**3. Wavelet-based fault detection**

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

temperature from -40℃ to 80℃ extends the usage of RUAV in various environments. The LPC3250 includes a USB 2.0 Full Speed interface, seven UARTs, two I2C interfaces, two SPI/SSP ports, and two I2S interfaces; Such a great number interfaces of LPC3250 makes it very suitable for navigation and control system with a plenty of sensors in standard

To decrease the developing work in programming, while increasing the system stability, a μC/OS-II embedded system is installed to organize the software development. This small sized embedded system is quite convenient to install; the hard-real-time architecture also makes it suitable for a time critical avionics system in RUAV. We divided the work of software into 5 parts. First, the OS Kernel is to maintain the whole system and arrange the task schedule. Second, the algorithms implements navigation and control theory. Third, the device interface process is to handle the task for sensor data acquire and drive the actuator. Fourth, the user interface carries out the job to display and receive necessary information to the user. Fifth, the log interface is to log the flight data for our experiment. To make sure that the algorithms can be calculated in time, a hardware timer is used instead of the software timer provided by operator system. With a proper design of the software architecture, the system's stability is maintained and the flexibility is also provided for other

The GPS receiver and magnetometer are in a separate part and the others are in the main navigation part. The flight control system and main navigation system are assembled in an anti-jamming aluminum box, and called flight control box. Such a separation is with the consideration that the GPS and magnetometers are susceptible to the install position because they may be influenced if it is covered by the airborne or near some magnetic material. The flight control box is mounted under fuselage of the RUAV. The separate part can be equipped in a proper place on the airframe. To avoid the disciplinary vibration about 20-22.5Hz caused by revolving of main rotor, ENIDINE aviation wire rope isolators are also used. They are comprised of stainless steel stranded cable, threaded through aluminum alloy retaining bars, crimped and mounted for effective vibration isolation. The assembled

RUAV system with the necessary components is shown in Figure 6.

interface. We designed interface circuit to drive the actuator and log the flight data.

algorithm implementations.

Fig. 6. Implemented ServoHeli-40 RUAV

**2.2.4 System realization** 

Without loss of generality, assume that the vehicle's sensor output *y*( )*t* is described as (Zhang & Yan, 2001):

$$y(t) = f[\mathbf{x}(t)] + n(t) \tag{1}$$

Where *n t*( ) is a noise signal and the measured *x t*( ) changes with a *k*-degree polynomial function *f*[ ( )] *x t* which describes the measured process changes. Stone-Weierstrass Theorem states that any continuous function on a compact set can be approximated to any degree of accuracy by a polynomial function (Rudin, 1976). Therefore, using a polynomial function to represent any function *f*[ ( )] *x t* will not lose the generality. Let ( )*t* be a wavelet function and ( ) (1/ ) ( / ) *<sup>s</sup> t s ts* be the dilation of ( )*t* by the scale factor *s*. The wavelet transform of *y t*( ) can be written as:

$$\mathcal{V}NT\_f(\mathbf{s},\tau) = \mathcal{y}(t) \* \boldsymbol{\nu}\_s(t) = f[\mathbf{x}(t) + \boldsymbol{n}(t)] \* \boldsymbol{\nu}\_s(t) \tag{2}$$

Where denotes the convolution and (, ) *WT s <sup>f</sup>* represents the wavelet transform. A wavelet ( )*t* is said to have *m* vanishing moments if and only for all positive integers *k<m*, the following equation is satisfied:

$$\int\_{-\infty}^{+\infty} t^k \boldsymbol{\nu}\_s(t) dt = 0\tag{3}$$

Application of Wavelets Transform in Rotorcraft UAV's Integrated Navigation System 623

Table 1 shows us that there are three sensors in the RUAV sensor system which have three channel separately. We design 12 wavelet analyzers for 12 channels of all sensors. The sensors data will directly send to the data fusion system when the data are in the normal states. However, if the sensors data is abnormal in one or some channels as a result of the failure of specific sensors, the alarms will send to the flight computer while the data link will be cut off. Then the navigation system will continue to compute with degraded sensors data.

The proposed wavelet-based fault detection system tested using the ServoHeli-20 RUAV

Fig. 8. Architecture of wavelet-based sensor system

**4.2 Experimental results and discussion** 

Fig. 9. ServoHeli-20 fault detection experiment

system in manual mode.

Now, let us call a smoothing function, any real function ( )*t* such that <sup>2</sup> ( ) (1 /1(1 )) *tO t* and whose integral is nonzero. A smoothing function can be viewed as the impulse response of a low-pass filter. Let *f*[ ( )] *x t* and ( ) (1 / ) ( / ) *<sup>s</sup> t s ts* be a real function in <sup>2</sup>*L R*( ) . The abrupt changes of the sensor data at scale *s* are defined as local sharp variation points *f*[ ( )] *x t* smoothed by ( ) *<sup>s</sup> t* . The method of detecting these sharp variation points with a wavelet transform is explained as follows.

Let <sup>1</sup> ( )*t* and <sup>2</sup> ( )*t* be the two wavelets defined by:

$$
\psi^{-1}(t) = \frac{d\theta(t)}{dt} \tag{4}
$$

$$
\psi^2(t) = \frac{d\theta^2(t)}{dt^2} \tag{5}
$$

The wavelet transform defined with respect to each of these wavelets are given by:

$$\mathcal{W}^1(t) = f \ast \boldsymbol{\nu}\_s^1(t) = f \ast (\mathbf{s} \frac{d\theta\_s}{dt})(t) = \mathbf{s} \frac{d}{dt} (f \ast \theta\_s)(t) \tag{6}$$

$$\mathcal{W}^1(t) = f \ast \nu\_s^1(t) = f \ast (s \frac{d\theta\_s}{dt})(t) = s \frac{d}{dt} (f \ast \theta\_s)(t) \tag{7}$$

The wavelet transforms of <sup>1</sup> *W f* (,) *s t* and <sup>2</sup> *W f* (,) *s t* is proportional respectively to the first and second derivatives of *f*[ ( )] *x t* smoothed by ( ) *<sup>s</sup> t* . As a result, the local maxima of <sup>1</sup> | ( , )| *W f s t* indicate the locations of sharp variation points and singularities of [ ( )] ( ) *<sup>s</sup> f xt t* (Mallat & Hwang, 1992).

From (4) to (7), it can be concluded that the wavelet transform of the signal (1) only includes some sharp variation points induced by sensor faults and random noise. Once a sharp variation point is claimed, and alarm will be triggered for a failure of the sensor.

#### **4. Fault detection experiment**

#### **4.1 Fault detection system design**

The sensors of the navigation system with different mechanism also have different performance. We cannot get the ideal fault detection results using the traditional fault detection techniques.

In order to accompany the short control period and the highly update rate, we use the parallel wavelet analyzer, which is shown as figure 8.

Fig. 8. Architecture of wavelet-based sensor system

() 0 *<sup>k</sup> <sup>s</sup> t t dt* 

and whose integral is nonzero. A smoothing function can be viewed as the impulse response

abrupt changes of the sensor data at scale *s* are defined as local sharp variation points

<sup>1</sup> ( ) ( ) *d t <sup>t</sup> dt* 

2

( ) ( ) ( )( ) ( )( ) *<sup>s</sup> s s <sup>d</sup> <sup>d</sup> W t <sup>f</sup> <sup>t</sup> <sup>f</sup> s ts <sup>f</sup> <sup>t</sup>*

( ) ( ) ( )( ) ( )( ) *<sup>s</sup> s s <sup>d</sup> <sup>d</sup> Wt f t f s t s f t dt dt* 

The wavelet transforms of <sup>1</sup> *W f* (,) *s t* and <sup>2</sup> *W f* (,) *s t* is proportional respectively to the first

<sup>1</sup> | ( , )| *W f s t* indicate the locations of sharp variation points and singularities of [ ( )] ( ) *<sup>s</sup> f xt t*

From (4) to (7), it can be concluded that the wavelet transform of the signal (1) only includes some sharp variation points induced by sensor faults and random noise. Once a sharp

The sensors of the navigation system with different mechanism also have different performance. We cannot get the ideal fault detection results using the traditional fault

In order to accompany the short control period and the highly update rate, we use the

variation point is claimed, and alarm will be triggered for a failure of the sensor.

2 ( ) ( ) *d t <sup>t</sup> dt* 

*dt dt*

 *t s ts* 

2

The wavelet transform defined with respect to each of these wavelets are given by:

Now, let us call a smoothing function, any real function

of a low-pass filter. Let *f*[ ( )] *x t* and ( ) (1 / ) ( / ) *<sup>s</sup>*

( )*t* be the two wavelets defined by:

1 1

1 1

and second derivatives of *f*[ ( )] *x t* smoothed by ( ) *<sup>s</sup>*

parallel wavelet analyzer, which is shown as figure 8.

wavelet transform is explained as follows.

*f*[ ( )] *x t* smoothed by ( ) *<sup>s</sup>*

( )*t* and <sup>2</sup>

(Mallat & Hwang, 1992).

detection techniques.

**4. Fault detection experiment 4.1 Fault detection system design** 

Let <sup>1</sup> 

(3)

(4)

(5)

 

 

*t* . As a result, the local maxima of

( )*t* such that <sup>2</sup> 

be a real function in <sup>2</sup>*L R*( ) . The

( ) (1 /1(1 )) *tO t*

(6)

(7)

*t* . The method of detecting these sharp variation points with a

Table 1 shows us that there are three sensors in the RUAV sensor system which have three channel separately. We design 12 wavelet analyzers for 12 channels of all sensors. The sensors data will directly send to the data fusion system when the data are in the normal states. However, if the sensors data is abnormal in one or some channels as a result of the failure of specific sensors, the alarms will send to the flight computer while the data link will be cut off. Then the navigation system will continue to compute with degraded sensors data.
