the accuracy of the estimates [7][8].
For military aircraft, initial alignment requires high accuracy and short time. When the aircraft is on the runway preparing for takeoff, a quick and accurate initial SINS alignment may be required. The aircraft is not absolutely stationary due to engine vibration, gusts, loading, fuel injection and uncertain human impact, so the output of the inertial sensor contains errors from this external disturbance.
Inertial measurement units (IMUs) have two types of errors: long-term error (low frequency noise) and short-term error (high frequency noise) [9]. Some short-term errors are introduced when measuring inertial output due to engine vibration and external disturbances. Pre-filtering the inertial sensor data for high-frequency noise can improve the initial alignment of the SINS. The wavelet denoising method has been well applied to the filtering of the original output of inertial sensors in alignment and inertial navigation [10]. This paper investigates wavelet multiresolution analysis (WMRA) used to remove short-term noise in initial alignment during aircraft engine startup. Its purpose is to calculate a more accurate initial rotation matrix in the coarse alignment stage and speed up the efficiency of fine alignment.
This paper is organized as follows: Section II briefly introduces a wavelet multiresolution analysis of a signal. Section 3 introduces the rough alignment theory, the system dynamics error model and the fine alignment measurement model. In Section IV, the realization equations of the Kalman filter are presented. The experimental procedure and result evaluation are completed in Section V. The final conclusion is drawn in Section VI.
2. Wavelet Multiresolution Analysis
Estimating signals corrupted by additional noise is of interest to many researchers for practical as well as theoretical reasons. We want the recovered signal to be as close as possible to the real signal, preserving most of its important properties (such as smoothness) [11]. Traditional denoising methods include moving average and low-pass filtering techniques. In recent years, as an emerging technology, wavelet decomposition is more used to deal with inertial sensor noise. While many other methods, such as neural network denoising, have also been extensively studied.
Wavelet analysis can therefore reveal information that other signal analysis techniques miss, such as trends, breaks and discontinuities in higher-order derivatives, and self-similarities. Wavelets are also capable of compressing or denoising the signal without destroying the original signal too much [12].
A. Fast Discrete Wavelet Transform and Reconstruction
Mallat proposed multi-resolution analysis and proposed a fast algorithm based on wavelet transform and reconstruction, which he named Mallat's algorithm. According to the algorithm, if the sampling information is included, and , the orthogonal wavelet decomposition formula is [13].
where and are the approximate and detailed coefficients, respectively, j is the level of decomposition, and h and g are the low-pass and high-pass filter coefficients.
Reconstruction is the inverse process of wavelet decomposition. The equation for fast wavelet reconstruction is:
b. Wavelet Threshold Selection and Algorithm
Wavelet thresholding technique is a signal estimation technique, which utilizes the ability of wavelet transform to de-noise the signal. The wavelet thresholding technique can reduce the noise level without distortion. Therefore, it can be used by preprocessing filters of inertial sensor signals, overcoming the shortcomings of existing low-pass filters. Let's consider the inertial sensor output corrupted by white noise. The output of the noise can be expressed as:
We want to estimate the original signal in order to minimize the mean squared error. Let w be the orthogonal wave transform operator. Then equation (4) can be written as:
Now, set as the wavelet threshold operator, the wavelet denoising scheme can be expressed as [11]:
Donoho et al. proposed two thresholding operators, soft thresholding and hard thresholding [14]. In this study, soft thresholding was adopted because the denoised signal obtained from soft thresholding was smoother. The soft threshold function is defined as [11]:
Donoho gave the optimal general threshold [14]:
where is the signal noise variance and N is the length of the signal. Signal denoising is performed in three steps:
1) Decompose the noise signal to the desired level using a specific wavelet to obtain the wavelet coefficients.
2) Choose an appropriate threshold function and threshold to calculate denoising coefficients.
3) Perform an inverse wavelet transform to reconstruct the denoised signal.
c. Reasonable Choice of Wavelet Decomposition and LOD
In wavelet decomposition, the correlated signal is passed through two complementary half-band filters: a low-pass filter and a high-pass filter. The output of the low-pass filter is called the "approximate part" and the output of the high-pass filter is called the "exact part", and according to the Nyquist theorem, if the sampling frequency of the measurement signal is , the maximum frequency component that can appear in the measurement Yes.
Therefore, when applying the wavelet discrete transform, the approximate part will contain components with frequencies less than and between and . In order to obtain lower resolutions, the decomposition compression can be iterative, approximating the decomposition successively until the limit of the desired value is reached. This process is called wavelet multiresolution decomposition (WMRD).
Earth's gravity and rotation rate are low frequency signals. Spectral analysis (not part of this study) showed that the frequency bands of these signals, as well as some long-term (low frequency) noise, were confined to 0.8Hz. A suitable value for the level of decomposition (LOD) can be calculated as follows:
where is the band limit value and n is the LOD.
The sampling frequency of the LINS800SINS inertial sensor used in this study is 200Hz. Therefore, the six-level decomposition will limit the frequency band to about 0.8Hz and is sufficient to remove high frequency noise. Further decomposition may lose useful information of the original signal.
In this study, Daubechies (db6) wavelets and the "Ribrsure" threshold principle based on Stein were used. "Ribrsure" is an unbiased likelihood estimate that adapts when a threshold is chosen.
3. Self-initial alignment method of SINS
The alignment problem in strapdown inertial navigation systems is basically the determination of the initial transformation matrix that can relate the instrumented airframe coordinate system to the reference computed coordinate system. Furthermore, if it is for commercial applications of inertial navigation systems, means of self-calibration will be essential [6]. The self-calibration of SINS is a two-stage process, that is, coarse alignment first, followed by fine alignment.
A. rough alignment
North, East, and Down (NED) is the navigation coordinate system used for navigation calculations in this paper. In order to continuously update the velocity and position of the main carrier in the navigation coordinate system, a transformation of the accelerometer measurements is required. The local gravity vector and the Earth's rotation rate in the NED system can be expressed as:
where , is the total Earth velocity, L is the latitude, and g is the magnitude of the local gravity.
The transformation matrix from the body frame (b frame) to the navigation frame (NED) is defined as:
where s and c represent the sine and cosine functions, respectively. This rotation matrix is an orthogonal matrix. Then the following relationship holds:
In stationary alignment, accelerometers and gyroscopes measure the Earth's gravity and angular velocity in the body coordinate system. During coarse alignment, accelerometer and gyroscope measurements are averaged over one to two minutes to estimate roll, pitch, and yaw angles [15]. The accelerometer measurements are related to the Earth's gravity as follows:
From (13), the roll and pitch can be calculated as:
From (14), the roll angle and pitch angle can be estimated from the output of the accelerometer without knowing the local earth gravity value. Once we have roll and pitch, the heading angle can be calculated as:
where , , are approximations for roll, pitch, and heading. An approximate rotation matrix is calculated based on the Euler angles estimated above.
b. Precisely aligned SINS error model
Inertial sensors have random biases and measurement errors. During coarse alignment, only a rough estimate of the rotation matrix is required. Fine alignment is the process of estimating small deviations between the real coordinate system and the calculated coordinate system. Kalman filters are often used for precise alignment.
(1) System dynamic error model
The SINS error model can be well derived using the perturbation method in [16]. The 10-level error model for precise alignment on the static base is expressed as:
in:
is the error state vector.
and is the velocity and attitude error, is the accelerometer bias, and is the gyro drift rate. G is the noise shaping matrix and w handles white noise with power spectral density Q(t).
(2) Measurement model
For static initial alignment, the position is constant and the velocity of the carrier is zero. Using navigation mechanics choreography, the velocity of the carrier can be calculated, which can be seen as an error in the velocity of the stationary object. This velocity error is used as a measure of fine alignment. The measurement model is expressed as:
where, is the measurement vector and v is zero-mean random white Gaussian noise.
4. Kalman filtering for fine alignment
The linear dynamic model described in (16) and (17) can be expressed in discrete form as:
in,. and are modeled as uncorrelated zero-mean white Gaussian noise sequences, such as:
is the covariance matrix of the process noise, which is calculated as:
is the covariance matrix of the measurement noise and is the filter period.
The standard Kalman filter equation is as follows:
One-step state and error covariance prediction:
Kalman gain calculation:
One-step state correction and error covariance update:
5. Experimental Results and Evaluation
To test the effectiveness of the wavelet multiresolution denoising (WMRD) proposed in this paper, we conduct initial alignment experiments on the LINS800 navigation-level system (developed by Beijing NavTechnology Co., Ltd.) for aircraft navigation. The LINS800 consists of three ring laser gyroscopes and three accelerometers along three mutually orthogonal axes. The LINS800 system is installed in a stationary carrier. Two sets of real data are collected in this way:
1) The first set of data collected when the carrier's engine is powered off.
2) A second set of data collected when the carrier's engine is started.
Figure 1 Flow chart of fine alignment
The calculation of navigation parameters and the Kalman filter algorithm for initial alignment including wavelet denoising are implemented in the MATLAB environment. Figure 1 shows the SINS initial alignment scheme. First, the raw inertial data are filtered with high frequency noise using the wavelet multiresolution method and coarsely aligned to calculate the attitude angle and initial quaternion. Then, the initial quaternion and pre-filtered inertial data are substituted into the strapdown mechanics choreography algorithm, which calculates the position, velocity and attitude of the navigation coordinate system after integration and transformation at 200Hz.
Because the carrier is stationary, the velocity calculated from the strap-down mechanics programming algorithm is considered the velocity error and is the measured input to the Kalman filter. The Kalman filter update interval is 1Hz. These estimation errors are used as feedback for strapdown mechanics programming to correct velocity and quaternion. In this way, the evolution of velocity errors used as filter observations is minimized.
Four sets of data were collected when the engine was powered off and an initial alignment test was performed. The alignment results are shown in Table 1. The average of these four results is considered to be the true pose angle. Short-term (high frequency) noise can affect the measurement of inertial data due to carrier engine startup, vibration, and other external disturbances. Therefore, proper pre-filtering techniques are required to obtain fast and precise alignment. Wavelet multi-resolution denoising is a good technique to remove high frequency noise from inertial sensors without affecting the original information. This technique is applicable to real data collected from the LINS800 under vibrating conditions. A 6-level decomposition (LOD) is sufficient to reduce short-term errors. The wavelet denoising results of one channel (vertical channel) of SINS are shown in Table 2 and Fig. 2. After denoising, the standard deviation of the inertial measurements is significantly reduced. The wavelet denoising method can remove the high frequency noise of the inertial sensor and obtain a more accurate pose in coarse alignment. However, eliminating low frequency noise still requires fine alignment.
Table 1 Results of four groups of alignment tests
group |
Roll (°) |
Pitch (°) |
Heading (°) |
1 |
-0.16657 |
-2.28360 |
59.38876 |
2 |
-0.16814 |
-2.28568 |
59.40585 |
3 |
-0.15480 |
-2.28988 |
59.43050 |
4 |
-0.16619 |
-2.29058 |
59.402867 |
average value |
-0.16393 |
-2.28744 |
59.406998 |
Table 2 Standard deviation of measured values
sensor |
average value |
Before denoising |
After denoising |
unit |
Gyro |
-9.878 |
197.035 |
9.36 |
degrees/hour |
Accelerometer |
-9.792 |
0.555 |
0.00191 |
m/ s2 |
Figures 3 and 4 show the results of the estimated pose angle during fine alignment. For comparison, we have performed good alignments with the data filtered during engine power-off, engine-on, and engine-on, respectively. These results show that after denoising the inertial measurements and applying a 10-state Kalman filter, the convergence speed and accuracy of the azimuth estimates are greatly improved. We also observe that the accuracy of coarse alignment also increases after denoising the finely aligned input. Figure 6 shows that the azimuth angle error converges faster after denoising.
Figure 2 Angular rate and acceleration measurements
Figure 3 Estimated values of roll and pitch angles in fine alignment
Fig. 4 Estimation of heading angle in fine alignment
Figure 5 Estimation error of roll and pitch
Figure 6 Estimation error of heading
6 Conclusion
This paper proposes a wavelet multi-resolution decomposition method and a precise alignment algorithm. When the aircraft is ready to take off on the runway, the engine is usually started. Wavelet denoising based pre-filtering is used to limit the noise level prior to fine alignment. Experimental results show that in pre-filtered data, the time for fine alignment is reduced and gives more accurate angle estimates compared to noisy data. The method presented here provides rapid and accurate initial alignment of the aircraft under vibration conditions resulting from engine start-up.
Thanks
This work was supported by Beijing NavTechnology Co., Ltd. The authors thank Dr. Yunchun Yang for his encouragement and support in completing this work. The authors also thank Mr. Xinchun Ding and Mr. Qiu for their help and support for the experimental data collection of this paper.
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