On-line Automatic Frequency Change Detection

The use of Wavelets for Determining Wing Flexure in Airborne GPS Multi-Antenna Attitude Determination Systems Michael Moore School of Surveying and Spatial Information Systems The University of New South Wales, Sydney NSW 2052 AUSTRALIA

BIOGRAPHY

1. INTRODUCTION

Michael Moore received a B.Geomatics and B.Sc. in 1998, from the University of Melbourne, and then joined the School of Surveying & SIS, The University of New South Wales (UNSW) as a PhD student in March 1999. His research interests include attitude determination with a multi-antenna GPS arrays, Kalman filtering, and the associated time series analysis using digital signal processing techniques.

The importance of wing flexure modeling to obtain precise aircraft attitude has been reported by several authors, for example, Cohen et al. (1993), Cannon et al. (1994), and Tsujji (1998). In Cohen et al. (1993) a multiantenna GPS array is able to not only resolve the attitude parameters but is also able to estimate the structural deformation of the wing during flight. The estimation of the wing flexure was done through incorporating an extra parameter to the Least Squares process when the attitude parameters are being resolved. The investigators also noted that even large deformations in excess of a GPS wavelength did not have a big impact upon the precision of the attitude parameters. However, this platform had baseline lengths in excess of 10m. Wing flexure occurring on a smaller aircraft will have a greater impact upon the level of precision of the estimated attitude parameters. This is because, in general, the attitude parameter accuracy increases with antenna separation.

ABSTRACT In the quest to obtain a precise set of aircraft attitude parameters the modeling of wing flexure can become an important issue, especially when short baselines are being used. Previous techniques for the estimation of wing flexure have relied on a constrained Least Squares process to determine the displacement at each epoch before the estimation of the attitude parameters. As will be shown in this paper, there are more effective techniques available for the estimation and removal of wing flexure.

In Tsujji (1998) not only is wing flexure modeled through a Least Squares estimation process, but also the lateral flexure of the aircraft fuselage. Cannon et al. (1994) drew attention to the fact that due to wing flexure the body frame can no longer be considered as being completely rigid. If wing flexure is not accounted for then the derived attitude angles will be with respect to the wrong body frame, thus affecting the accuracy of the estimated attitude parameters. To avoid this, the wing flexure component must be removed before the attitude parameters are calculated. Cannon et al. (1994) and Tsujii (1998) also pointed out that the largest wing flexure occurred during significant maneuvers such as take-off and landing.

If the variation in the height of the antenna is viewed as a signal, digital signal processing techniques can be implemented, such as Discrete Fast Fourier Transforms (DFFT), to spectrally analyze the signal. This information can then be used to construct an equiripple FIR filter to isolate/remove the wing flexure component from the original signal. Another approach is to use a wavelet technique. The coefficients obtained from a wavelet transformation can be manipulated to filter the noise from the wing flexure signal. Only the coefficients corresponding to the wing flexure are used in the reconstruction process thus effectively filtering the unwanted noise from the original signal.

The capacity of GPS to estimate structural deformation adds an additional dimension to the process of attitude determination for airborne vehicles. Three different techniques will be investigated in this paper. The first is based on the estimation of the instantaneous wing flexure through the augmentation of the Least Squares process. The second makes use of Fast Fourier Transform (FFT) techniques that filter the signal via a Finite Impulse

This paper will assess the constrained Least Squares solution compared with the DFFT and wavelet techniques through a series of simulations and experiments. The results obtained highlight the superior ability of wavelet techniques to denoise stationary and in particular nonstationary signals.

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domain the time information is 'lost', and it therefore becomes impossible to tell when a particular event took place. If the signal happens to contain numerous nonstationary signatures, then these important characteristics of the signal will be missed.

Response (FIR) filter. The third uses a wavelet technique to de-noise the signal of interest. If the wing flexure has a periodic signature, as may be the case during turbulence or after a manoeuvre has been completed, then the discrete FFT technique may be applied. The frequency of the oscillation of the wing flexure can be estimated by applying a DFFT to the change in antenna height. A power spectral density plot can then be obtained. From this information two filters can be constructed: a band pass and a band stop filter. The purpose of the band pass filter would be to allow the wing flexure signal to be 'passed through' with a reduction in noise level for later analysis. The band stop filter would eliminate/reduce the wing flexure from the original set of data. This 'cleaned' data can then be used in a standard Least Squares process to calculate a more precise set of attitude parameters.

The Fourier transform that is applied to discrete sequences is referred to as the Discrete Time Fourier Transform (DFFT). There are many different formulations of the DFFT, but the one most commonly used for the analysis of signals is defined below in eqn.(1), along with its inverse (Han,1998): N −1

X [k ] = ∑ x[ n]e

− j 2 πnk / N

for _ k = 0,1,2,L,N − 1

n =0

x[n ] =

The main assumption in the case of the DFFT technique is that the signal must be stationary during the observation period. With regards to wing flexure this may not be a realistic assumption. The non-stationary behavior of wing flexure could be due to different levels of turbulence being experienced by the aircraft, a change in the aircraft dynamics, or just decay in the amount of wing flexure being experienced. Wavelets are very good at dealing with transient, non-stationary signals. Therefore the use of wavelets to analyze and de-noise the wing flexure signal is feasible and is investigated in this paper.

1 N −1 ∑ X[k ]e j2πnk / N for_ n = 0,1,2,L, N −1 ( 1) N n= 0

where x[n] is the discrete function and X[k] is the DFFT, both of which are periodic with the same period N, and j = √-1. One of the main assumptions in using the DFFT for the calculation of the spectrum of a discrete signal is that the observed signal is stationary. Spectral analysis aims to describe the frequency content of a signal based on a finite set of data. The power spectral density (PSD) of a stationary random variable is related mathematically to the correlation sequence by the discrete-time Fourier transform:

2. WING FLEXURE ESTIMATION TECHNIQUES If wing flexure is occurring, then the rigid body assumption for the body frame coordinate system is no longer true. As the wing flexure is changing, the body frame coordinate system in which the derived attitude is expressed is not rigidly attached to the aircraft. In order to overcome this wing flexure has to be removed before the computation of the attitude parameters.

Γxx (ω ) =

∞

∑γ

m =−∞

xx

(m)e

jω m

( 2)

where Γxx(f) is the distribution of power as a function of frequency, and γxx(m) is the discrete auto-correlation sequence. The basic goal of filter design is to perform a frequency dependent alteration of a data sequence. This can be to remove noise above a certain level, or to pass a specific set of frequencies through. A FIR filter has a number of advantages they have a linear phase, are always stable, and the filter start-up time has a finite duration. The equiripple filter has the advantage that the gain is very close to one, and has therefore been used in this investigation to maintain the original magnitude of the signal being filtered.

2.1 Least Squares Typically the attitude components, i.e. heading, pitch and roll, can be estimated through a Least Squares estimation process in which the vectors between the antennas are the observables. This is done by rotating the results obtained from the GPS antenna array, expressed in the local level system, back onto the body frame system in order to determine the attitude parameters. Usually in a Least Squares process the wing flexure is accounted for by an extra parameter, constrained to 'move' only in the z-axis of the body frame (Cannon et al., 1994).

In the following simulations and experiments, the original signal of the height difference between the master station and the two wing antennas are analyzed and expressed in the form of a PSD plot using the Welch method. From the PSD it is possible to determine the dominant frequencies at which wing flexure is occurring. From this information one can then construct two equiripple FIR filters. The first one being a band stop filter, used to filter the unwanted wing flexure signal from the original signal. This filtered signal can then be used in the computation of the attitude

2.2 The DFFT and Equiripple FIR Technique If the wing flexure has a periodic signature then the Fast Fourier Transform can be used to break down the signal into sinusoidal components of different frequencies. This is equivalent to transforming a signal from a time-based representation to a frequency-based one. The main disadvantage of this approach is that in the frequency

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parameters. The second is a band pass filter, which will be designed to pass the wing flexure signal through for analysis with a reduced receiver noise level.

The main modification of the Daubechies wavelet to construct the Symlet wavelet involves reusing the function m0, by considering |m0(ω)|2 as a function W of z = eiω (Misiti et al., 2000).

2.3 Wavelet Techniques The main advantage of wavelet analysis is that it allows the use of long time intervals where more precise low frequency information is wanted, and shorter intervals where high frequency information is sought. Wavelet analysis is therefore capable of revealing aspects of data that other signal analysis techniques miss, such as trends, breakdown points, and discontinuities in higher derivatives and self-similarity. Wavelets are also capable of compressing or de-noising a signal without appreciable degradation of the original signal. A Wavelet is basically a waveform of effectively limited duration that has an average of zero, that tend to be asymmetric and irregular. Wavelets are grouped into families that have a number of common characteristics, which are then scaled and translated to represent the signal as fixed series of building blocks (Wickerhauser, 1994; Daubechies, 1990). A wavelet family is defined by two wavelets: a father wavelet (φ) and a mother wavelet (ϕ). For the Haar family:

1,if 0 ≤ t ≤ 1  0,otherwise

φ (t) = 

Figure 1 The father and mother wavelet Haar (top) and Symlet wavelet (bottom). There are a great deal of wavelets that can be used, and are continually being discovered. This investigation has concentrated on using two well known wavelets the Haar and Symlet, as shown in Figure 1. Finding the optimal wavelet for a particular de-noising application is a research topic of its own and is not dealt with in this paper.

( 3)

The father wavelet (defined below for the Haar wavelet) is also referred to as the scaling function:

 1,if 0 ≤ t < 1/ 2    ϕ (t) = −1,if 0 ≤ 1/ 2 < 1    0,otherwise 

The wavelet transform turns the information of a signal into coefficients that can be manipulated, stored, transmitted, analyzed or used to reconstruct the original signal. The discrete wavelet transform (DWT) and its inverse are given by:

( 4)

W[ k, j ] = ∫ x(t)ψ j, k (t )dt x(t) = ∑ ∑ W[ k, j ]ψ˜ j ,k (t)

The Symlet wavelet is only nearly symmetric, and is not exactly symmetrical as the name would suggest (see Figure 1). These wavelets are a product of some modifications suggested by Daubechies (1990) to her wavelets in order to increase the symmetry, while retaining their relative simplicity. The Daubechies wavelet does not have an explicit form, but the square modulus of the transfer function is: N −1

P(y) = ∑ Ck

N −1+ k k

y

j

where ψ(t) is the mother wavelet (for analysis), ψ~ is the wavelet used for synthesis, j is the scale factor, and k is the time delay. In this case the basis for a wavelet expansion system is generated from simple scaling and translation. The mother wavelet is then represented by (Reza, 1999):

( 5)

k= 0

N-1+k

where Ck

denotes the binomial coefficients.

ψ j ,k (t) = 2 j / 2 ψ (2 j t − k); for _ j,k = 1,2,L

Then:

  ω  N  ω 2 m0 (ω ) =  cos 2    P  sin2     2    2   where,

m0 (ω ) =

1 2 N−1 −ikω ∑ hk e 2 k =0

( 8)

k

( 9)

The 2j/2 factor in eqn.(6) normalizes each wavelet to maintain a constant norm, independent of scale j. In this case the lower resolution coefficients can be calculated from the higher resolution coefficients by a tree-structure algorithm, called a filter bank, as shown in Figure 2.

( 6)

From the DWT, a set of coefficients can be calculated which, under certain conditions, can completely describe the original function. These coefficients can then be used for analysis, description, approximation and filtering. The

( 7)

3

DWT coefficients are obtained by using the following set of equations:

larger than the coefficients corresponding to white noise as they are dispersed throughout the transform, resulting in small coefficients, which may be suppressed by the procedure described above.

c j −1 (k) = ∑ h(m − 2k)c j (m ) m

d j −1 (k) = ∑ h' (m − 2k)c j (m)

( 10)

Applying the wavelet transform in real time introduces a number of problems. These are mainly caused by the usual formulation of the wavelet transformation having references that extend backward and forward in time. There is also the problem of dealing with a finite set of data, which introduces boundary problems. There are a number of ways to reformulate the transformation to avoid references to the signal forward in time, and there are a number of well documented techniques of dealing with the boundary problems. However, there is no universal solution to these problems, and it is outside the scope of this paper to deal with them in any detail. For more background information concerning these issues readers are referred to Jensen (2001).

m

j = J , J − 1,L, j 0 + 1 where h(k) is the scaling function coefficients. The collection of these coefficients is called the DWT of the original signal x(k). The number of coefficients obtained from the DWT is equal to the number of points in the discrete signal. Original Signal High Frequency Term

HPF

LPF

Four different scenarios for wing flexure are considered. The results are analyzed to assess the performance of a number of different possible modeling techniques. The four simulated scenarios are: 1. No wing flexure 2. Constant bias (uplift of the wings) 3. Constant periodic oscillation 4. Exponentially decaying periodic oscillation

HPF

LPF Low Frequency Term

3. SIMULATIONS AND ANALYSIS

HPF

LPF

In each of the simulated scenarios the following techniques are assessed: Standard Combined Least Squares solution without any extra parameters ('do nothing') Combined Least Squares with a constrained wing flexure parameter FIR filter Wavelets and/or Wavelet Packets

Figure 2 Three-stage wavelet decomposition tree (adapted from Satirapod et al., 2001). In the discrete wavelet transform a wavelet is translated and dilated only by discrete values, usually by a power of 2. The continuous wavelet transform is very good for the analysis of a signal, but is too computationally intensive for the manipulation of a signal. Therefore for the purposes of de-noising the discrete wavelet transform is used in this investigation.

In all of the simulated scenarios a common heading of 23° and a common pitch and roll of 0° was assumed. In all of the simulations a range of receiver noise (assumed to be white) from 0.005m to 0.02m were considered, while the magnitude of the wing deflection was constant at 0.005m. Then the noise level was held fixed at 0.005m while the wing deflection was varied from 0.005m to 0.02m; the purpose of which was to simulate a good range of signalto-noise conditions (SNR ranging between 0.25 to 4).

Wavelets can be very effective when used for the denoising of a signal. In general there are three principle steps to recover a signal from noisy data using wavelets: 1. Decompose the signal with a specified wavelet, at a chosen level. 2. Use a threshold on the detail coefficients at each level to decide which coefficients should be passed through. 3. Reconstruct the signal using the wavelet coefficients that have been passed through at each level.

3.1 No Wing Flexure Simulation No wing flexure was the condition simulated in this scenario, with receiver noise levels varying from 0.005m to 0.02m. The techniques tested here are the Least Squares with, and without, the estimation of a wing flexure parameter. (The results from Wavelets and the FIR technique are not included here as they produced the same results as the Least Squares solution without wing flexure parameter.) The main aim of this first simulation was to

An advantage of using the wavelet transform to de-noise a signal is that it is relatively easy to reduce the amount of white noise. The wavelet transform effectively reduces the signal into relatively few coefficients, based upon the frequency and time of occurrence. They will be much

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investigate the effect of modeling for wing flexure when there was none present.

3.2 Constant Bias Simulation In this scenario the wing flexure is considered a constant offset from its rigid body frame, simulating an uplift of the wings from their rest position on the ground. Both the level of receiver noise and the amount of wing flexure were varied between 0.005m and 0.02m. The techniques used here to obtain the wing flexure signal were a standard Least Squares solution with/without the extra wing flexure parameter, and the Haar wavelet for denoising the signal.

Yaw Error (deg)

Simulation1 RMS Yaw 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

No Model Least Squares

1

2

3

4

Noise level (*0.005m)

Simulation2 RMS Wing Flex

Wing Flex Error (m)

Simulation1 RMS Pitch

Pitch Error (deg)

0.5 0.4 0.3

No Model Least Squares

0.2

0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0

Least Squares Wavelets

1

0.1

2

3

4

5

6

7

Signal to Noise (0.25-4)

0 1

2

3

4

Figure 4 RMS values for wing flexure for different signal-to-noise values - simulation 2.

Noise Level (*0.005m)

Roll Error (deg)

Simulation1 RMS Roll 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

From Figure 4 one can see that using the Haar wavelet produces very consistent results in overall performance, in estimating the constant bias in the wing flexure. In the case of the Least Squares techniques, a significant improvement in performance is gained as the signal-tonoise ratio increases. However, this levels off as the SNR increases above 1. It is therefore clear that the wavelet technique produces a significantly better result than the Least Squares technique for low SNR values, while having a smaller improvement for higher SNR values (approx 2mm).

No Model Least Squares

1

2

3

4

Noise Level (*0.005m)

Figure 3 RMS values for yaw, pitch and roll for different receiver noise levels - simulation 1. From Figure 3 one can see the influence of incorporating an extra parameter to the Least Squares solution. From the yaw and pitch plots it can be seen that estimation of the extra parameter decreases the precision. These differences range from 0.03° for low noise, to 0.12° for higher receiver noise levels. The roll plot, on the other hand, shows a significant increase in precision (the differences between the two range from 0.05° to 0.25° with increasing simulated noise), this gives some indication of the influence wing flexure has upon the roll component. The differences between the two techniques is due to the wing flexure constraining the solution to take more noise from the roll component, and translate it into the yaw and pitch components.

3.3 Constant Oscillation Simulation In the third scenario the wing flexure was assumed to have a periodic nature, oscillating at a constant frequency and with constant amplitude. All of the techniques can be applied to this scenario. In this simulation, and the following simulation, a Symlet wavelet is used to de-noise the signal. The wing flexure was an oscillation with different magnitudes, ranging from 0.005m to 0.020m. Different levels of receiver noise, varying from 0.005m to 0.020m, were considered. Esitimation Error of Wing Flexure (m)

Simulation3 RMS Wing Flex

This pattern was repeated in all of the following simulations. The Least Squares solution always produces the lowest rms values for the roll component (by differing margins), but at the cost of an increase in noise in the estimated yaw and pitch components. This is because the roll component is closely coupled with the constraint to the z-axis. The remaining systematic bias which has not been completely removed, or reliably estimated, therefore contaminates the yaw and pitch component residuals.

0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0

Least Squares FIR Wavelets

1

2

3

4

5

6

7

SNR (0.25-4)

Figure 5 RMS values for wing flexure for different SNR values - simulation 3. From Figure 5 one can see that the FIR filter performs better than the Least Squares technique for all of the SNR values. The Least Squares solution does not perform very

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well at low SNR values, but the performance increases substantially as the signal level increases until the SNR value reaches 1. From Figure 5 it is apparent that the wavelet technique outperforms all the other techniques, for all of the SNR values, especially in the low SNR range. 3.4 Exponentially Decaying Oscillation Simulation In the fourth scenario the wing flexure is assumed to start with a relatively high frequency, which then undergoes an exponential decay while maintaining constant amplitude. The FIR filter was unable to be applied successfully for small SNR values, as a distinctive frequency range could not be deduced from any of the power spectral density plots. However as the signal strength increased, the FIR filter was able to be applied, with fairly good results, by using a broader pass and stop band to cope with the variation in frequencies. Wing Flexure Error (in m)

Wing Flexure RMS 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0

Least Squares FIR Wavelets Packets

1

2

3

4

5

6

7

SNR (0.25-4)

Figure 6 RMS values for wing flexure for different SNR values - simulation 4.

Figure 7 De-noised signal (top) and manual thresholding of the wavelet coefficients applied to simulated data (noise =0.005m, wf=0.005m).

From Figure 6 one can clearly see that wavelet, and in particular the wavelet packet, technique performs very well (See Figure 7 for an example of de-noising the signal by thresholding of the wavelet coefficients). The wavelet packet technique was better than the Least Squares technique by 3mm through to 1cm, and better than the standard wavelet technique by 2mm through to 6mm.

4. TEST EXPERIMENTS AND RESULTS 4.1 Mechanical Shaker Experiment Data collected from a previous experiment carried out on the 24th January 2001, on the rooftop of the Geography and Surveying (GAS) building, at The University of New South Wales, was used in for the following investigation. A baseline of approximately 10m was established and observations were collected for approximately 2 hours.

The wavelet packet technique is better than the standard discrete wavelet technique in this simulation as it is a product of a wavelet and an oscillating function. It is therefore well suited to a situation where there are stationary and non-stationary signal characteristics.

The set up involved two Leica CRS1000 GPS receivers, a signal generator and a mechanical shaker. The base receiver was set up on one of the coordinated survey pillars while the second antenna was mounted atop the mechanical shaker (see Figure 8).

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From Table 2 one can see that the wavelet technique did a better job at removing the shaker induced noise from the system compared to the FIR technique. 4.2 Wing Flexure Experiment This experiment was carried out on a test rig set up with similar dimensions to a tandem wing unmanned airborne vehicle. The longitudinal axis was defined by a steel pole, at which at one end the master antenna was mounted. The 'wings' were represented by a flexible PVC pipe. The four antennas were mounted with their antenna phase centers pointed in the same direction. Figure 8 Mechanical Shaker on top of GAS building. Corrections generated at the base station were transmitted to the second 'rover' receiver through a RS232 cable connection, with the resulting RTK positions recorded at a 10Hz sample rate. The signal generator supplied the mechanical shaker with frequencies between 0.2Hz and 5Hz. Table 1. Test Frequencies for the experiment at UNSW. Shaker Freq. (Hz) DC 2.0 3.0 DC 0.2 0.3 0.5 0.7 1.2 1.4 1.7

Period 0 6001 12001 18001 24001 30001 36001 42001 48001 54001 60001

– – – – – – – – – – –

6000 12000 18000 24000 30000 36000 42000 48000 54000 60000 66000

Figure 9 The Wing Flexure Experiment set up. Four Leica 500 GPS receivers, that are capable of recording raw carrier phase data at 10Hz, were used to gather the data for this experiment (see Figure 9). The platform was kept stationary for 10 minutes allowing the PVC pipes to flex in the wind, then for a period of approximately 5 minutes the pipes were pulled down occasionally to simulate a sudden change of wing flexure. This was then followed by a further 5 minutes of data collection with no impulse. The time series of the baseline master-to-rightwing height component, along with its corresponding PSD, is shown in Figure 10.

The data was subdivided into the corresponding frequencies as listed in Table 1. This data was then used to test how well the FIR and wavelet techniques could extract the constant frequency signal from the original signal. The results are summarized in Table 2. Table 2 Results from the mechanical shaker experiment. Shaker Freq. (Hz) 0.2 0.3 0.5 0.7 1.2 1.4 1.7 2 3

Original σ 0.006 0.007 0.007 0.007 0.007 0.007 0.009 0.012 0.010

FIR Technique σ 0.006 0.006 0.006 0.006 0.006 0.005 0.009 0.010 0.009

Wavelet Technique σ 0.004 0.004 0.004 0.004 0.004 0.005 0.006 0.006 0.006

Figure 10 Mean centred time series plot of the height component for the rightwing (top) and associated PSD plot of the time series (bottom). The GPS carrier phase ambiguities were resolved through post-processing of the logged data using a dual-frequency

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epoch-by-epoch solution. The baseline components between the master antenna and the other antennas were obtained. As can be seen from the time series plot in Figure 10, there are some periods when wing flexure is clearly present. The time series for the height component was subdivided into five sections that had obvious wing flexure present and these sections were processed using the FIR filter and the wavelet packets technique (to denoise the wing flexure signal).

wing flexure component from the original signal, is very successful. This technique of de-noising can be used for a wide range of frequencies, and especially for signals with time varying frequencies and amplitudes. Also it was noticed that the wavelet packet technique worked best in situations when the signal was between being periodic and highly non-stationary. Whereas the standard wavelet techniques worked best in situations that dealt with signals that had less stationary characteristics.

Table 3 Standard deviations of the derived signal (with wing flexure removed) from the Wing Flexure Experiment using the FIR and wavelet packets techniques Epochs Original σ FIR σ Wavelet σ 2900-3400 0.028 0.010 0.009 3925-4800 0.050 0.014 0.013 5350-6200 0.017 0.009 0.008 6815-9600 0.073 0.030 0.024 10000-12000 0.023 0.014 0.008 1-12923 0.042 0.023 0.011

ACKNOWLEDGMENTS The author is supported in his PhD studies by a Faculty of Engineering Scholarship provided by The University of New South Wales (UNSW). The guidance and support provided by his supervisors Prof. Chris Rizos and Dr. Jinling Wang during this research are greatly appreciated. Clement Ogaja provided the experimental data from the mechanical shaker. Jun Zhang, along with various members of SNAP (Satellite Navigation and Positioning) Group, at UNSW, provided invaluable support during the experiment.

From Table 3 it is clear that the wavelet packets technique is able to extract the wing flexure signal from the original data, leaving considerably less noise in the derived signal (as indicated by the much lower standard deviations). The wavelet packets technique outperforms the FIR band pass filter.

REFERENCES Cannon M.E., H. Sun, T. Owen & M. Meindl (1994), Assessment of a non-dedicated GPS receiver system for precise airborne attitude determination, 7th Int. Tech. Meeting of the Satellite Division of the U.S. Inst. of Navigation, Salt Lake City, 20-23 September, 645-654.

5. CONCLUDING REMARKS In the simulation without any wing flexure it was found that the combined Least Squares solution with estimated wing flexure parameter was forced to shift the noise associated with the roll component into the estimated yaw and pitch components. In the second simulation, of a constant offset in the wing's height, it was found that the Least Squares solution did not perform very well in a high noise environment, though it did improve as the SNR values increased. The wavelet techniques performed consistently, and were especially useful in high noise environments for de-noising the signal. The simulation of a constant periodic oscillation showed that the FIR technique performed better than the Least Squares method for the complete range of SNR values simulated. The wavelet technique performed the best for all of the SNR values simulated. In the final simulation it was found that the FIR, wavelets, and wavelet packets techniques had similar levels of performance for high signal environments, however the wavelet packets technique was superior for all of the SNR values simulated.

Cohen C.E., B.W. Parkinson & B.D. McNally (1994), Flight tests of attitude determination using GPS compared against an inertial navigation unit, Navigation: Journal of the U.S. Institute of Navigation, 41(1), 83-97. Daubechies I. (1990), The wavelet transform, timefrequency localisation and signal anlaysis, IEEE trans. IT, 36(5). Han S., (1998), Digital Signal Processing Techniques and Their Applications in Geomatic Engineering, Course notes, UNSW, pp 45. Jensen A. (2001) Ripples in Mathematics, The Discrete Wavelet Transform, Springer Verlag, Germany, pg. 127169. Misiti M., Y. Misiti, G. Oppenheim & J. Poggi (2000), Wavelet Toolbox User's Guide, Version 2, The Maths Work Inc.

In the analysis of the mechanical shaker experiment, the wavelet technique was able to outperform the FIR band pass filter for all of the frequencies tested. This was also the case for the wing flexure.

Satirapod C., C. Ogaja, J. Wang & C. Rizos (2001), An approach to GPS analysis incorporating Wavelet Decomposition, Artificial Satellites, 36(2), 27-35.

From these experiments and simulations one can conclude that the use of wavelets to de-noise the wing flexure signal, and then to use the derived signal to remove the

Reza A., (1999), From Fourier Transform to Wavelet Transform, White Paper, Spire Lab, UWM, http://www.xilinx.com

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Tsujji T. (1998), Precise determination of aircraft position and attitude using GPS carrier phase measurement, PhD Dissertation for Applied Mathematics & Physics, Kyoto University.

Wickerhauser, M. (1994), Adapted Wavelet Analysis from Theory to Software, AK Peters Ltd., Wellesley, Mass., 486pp.

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