**5.1 MEMS gyroscope signal analysis**

With the development of microelectronics technology, low cost MEMS gyroscopes begin to be used widely. It makes the great development of integrated navigation system, especially in UAV system. Compared with high costs gyroscope, the MEMS gyroscope devices have

Application of Wavelets Transform in Rotorcraft UAV's Integrated Navigation System 627

\* ,,, () () , *C f t t dt f <sup>j</sup> <sup>k</sup>* 

, , ( ) ( ) *jk jk ft C C t*

The purpose of wavelet transform for denoising is to extract useful signal and remove the interference signal in the output signal. In other words, the useful signal and noise signal are separated by the method of wavelet transform. There are 4 common methods of wavelet

1. Thresholding Denoising Method. It is also called wavelet shrinkage. The basic idea of this method can be described as: The wavelet coefficients have different characters in particular wavelet scales. According to this characteristic of the signal and noise, the noise signals are converted by wavelet transform in certain wavelet scales. According to a certain threshold processing strategies for treatment of wavelet coefficients, the coefficients, which are greater than the threshold, are kept (hard thresholding method) or shrunk (soft thresholding method). The coefficients, which are less than the threshold, are considered to be noise and set to zero directly. Then based on these wavelet coefficients, the original signal is reconstructed using inverse wavelet transform. And this method requires the assumption that the noise signal is Gaussian

2. Wavelet Decomposition and Reconstruction Method. It is also known as the Mallet method. It decomposes the signal with noise in scale into different frequency bands, sets the bands with noise to zero and reconstructs the signal using wavelet method. This method will remove the noise signals with the useful signals. So it may distort the

3. Modulus Maximum Method. In different wavelet scale, this method uses the variation features of wavelet transform modulus' maxima value to denoising the signal. The extreme points, whose amplitude decrease with scale increasing in signal, are removed. The extreme points, whose amplitude increase with scale increasing in signal, are retained. Using alternating projection method, the original signal is reconstructed from de-noised diagram of maxima modulus. And the noise signal is

4. Translation Invariant Method. It is a method improved from the basis of the Thresholding Denoising method. The noise signals are taken n times cycles shift by this method. And the translated signals are de-noised using thresholding denoising method. In the end, de-noised signal are equilibrated. This method has a smaller mean square

The reconstruction function of Discrete Wavelet transform can be expressed as :

 

(9)

(10)

denotes the translate factor. The *f* ( )*t* represents the

*j k j k*

The discrete wavelet translate factor can be expressed as :

Where *s* denotes the scale factor,

denoising (Burrus et al., 1998; Guo et al., 2003):

signal function.

white noise.

de-noised.

reconstructed signal.

error and improves signal-to-noise ratio.

some drawbacks, such as large bias stability, big temperature noise, high noise density. And all these disadvantages lead to that the long-term accuracy of navigation system is very low. While random vibration due to main rotor of RUAV, have an impact on gyroscopes measurements. Simple passive vibration damping measures cannot be completely filtering the vibration. And the measurement error is unacceptable (Ma et al., 2007). In order to eliminate noise of MEMS gyroscope, error analysis of signals need to be done. This is very important to improve the performance of integrated navigation system and increase the stability of the RUAV system.

According to the frequency spectrum characteristics of MEMS gyroscope, its errors can be divided into long-term errors and short-term errors. Long-term errors include bias stability, scale non-linearity, angular random walk, bias variation over temperature, rate noise density, and so on. These errors can be predicted by the mathematic model and adjusted. According to the error mechanism of MEMS gyroscope, an ARMA model is established. Then, using parameter identification, the parameter of the ARMA model is identified. So the long-term errors can be compensated. The short-term errors include random interference noise, measurement noise, and so on. It is a tricky problem to deal with errors. Usually, we use digital filters to compensate the error. These conventional denoising methods include Low-Pass filter, Kalman filter, and wavelet filter. Under the principle of linear least mean squares error, angular velocity estimation is recursively calculated by Kalman filtering in literature (Shi & Zhang, 2000). Although the approach is successfully used in reducing gyroscope noise on the stationary platforms, it is based on the assumption that the signal is corrupted by Gaussian noise and model is exactitude. Unfortunately, for imprecise model and colored noise, this method may yield worse results. Low-Pass filter passes low-frequency signals but reduces the amplitude of signals with frequencies higher than the cutoff frequency. And greater accuracy in approximation requires a longer delay. It can be realized by cheap hardware, while its low quality, however, is not very satisfactory. Wavelet transforms have excellent multi-resolution analysis feature and do not need model. So it is suitable for non-stationary signals processing. And wavelet transforms have been successfully applied to the denoising of signals or images in recent years. This method has achieved good results in gyroscope signal denoising process (Imola et al., 2001; Qu et al., 2009).

#### **5.2 Wavelet for denoising**

Wavelet transform is method of time-frequency localization analysis. Its window size (area) is fixed, but the shape can be changed. Wavelet transform developed the short - time Fourier transform of localized. It has a high frequency resolution and lower time resolution in low frequency part of the signal. And in high frequency part of the signal, it has a high time resolution and lower frequency resolution. Wavelet transform method has the character of frequency analysis, and it can indicate the occurred time. It is very suitable for noise reduction of MEMS gyroscopes.

The discrete wavelet function , ( ) *j k t* in Discrete Wavelet transform (DWT) can be expressed as :

$$
\Psi \,\nu\_{j,k}(t) = s\_0^{-j/2} \nu \left( \frac{t - ks\_0^j \tau\_0}{s\_0^j} \right) = s\_0^{-j/2} \nu \left( s\_0^{-j} t - k \tau\_0 \right) \tag{8}
$$

The discrete wavelet translate factor can be expressed as :

626 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

some drawbacks, such as large bias stability, big temperature noise, high noise density. And all these disadvantages lead to that the long-term accuracy of navigation system is very low. While random vibration due to main rotor of RUAV, have an impact on gyroscopes measurements. Simple passive vibration damping measures cannot be completely filtering the vibration. And the measurement error is unacceptable (Ma et al., 2007). In order to eliminate noise of MEMS gyroscope, error analysis of signals need to be done. This is very important to improve the performance of integrated navigation system and increase the

According to the frequency spectrum characteristics of MEMS gyroscope, its errors can be divided into long-term errors and short-term errors. Long-term errors include bias stability, scale non-linearity, angular random walk, bias variation over temperature, rate noise density, and so on. These errors can be predicted by the mathematic model and adjusted. According to the error mechanism of MEMS gyroscope, an ARMA model is established. Then, using parameter identification, the parameter of the ARMA model is identified. So the long-term errors can be compensated. The short-term errors include random interference noise, measurement noise, and so on. It is a tricky problem to deal with errors. Usually, we use digital filters to compensate the error. These conventional denoising methods include Low-Pass filter, Kalman filter, and wavelet filter. Under the principle of linear least mean squares error, angular velocity estimation is recursively calculated by Kalman filtering in literature (Shi & Zhang, 2000). Although the approach is successfully used in reducing gyroscope noise on the stationary platforms, it is based on the assumption that the signal is corrupted by Gaussian noise and model is exactitude. Unfortunately, for imprecise model and colored noise, this method may yield worse results. Low-Pass filter passes low-frequency signals but reduces the amplitude of signals with frequencies higher than the cutoff frequency. And greater accuracy in approximation requires a longer delay. It can be realized by cheap hardware, while its low quality, however, is not very satisfactory. Wavelet transforms have excellent multi-resolution analysis feature and do not need model. So it is suitable for non-stationary signals processing. And wavelet transforms have been successfully applied to the denoising of signals or images in recent years. This method has achieved good results in gyroscope signal denoising process

Wavelet transform is method of time-frequency localization analysis. Its window size (area) is fixed, but the shape can be changed. Wavelet transform developed the short - time Fourier transform of localized. It has a high frequency resolution and lower time resolution in low frequency part of the signal. And in high frequency part of the signal, it has a high time resolution and lower frequency resolution. Wavelet transform method has the character of frequency analysis, and it can indicate the occurred time. It is very suitable for noise

/2 <sup>0</sup> <sup>0</sup> /2

, 0 0 0 0 0

*t ks t s s st k s* 

*<sup>j</sup> <sup>j</sup> j j*

*t* in Discrete Wavelet transform (DWT) can be expressed

(8)

stability of the RUAV system.

(Imola et al., 2001; Qu et al., 2009).

reduction of MEMS gyroscopes.

as :

The discrete wavelet function , ( ) *j k*

( )

*j k j*

**5.2 Wavelet for denoising** 

$$C\_{j,k} = \int\_{-\infty}^{\infty} f(t) \boldsymbol{\nu}^\* \, {}\_{j,k}(t) dt = \left\langle f, \boldsymbol{\nu} \boldsymbol{\nu}\_{j,k} \right\rangle \tag{9}$$

The reconstruction function of Discrete Wavelet transform can be expressed as :

$$f(t) = \mathbb{C} \sum\_{-\infty}^{\infty} \sum\_{-\infty}^{\infty} \mathbb{C}\_{j,k} \boldsymbol{\nu}\_{j,k}(t) \tag{10}$$

Where *s* denotes the scale factor, denotes the translate factor. The *f* ( )*t* represents the signal function.

The purpose of wavelet transform for denoising is to extract useful signal and remove the interference signal in the output signal. In other words, the useful signal and noise signal are separated by the method of wavelet transform. There are 4 common methods of wavelet denoising (Burrus et al., 1998; Guo et al., 2003):


### **5.3 Thresholding denoising method**

Through the above analysis, the modulus maximum method and translation invariant method have large calculation amounts. And this will affect the real-time calculation of integrated navigation system. So considering the speed of calculation and the ease of implementation, thresholding denoising method is used in our navigation system.

The step of thresholding denoising method is as follows (Song et al., 2009; Su & Zhou, 2009):


The hard threshold estimation is defined as follows:

$$
\hat{d}\_{j,k} = \begin{cases}
 d\_{j,k'} & \left| d\_{j,k} \right| \ge \mathcal{A}\_j \\
 0, & \left| d\_{j,k} \right| < \mathcal{A}\_j
\end{cases} \tag{11}
$$

Application of Wavelets Transform in Rotorcraft UAV's Integrated Navigation System 629

Y axis gyroscope data

gyroy

0 2000 4000 6000 8000 10000 12000 14000

In order to find a appropriate wavelet functions and decomposition levels, the simulation compared the thresholding denoising method using harr, db2, db4, db6, sym2, sym4, coif2, bior1.5 and bior5.5 wavelet functions. The decomposition levels are respectively 2, 5 and 8. The standard deviation of de-noised signal's residuals is calculated to compare the wavelet denoising results. The results are shown in Table 3. And the standard deviation of the

Level haar db2 db4 db6 sym2 sym4 coif2 bior1.5 bior5.5 2 0.01841 0.1378 0.01323 0.01285 0.01378 0.01329 0.01288 0.01906 0.01271

5 0.04229 0.05397 0.05467 0.05489 0.05397 0.05561 0.05558 0.05514 0.05572

8 0.06552 0.06537 0.06576 0.0658 0.06573 0.06578 0.06582 0.06573 0.06583

In Table 3, when decomposition level increases to more than 5 layers, improvement in denoised signal's residuals is unobvious. When the de-noised signal's residuals approach to 0.06606, the de-noised signal is close to straight line. And the computation cost is increased

Engine ignition Speed idle Hovering

Fig. 12. The original signal of ServoHeli-40's Y axis gyroscope

Table 3. Standard deviation of de-noised signal's residuals

as layers increasing. So decomposition level of 5 is a good choice.

Sampling time (t=10ms)

Trajectory tracking


original data is 0.06606.

wavelet



Standing still


Angular veloctiy deg/s

0

0.5

The soft threshold estimation is defined as follows:

$$
\hat{d}\_{j,k} = \begin{cases}
\text{sgn}(d\_{j,k})(\left|d\_{j,k}\right| - \mathcal{A}\_j), & \left|d\_{j,k}\right| \ge \mathcal{A}\_j \\
0, & \left|d\_{j,k}\right| < \mathcal{A}\_j
\end{cases} \tag{12}
$$

Where *<sup>j</sup>* is the threshold constant.

3. Wavelet reconstruction. Using the inverse of discrete wavelet transform formulas, we can get the de-noised signal ˆ*<sup>i</sup> y* .
