INTRODUCTION
The Global Positioning System (GPS) has been at the head of current navigation
technologies. However, the surveyors are still facing problems in places where
the GPS signal gets lost due to different factors such as blockage by building,
canopy and other natural and nonnatural obstructions. Many of the studies are
being carried out to address the issue of signal loss. Where some high sensitivity
receivers can detect the reflected signals as well as direct signals but the
accuracy degrades significantly if multipath is stronger than the direct signals
(Syed et al., 2006). However, Inertial Navigation
System (INS) is one of the most popular mechanical navigation systems that can
provide a navigation solution in case of GPS signal loss. It is worth to mention
that the output given by the inertial sensors is in terms of accelerations and
rotational velocity that need to be processed to get position and velocity information.
Inertial Navigation Systems (INSs) has been around since the mid of twentieth
century and now gaining popularity due to technological advancements in micromachined
sensors that reduce the size of IMU as well as the associated cost. The good
thing about INS is its independent and jam proof navigation data, as compared
to GPS that is dependent on satellite signal; however, INS accuracy degrades
with respect to time making it a major drawback. Most of the errors in the INS
are caused by sensor imperfections (instrumental errors); therefore, accuracy
mostly depends on the type of sensors available. However, the cost of the INS
is directly proportional to the accuracy, implying that high performance accurate
sensors are still very expensive and limited to certain applications (Shaikh
et al., 2003; Miskam et al., 2009).
An IMU is a black box housing inertial sensors that are mounted on three orthogonal
axes. The combination of three accelerometers and three gyroscopes provide linear
accelerations and angular velocity, respectively along three orthogonal axes.
Both accelerometer and gyroscope operate on the inertial principles (Newton's
Laws of Motion) that could be used to provide navigation solution (Mostafa,
2001). The measurements from the IMU are mathematically integrated to obtain
position information and orientation (rotation about an axis). By tracking both
the current linear accelerations and angular velocity it is possible to determine
the position of the body in fixed coordinate system (Mostafa,
2001; Schmidt and Barbour, 2001). The advantage
of INS is that it could operate independently and provide reliable navigation
data, as compared to GPS that is dependent on satellite signal. However, INS
accuracy degrades with respect to time due to integration drift. Sensor imperfections
coupled with instrumental errors also aggravate accuracy of the INS. The cost
of high performance accurate sensors is still high and beyond the reach for
certain civilian applications. Also, it is worth to mention that Tactical and
navigation grade sensors are limited to commercial and military applications
(Shaikh et al., 2003; Skaloud
and Schwarz, 1999).
The first INS was built and based on mechanical gyros with very complex and
power consuming architecture. Later on strapdown solutions have been realized
by using modern integrated electromechanical or electrooptical sensors (Shaikh
et al., 2003). These strapdown systems are mostly based on the MEMS
(Micro ElectroMechanical System) technology that is relatively inexpensive
and compact. These costeffective sensors, due to their shortterm sustainability
and opposite characteristics, are widely used in inertial navigation systems.
This study focuses on developing and implementing the strapdown INS algorithm
by effectively bandlimit the INS signal prior to its mechanization and further
processing. The motivation behind this concept is schematically depicted in
Fig. 1. Where Fig. 1a shows the main two
types of errors in the inertial measurement unit are long and short term errors
(Skaloud and Schwarz, 1999). Where multilevel decomposition
is utilized to improve the performance of the inertial sensors (gyroscopes and
accelerometers) by removing their short term errors as shown in Fig.
1b. The separation of the high and low frequency inertial sensor noise components
can be done by deionizing the inertial measurements before using them as input
to the SDINS algorithm as done in this study. It must be mentioned that when
using multilevel of decomposition the short term error was extensively reduced
by wavelet deionizing (optimal low pass filtering) while the long term error
still affect the INS performance for long time processing. In this study we
can remove the effect of the short term error of the stand alone INS only, while
the long term error can be removed by aiding the INS with another navigation
devices such as GPS to eliminate or reduce the longterm error as shown in Fig.
1c.
BACKGROUND ON INERTIAL NAVIGATION SYSTEMS
Types of inertial navigation systems: Inertial navigation can be classified into three basic categories:
Geometric: In this type, the navigation information was available in
analog fashion directly from the gimbals angles. It is necessary to physically
instrument two reference frames to provide this information and these two frames
are an inertially nonrotating frame and a local navigation frame. In this kind
of navigation system minimal computation capacity is required. At least five
gimbals are necessary to provide the navigational quantities of interest, called
latitude, longitude and vehicle roll, pitch and yaw (Lin,
1991).
Semianalytic: Semianalytic systems physically instrument only one
reference frame, either an inertially nonrotating frame or a local navigational
frame.

Fig. 1: 
A schematic plot of inertial noise in frequency domain: (a)
before filtering, (b) after optimal low pass filtering only and (c) lowpass
filtering with estimated INS/GPS error algorithm 
Three gimbals are required to affect this coordinate navigational frame, but
it is highly recommended to install four gimbals because there is a case of
singularity in the first implementation for the computation of latitude and
longitude being accomplished in a computer (Britting, 1971).
Analytic: Which is also called a strapdown inertial navigation system,
which does not physically instrument a reference frame but rather use the gyroscopes
outputs to calculate analytically the relative orientation between the system’s
initial and present state (Lin, 1991).
It is important to make a distinction between the primary types of inertial navigation systems: Gimbaled and Strapdown.
In inertialplatform gimbal mechanization, the gyroscopes mounted on a stable
element measure angular rates and gimbaldrive systems use the angular rate
information to null the angular motion sensed by the gyroscopes. In this manner,
the gyroscopes and accelerometers on the stable element are inertially stabilized
from the vehicle motion and the stable member physically represents an inertial
reference frame. By double integrating the specific force taken from the accelerometers
with a correction for gravity, position determination is possible (Bielas,
1994).
In strapdown inertial systems, sensors are mounted directly (or perhaps with
vibration isolators) on the vehicle. Inertial sensors outputs now represent
specific force and angular rate with respect to inertial space coordinatized
in vehicle body axes. Therefore, to maintain an inertial reference frame, a
computergenerated transformation matrix algorithm between body and inertial
frame must be used to process the rate gyroscope outputs as the vehicle moves
and its orientation changes. Then the accelerometer information must be transformed
from the body frame to the inertial reference frame (Lin,
1991).
Errors in INS: Most of the error sources that distort the navigation
solution are sensor errors or random disturbances. These are the residual errors
exhibited by the installed gyros and accelerometers following calibration of
the INS. The dominant error sources that affect the accuracy of the navigation
solution obtained from INS such as alignment, scale factor, biasing, nonorthogonality
and random noise as illustrated in Table 1 (GrejnerBrzezinska
and Wang, 1998).
There are nine navigation errors caused by the accelerometers and gyroscopes.
These errors in the accelerations and angular rates lead to steadily growing
errors in position, velocity and attitude information.
These navigation errors caused by the mathematical integration operation in
the INS algorithm. The GPS navigation system can be used to aid the INS and
prevent these time drift errors (Chiang et al., 2008).
In addition to these navigation errors the INS algorithm also suffers from acceleration
and angular rates sensor reading inaccuracies caused by the earth gravity and
rotation. These errors must be handled carefully especially in strapdown system
rather than gimbaled inertial sensors (Britting, 1971).
Knowledge of the error sources enables the system to cancel their effects as
it navigates. In a strapdown system, however, only few of the sensor errors
can be calibrated. Errors that cannot be calibrated will propagate into navigation
errors when the system begins to navigate. These systems also require lengthy
alignment time. If both of these necessities are not met, even the most accurate
INS can become worthless (Noureldin et al., 2004;
ElSheimy et al., 2004). Figure
2 shows the effect of onedegree INS sensor errors on the position of the
moving body noise, bias, scale factor, combined and initial condition accumulated
error.
A row IMU data collected from a low cost inertial sensor (MotionPakII) was
used in this paper for analysis and denoising the sensors outputs in order
to improve the accuracy of the position and velocity components. Table
2 shows the specification of MotionPak II used in this study to provide
real data for further manipulation and denoising for both the accelerometer
and gyroscope in terms of sensitivity range, bias, alignment and resolution
obtained.

Fig. 2: 
Effect of onedegree INS sensor errors on the position of
the moving body 
WAVELET MULTIRESOLUTION ANALYSIS
The concept of multiresolution approximation of functions provides a powerful
framework to understand wavelet decompositions. The basic idea is that of successive
approximation, together with that of added details as one goes from one approximation
to the next finer one (Donoho and Johnstone, 1995).
The main advantage of wavelet analysis is that it allows the use of longtime
wavelet intervals where more precise lowfrequency information is needed and
shorter intervals where highfrequency information is sought (Burrus
et al., 1998). 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 selfsimilarity
(Burrus et al., 1998). Wavelets are also capable
of compressing or denoising a signal without appreciable degradation of the
original signal. In general, the wavelet transformation of a timedomain signal
is defined in terms of the projections of this signal into a family of functions
that are all normalized dilations and translations of a wavelet function (Jaideva
and Chan, 1999).
Discrete Wavelet Transform (DWT): Since dealing with discretetime inertial
sensor signals, the (DWT) is implemented instead of the Continuous Wavelet Transform
(CWT). The DWT of a discrete time sequence x(n) is given as (Jaideva
and Chan, 1999; Ahmed et al., 2008; Burrus
et al., 1998):
where, Φ_{s,k} is the scale function and ψ_{s,k} is the wavelet function and 2^{(s/2)} Φ_{s,k} (2^{s} nk), 2^{(s/2)}ψ_{s,k} (2^{s} nk) are the scaled and shifted versions of Φ_{s,k} and ψ_{s,k}, respectively, based on the values of s (scaling coefficient) and k (shifting coefficient). The s and k coefficients acquire integer values for different scaling and shifted versions of Φ_{s,k} (n), ψ_{s,k} (n) and C_{s,k}, d_{s,k}, respectively.
The original signal x(n) can be generated from the matching wavelet function using the following equation:
The wavelet function ψ_{s,k} is not limited to exponential
functions as in the case of Fourier Transform (FT) or Short Time Fourier Transform
(STFT). The only restriction on ψ_{s,k} is that it must be short
and oscillatory (it must have zero average and decay quickly at both ends).
This restriction ensures that the summation in the DWT transform equation is
finite (Jaideva and Chan, 1999; Alnuaimy
et al., 2009; Putra et al., 2010).
Since the low frequency fraction of the inertial measurement reading contain
the majority of the inertial sensor dynamics during the static alignment phase,
these inertial measurement readings can be denoised using wavelet multilevel
of decomposition to separate the low and high frequencies (Skaloud
and Schwarz, 1999). Wavelet multilevel of decomposition separates each
of the IMU reading (three for both the accelerometer and gyroscopes) into two
parts. The first part is called approximation; this part is the output of lowpass
filter of wavelet multilevel of decomposition, which includes the longterm
noises, in addition to the earth gravity and rotation rate frequency components.
Both of these two components exist together within very small frequency band
at low frequency. The wavelet multilevel of decomposition are unable to separate
earth gravity and rotation rate from the IMU readings and thus it will propagate
into the INS algorithm computation. The second part which is called the details
obtained from the high pass filter of wavelet multilevel of decomposition includes
the undesirable high frequency noise components of the SDINS and a lot of noise
disturbances such as vehicle vibration.
Equations 13 are referred to as the analysis and synthesis equations. The
wavelet transform offers advantages over its Fourier domain counterpart, where
the basis function offers only a fixed frequency resolution and no localization
in time (Burrus et al., 1998; Chik
et al., 2009).
Theoretically, wavelet decomposition process can be continued for ever. Basically,
the decomposition process can continue until the individual coefficients consist
of a single frequency. On the other hand, an appropriate level of decomposition
is elected based on the nature of the signal on a suitable criterion (Skaloud
and Schwarz, 1999). In this study the data rate of the inertial sensors
of MotionPak II is 32 Hz. Consequently, five levels of decomposition will limit
the frequency band to 0.5 Hz. We conclude that five Level of Decomposition (LOD)
are adequate to reduce the high frequency noise from the real inertial sensor
measurement.
The proposed IMU denoising procedure consists of (1) performing a wavelet analysis, using the analysis equations, (2) applying a thresholding of the wavelet coefficients and (3) recovering the denoised signal using the synthesis equation. It is obvious that the choice of threshold in the second step above is crucial to the quality of the denoising process and should be made carefully in addition to the selection of the type of wavelet function and its order.
Selection of the appropriate filter: The wavelet transform has a flexible feature of using a variety of filters that differ by their coefficients. After using all types of the wavelet filters such as (Daubechies, Coiflet, Biorsplines, Symlets). The deionizing result shows that Db3 wavelet filter is the best filter type used to remove the high frequency noise from the accelerometer and gyroscopes of the IMU which reduce the mean square error.
Performance analysis of different thresholding algorithm: Thresholding operations are applied on the coefficients of the wavelet and wavelet packet transforms and generally can be classified into Hardthresholding and Softthresholding as described by Burrus et al., 1998.
The choice of threshold is crucial to the quality of the deionizing process
and should be made carefully. In thresholding process coefficients smaller than
threshold value (Thrv) are judged negligible, or noise other than signal
(Rizzi et al., 2009).
In this study six methods are used to select the value of Thrv. the
first method is based on estimating the standard deviation σ_{x},
of the noise at each scale by dividing the noise power for the noisy signal
over the standard deviation for the details coefficients as in Thrv =
σ^{2}/σ_{x} (Ma et al.,
2002; Li and Zhao, 2009), another approach is used this relationship (Veterli
et al., 2000; Li and Zhao, 2009).
Where:
σ^{2} 
= 
Represents Noise Power for noisy signal 
σ_{x} 
= 
Standard deviation for the detail coefficients 
N 
= 
Sequence length 
Third method is stein's unbiased risk estimate (SURE) with MatLab code rigrsure,
selection using fixed form threshold with MatLab code sqtwolog, selection using
mixture of the last previous two selection rules with MatLab code heursure and
the last selection rule use minimax principle with MatLab code minimaxi (Misite
et al., 2002).
Hard and soft threshold functions are widely used in practice, resulting in good effect. Hard thresholding function can preserve the accelerometer and gyroscopes output signals and characteristic but results in unsmooth accelerometer and gyroscope denoised signal. However, soft threshold function can achieve smooth accelerometer and gyroscopes signal.
In this study soft thresholding was used to remove some of the noise of the details part of the signals with keeping the original features of the signal and improve the Signal to Noise Ratio (SNR). Where, Fig. 3 shows the Root Mean Square Error (RMSE) after applying soft thresholding using the six methods mentioned previously for the IMU accelerometers and gyroscopes, the lowest value for RMSE would have the highest value of SNR and the corresponding method is optimized to select the threshold value. Analysis shows that Steins Unbiased Risk Estimation (SURE) method is the best selection technique for the IMU output. An optimum selection rule is important to choose the threshold value for the wavelet analysis as it has a significant effect on position and velocity components and enhance the denoising algorithm performance.

Fig. 3: 
Performance comparison after using six threshold selection
rules 
PROPOSED IMU DEIONIZING FOR SDINS
A strapdown INS (SDINS) algorithm has been implemented using Matlab for the wavelet multiresolution algorithm to denoise the IMU outputs and provide reliable navigation information. Wavelet deionizing analysis was conducted for kinematic inertial data over 2500 sec. Comparison has been made with relatively accurate GPS information as shown in Fig. 4 to compare the appropriate wavelet Level of Decomposition (LOD) required for removing the high frequency noise and disturbances from the IMU device.
Terrestrial strapdown system dynamic equation: The differential equation
of the relative quaternion between body coordinate and geographic coordinate
(Britting, 1971):
where, the angular velocity skewsymmetric matrix Ω_{ib}^{b} and Ω_{in}^{b} are given by:
and
where, [L, l, h]: are geodetic positions (latitude, longitude and height). w_{R}, w_{P}, w_{Y}: are the body angular velocities in the body coordinate (roll, pitch and yaw), respectively.
Body fixed coordinate to navigation coordinate (C_{b}^{n})
can be described in terms of the quaternion parameters:
The differential equations of the vehicle position in terms of latitude, longitude and heading can be arranged in matrix form:
where, [V_{N} V_{E} V_{D}] = V^{n}: Geodetic velocity vector (North, East and down). R_{N} and R_{E}: are the radii of curvature in the north and east direction and given by:
and e: eccentricity (= 0.0818)

Fig. 4: 
Schematic diagram of development strapdown inertial navigation
system 
The differential equations relating the second derivative of the geodetic position
and velocities can be derived as:
where, f ^{b} is specific force outputs in the body coordinate = [f_{x} f_{y} f_{z}]^{T}. g_{e} is gravity force applied on down direction.
Gravity force (g_{e}) can be found from initial gravity g_{o}:
and
Equation 4, 9 and 12
represent the mechanization equation for the terrestrial navigation system.
The raw data obtained from the inertial sensor contains substantial noise that needs to be filtered.
Vibration in the INS data can cause a lot of problems if not well taken care of. Vibration of the vehicle contributes to the noise in the data making it inaccurate; therefore, proper filtering techniques should be devised to get accurate and worth while results. As mentioned before, wavelet deionizing technique was used to filter the noise of INS data, which is widely used in filtering technique in the field of signal processing. It can be seen that waveletdeionizing result is fairly smooth. It is found that five level of decomposition is adequate to reduce the short term error of the INS position and velocity components.
The anticipated denoising procedure was applied to a real data collected from the MultiAxis Inertial Sensing System (MotionPak II) MEMSgrade IMU. The MotionPakII consists of three orthogonally mounted micromachined quartz angular rate sensors and three silicon based accelerometers. The specifications of the MotionPak II IMU are given in Table 2.
The outputs of the inertial measurement unit were denoised by applying five LOD to bound the output high frequency noise. As the MotionPak II measurements are supplied at a data rate of 32 Hz, the five decomposition levels bound the frequency band of the original signal from 16 to 0.5 Hz.
Table 3: 
The standard deviation and mean square error of the inertial
sensors (before and after wavelet Denoising) 

It must be mentioned that increasing the number of decomposition level could
possibly lead to remove some of the useful frequency components such vehicle
motion dynamic. It is clear that applying wavelet multiresolution analysis
to denoise the inertial sensor outputs has proven its achievement in enhancing
the output of the INS algorithm by reducing the estimated position and velocity
errors as shown in Fig. 5af.
We found that five level is adequate to restrain most of the high frequency
noise (shortterm errors) existing in the inertial sensor measurement to keep
away from removing part of the earth's rotation and gravity components. Figure
6ac and 7ac show
the MotionPak II raw measurements for force and angular velocity measurements
before and after five level of decomposition process, respectively. It is obvious
that most of the high frequency noise components are suppressed after the fifth
level and hence reducing the measurement uncertainty. Table 3
shows the mean values and standard deviation of the IMU output for five level
of decomposition.
Noise was also observed in the INS data during static mode. Since the equipment
is sensitive and logs data with a sampling rate of 32 Hz, even the minor variation
in the environmental affects the data.
A combination of several filtering techniques can remove the INS noise to quite
some extent. Figure 8af show the position
and velocity of the INS algorithm after denoising the accelerometers and gyroscopes
output for five LOD compared with the reference GPS data. Also, from this figure
we can observe that five LOD is adequate to remove the short term error existed
in the accelerometers and gyroscopes reading from the IMU. Figure
5 shows the resultant error in position and velocity after denoising and
indicates that five level of decomposition are suitable to remove the high frequency
error of the IMU measurement. Increasing the level of decomposition results
in undesired features of the navigation solution since the original features
of the IMU data will be lost and from this results we can conclude that appropriate
LOD can be optimized using an optimization technique such as genetic algorithm,
particle swarm optimization and other optimization techniques without using
reference GPS data for comparison to obtain accurate results.
CONCLUSIONS
The intuition of filtering shortterm noise from an IMU defined by the motion of the vehicle has been studied. A denoising algorithm based on wavelet multiresolution analysis has been introduced. In addition the results shoed that the proposed algorithm procedure could be performed and reduce the error for acceptable range of INS operating period and reduce the short term error to provides more accurate position and velocity if compared to the results obtained from nondenoised inertial data before the error will growth gradually.
Most of the current inertial denoising methods suffer from the comparatively
high noise levels of the inertial measurement unit. While, the anticipated technique
is highly beneficial in providing fast and accurate navigation solution for
several applications and improves the short term error of the low cost inertial
measurement device. It was demonstrated that wavelet as a tool can be useful
for analysis of the measurements.

Fig. 5: 
Error in Position and velocity after using wavelet Denoising for 5levels of Decomposition data. (a) Xaxis, (b) Yaxis, (c) Zaxis, (d) North, (e) East and (f) Down 

Fig. 6: 
Force measurements (m sec¯^{2}) before and after five level of decomposition Wavelet Denoising for (a) Xaxis, (b) Yaxis, and (c) Zaxis respectively 

Fig. 7: 
Angular velocity Measurements (deg sec¯^{1}) before and after five level of decomposition Wavelet Denoising for (a) Xaxis, (b) Yaxis and (c) Zaxis, respectively 

Fig. 8: 
Strapdown INS Position and velocity in ECEFFrame after using wavelet denoising with reference GPS data along (a) Xaxis, (b) Yaxis, (c) Zaxis, (d) North, (e) East, and (f) Down, respectively 
It also showed that wavelet based denoising can be used as excellent tool
to remove the noise from the measurements reading of the IMU. Finally, experimental
results obviously indicates the capability of the proposed prefiltering approach
to reduce the standard deviation of the estimated error and increase the SNR
of the accelerometer and gyroscopes measurements as the RMSE reduced and provide
an accurate navigation solution for several navigation application. The proposed
method contribute positively in reducing the high frequency noise of the inertial
sensors where GPS aiding can not provide the predictable reduction in the high
frequency noise of the inertial sensors.
ACKNOWLEDGMENT
This study was supported in part by University Putra Malaysia/Kuala Lumpur, Malaysia. Authors would like to thank the Mobile Multisensor Research Group at the University of Calgary, Calgary, AB, Canada, for providing the experimental data. Also acknowledgment for Dr. Naser ElSheimy and Dr. Sameh Nassar for their helping to provide and use the test data used in this study.