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Abstract -Navigation and guidance of an autonomous vehicle requires determination of the position and velocity of the vehicle. Therefore, fusing Inertial Navigation System (INS) and Global Positioning System (GPS) is important. Various methods have been applied to smooth and predict the INS and GPS errors. Recently wavelet de-noising methodologies have been applied to improve the accuracy and reliability of GPS/INS system. In this work analysis of real data to identify the optimal wavelet filter for each GPS and INS component for high quality error estimation is presented. A comprehensive comparison of various wavelet thresholding selections with different level of decomposition is conducted to study the effect on INS/GPS error estimation while maintaining the original features of the signal. Results shows that while some wavelet filters and thresholding selection algorithms perform better than others on each of the GPS and INS components, no specific parameter selection perform uniformly better than others.
Index Terms - Vehicular navigation; inertial navigation; GPS; Wavelet multi-resolution Analysis; level of decomposition (LOD); Thresholding.
Integration of GPS and INS system provides consistent navigation solutions by overcoming each of their respective shortcomings, including signal jamming for GPS and increase of position errors as a function of time for INS. GPS relies on the technique of comparing signals from orbiting satellites to calculate position, velocity (and possibly attitude) at regular time intervals, but being dependent on the satellites signals makes GPS less reliable than self contained INS due to the possibility of drop-outs or jamming,1. Strapdown Inertial Navigation System (SDINS) technologies are based on the principle of integrating specific forces and rates measured by accelerometers and rate gyros of an Inertial Measurement Unit (IMU) fixed to the navigating body. Given the initial conditions of position, velocity, and attitude, accurate real time integration of IMU output will produce position and attitude information in some given navigation coordinate system, 2.
Typically, the dynamic error model for a terrestrial INS algorithm requires the position, velocity, and attitude errors in an INS (i.e. the system error states). These errors are also augmented by some sensor error states such as accelerometers biases and gyroscopes drifts, which can be modeled as stochastic processes. Actually, there are numerous random errors related with each inertial sensor. Therefore, it is usually difficult to set a firm stochastic model for each inertial sensor that works proficiently in all environments and reflects the long-term actions of sensor errors, 2, 3.
Most of the current navigation systems rely on Kalman filtering to combine data from inertial navigation system (INS) and global positioning system (GPS). However, Kalman filtering have some drawbacks associated to the stochastic error models of inertial sensors, immunity to noise, sensor dependency, linearization dependency and observability,4. GPS errors can be denoised by wavelet,5, while only short-term errors of the INS can be eliminated by wavelet de-noising (optimal low pass filtering). Long-term error of INS can be predicted by GPS/INS integration,1, 6. The impact of the INS long term error can be removed by comparing the denoised INS signal with the GPS data as will be discussed later in this paper.
In this paper, the effect of different wavelet filter on the INS/GPS error will be studied by considering the effect on predictibility of the INS/GPS error. The effect of various thresholding selection methods on the INS/GPS error will be discussed also. The optimal wavelet decomposition level for each INS and GPS data components will be determined without sacrificing original features of the signal. To the best of the author's knowledge, no other work in the literature had investigated the effect of wavelet filter and threshold selection techniques for GPS/INS integration application.
To test the performance of the proposed WMRA algorithm a trajectory from GPS system is used as a reference since GPS signals were available all the time. The de-noising algorithm will be verified using the collected real data from the field test. The whole methodology including the wavelet decomposition, multi-resolution analysis algorithm and wavelet reconstruction to estimate the INS error will be implemented and compared to the reference data as will be shown in the results.
This paper is organized as follows: in section 2 we describe the basic information of GPS and INS systems and the type of errors of these navigation systems and the motivation towards integrating GPS and INS systems. Section 3 presents the multi-resolution analysis algorithm (MRA) to estimate the INS error. The effect of different wavelet filters type affect the estimated INS error and the best selection rule to select the thresholding value are discussed in section 4 and 5 respectively; finally, conclusions are given in section 6.
2.1. GPS System Overview
Global Positioning System (GPS) is a satellite-based navigation system that allows a user with the appropriate apparatus access to valuable and precise positioning information anywhere on the globe. Position and time determination is accomplished by the reception of GPS signals to obtain ranging information as well as messages transmitted by the satellites. GPS receivers work passively, therefore allowing an unlimited number of concurrent users. Additional features of GPS also provide accurate service to unauthorized users; avoid spoofing and reduce-receiver susceptibility to jamming,7. However, the measurement of the satellite-to-receiver range in the GPS system is inaccurate due to several forms of errors, including receiver clock bias, satellite clock bias, atmospheric delay, ephermeris errors, multipath and receiver noise,2, 8. Most of the GPS error can be removed using low-pass filtering technique including kalman and wavelet filtering,4,9,10.
2.2. Strapdown Terrestrial INS System
Strapdown system algorithms are the mathematical definition of processes which convert the measured outputs of inertial sensors that are fixed to a vehicle body axis into quantities which can be used to control the vehicle. There are two SDINS algorithms (Celestial and Terrestrial).
The celestial strapdown inertial navigation system is mechanized in inertial frame. This frame is widely used for spacecraft applications in which geographical information is not required. But, for terrestrial navigation, the inherent time-varying relationship between the inertial and geographic frames complicates the space-stable system design,2.
Thus, the inertial frame implementation results in the most straight-forward navigation-state differential equations, which is not commonly used.
The reasons for this lack of use are the difficulty in calculating gravitational forces, and the terrestrial navigation has the same coordinate used by the GPS system. Further detail of the terrestrial algorithm can be obtained from,11, 12.
Knowledge of the error sources enables the system to cancel their effects as it navigates. In a strapdwon 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. There are several sources of these errors in SDINS are discussed in references,2, 6.
2.3. GPS/INS System Integration
Strapdown Inertial Navigation System (SDINS) and Global Positioning System (GPS) can be incorporated together to provide a consistent navigation solution. GPS provides position information and possibly velocity when there is direct line of sight to at least four or more satellites; and SDINS utilizes the local measurements of angular velocity and linear acceleration to determine both the vehicle's position and velocity. Therefore, integration leads to consistent navigation solution by minimizing their disadvantages and maximizing their advantages. Integration of SDINS and GPS are expected to become more extensive as a result of low cost MEMS inertial sensors.
Several researchers have introduced various methods to integrate GPS and INS data for improving accuracy of navigation system. In terms of integration schemes there are several integration strategies applied to GPS/INS integration that is characterized by the type of information that is shared between the individual systems such as uncoupled mode, loosely coupled mode, tightly coupled mode, and ultra-tightly coupled mode,13. However, choosing an appropriate estimation method is a key problem in the field of GPS/INS integration systems in developing an aided INS. Three approaches that have been applied include linearized or extended kalman filter,4, 14, sampling-based filters such as the unscented kalman filter and particle filters,15 and artificial intelligence based methods incorporated with wavelet filtering,3, 16.
Most integrated GPS and INS systems have been implemented using the kalman filtering technique. These system suffers from the need for predefined INS error model and influenced by noise effects. Wavelet multiresolution analysis could overcome these limitations and provide detail observability unlike kalman filtering,10, 17. The INS and GPS data are initially processed using wavelet analysis before being feed into other compensation techniques such as neural network,9. Therefore it is important to clearly identify the most effective wavelet filter for the GPS/INS system. Although a handful of studies have been conducted in this area,5, 10, none of them has provided detail analysis on the performance of de-noising with various wavelet filters and threshold selection technique. Our results demonstrated that substantial performance improvement in de-noising of GPS/INS by careful selection of the wavelet filter and threshold using real data collected from BEI MotionPak II inertial sensing system and Novatel OEM4 GPS module.
3. Multi-Resolution Analysis Algorithm (MRA)
In order to estimate the GPS/INS error that can be used to model the INS position and velocity error a wavelet multi-level decomposition must be performed for each component of the INS and GPS output signals as described in the flowchart in Figure (1).
The following steps describe the mathematical wavelet decomposition procedure:
Step1: For each one of INS and GPS outputs signals, calculate the approximation coefficient at Sth resolution level using:
Where †(n) is the wavelet function (the basis function utilized in the wavelet transform) and are scaled and shifted versions of †(n) based on the values of s (scaling coefficient) and k (shifting coefficient), Cs, k are the corresponding wavelet coefficients and x (n) is the original signal,18,19.
This operation is equivalent to low pass filtering.
Step2: Obtain the approximation from the approximation coefficient obtained from step1 above using,18:
Step3: Calculate the details coefficient at resolution level using:
This operation is equivalent to high pass filtering.
Step4: Obtain the detail from the details coefficient obtained from step3 above using:
Step5: Return to step one and continue the wavelet decomposition process until appropriate level of decomposition (LOD) is reached which is different from one IMU to another (the appropriate LOD selection will be described later in this paper). It must be noted that the next wavelet decomposition process must be performed on the approximation obtained from the previous wavelet decomposition process and so on.
Step6: De-noising the details of the INS and GPS signals by applying the thresholding technique which is described later.
Step7: Evaluate the GPS and INS position and velocity outputs at several resolution levels (by subtracting the wavelet coefficients of each of the GPS outputs from the corresponding wavelet coefficients of each of the INS outputs) as follows:
GPS and INS approximations respectively.
GPS and INS details respectively.
Difference between the GPS and INS approximations for different levels.
Difference between the GPS and INS details for different levels.
Step8: Reconstruct the INS/GPS error (INS error) signal from the wavelet coefficients differences obtained in step7 as follows:
Difference between the GPS and INS details of the first LOD.
Addition of the difference between the GPS and INS details of the second LOD with the difference of previous LOD (s1st).
Addition of difference between the GPS and INS approximation at sth level and the difference between the GPS and INS details of the previous LOD ().
The output of MRA is generally the INS errors which is used to amend the raw INS outputs during GPS outages. Different criteria need to be considered such as number of wavelet decomposition (LOD), the type of the filters to be used, and the thresholding technique used to denoise the signal details.
4. Selection of Wavelet Filter
The wavelet transform has a flexible feature of using a variety of filters that differ only by their coefficients. The filters are selected based on the lowest standard deviation of INS/GPS error value. Note that to choose the best filter for each component of position and velocity; calculations are performed only for the first level of decomposition. Table 1 summarizes the best filters used for each component of position and velocity using GPS and INS data.
5. Analysis of Wavelet Thresholding Algorithm
Kinematic inertial data (real data) contains both effects of the actual vehicle motion dynamics and the sensor noise as well as some other undesirable effects (e.g. vehicle engine vibrations). Therefore, the criterion for the selection of the appropriate wavelet level of decomposition (LOD) will be different from the static data case,1. Before applying the wavelet multi-resolution analysis on kinematic SDINS and GPS data, particular care should be taken to ensure that the decomposition or de-noising process does not remove any actual motion information,6, 20.
To select an appropriate LOD, the GPS and INS outputs data are decomposed into several levels and the standard deviation (STD) for each approximation difference of the INS/GPS error compared with the real INS error is computed. Figure (2) and figure (3) illustrate the INS/GPS error and the real INS error without thresholding. The comparison result reveals that wavelet de-noising method could not achieve satisfactory results for GPS/INS integration system.
In order to improve the signal-to-noise ratio (SNR), thresholding operations are applied on the coefficients of the wavelet and wavelet packet transforms. Thresholding operation can be classified into hard-thresholding and soft-thresholding as described by Veterli et. al,21 and Ref. 22. In this paper soft thresholding was used to remove some of the noise while keeping the original features of the signal by improving the SNR.
The choice of threshold is crucial maintain the quality of the de-noising process and should be made carefully. In thresholding process coefficients smaller than the threshold value (Thrv) are considered negligible. In this paper six methods are used to select the value of Thrv. The first method estimate the standard deviation ³x , of the noise at each scale by dividing the noise power for the noisy signal over the standard deviation as follows:
The second approach relies on the standard deviation as shown in the following relationship
³2 = Represents Noise Power for noisy signal.
³x = Standard deviation for the detail coefficients.
N = the sequence length.
Details of both wavelet thresholding can be obtained from Zhong et. al,23. The rest of the wavelet thresholding relies on MATLAB based routine including Stein's unbiased risk estimate (SURE) from "rigrsure", fixed form threshold from "sqtwolog", mixture of the last previous two selection rules from "heursure", and selection rule using minimax principle from "minimaxi",24,25.
The Root Mean Square Error (RMSE) and Signal to Noise Ratio (SNR) can be evaluated using equation (9) and (10), respectively,6, 22:
and X (n) = Denoised and the clean signal, respectively.
Figures (4) shows the SNR and RMSE after applying thresholding using the six methods mentioned previously for the GPS and INS position and velocity outputs. The lowest value for RMSE would have the highest values of SNR and the corresponding method is optimized to select the threshold value.
Analysis shows that the fixed form thresholding (sqtwolog) is the best selection technique for all of the position components. There appeared to be two suitable threshold selection technique for velocity components which are the Stein's unbiased risk estimator (rigrsure) and selection rule using minimax principle from "minimax". An optimum selection rule is important to choose the threshold value for the wavelet analysis as it has a significant effect on velocity components de-noising performance as shown in figure (5) and figure (6). Table 2 shows the results using different LOD on GPS and INS data. From observation, it is unnecessary to increase the order of LOD because the features of the INS/GPS-error will disappear. The chosen wavelet filter and thresholding technique contributed positively to remove the undesired noise from the GPS and INS data.
The kinematic data used in this study were collected in University of Calgary (UoC). In this test, a Novatel OEM4 GPS receiver and a MEMS-grade INS (BEI MotionPak II) solid state inertial sensing system were used. The minimum number of available satellite was 7 and the average Van speed was 50 km/h. All the development made in this paper was implemented using MATLAB computer-aided design software (MathWorks, Natick, MA) including the wavelet analysis.
Wavelet analysis is able to filter the noise and disturbances that may possibly exist at the GPS and INS outputs. WMRA algorithm provides the advantage of comparing the GPS and INS position and velocity outputs for different levels of resolution. Evaluation of different types of wavelet filters identified the best filters to be used for each of the GPS/INS system. Various selection rules have been examined to select the threshold value for the wavelet analysis. The thresholding technique selected combined with suitable wavelet decomposition level can improve the de-noising process and maintain the original features of the signal.