| Asian GPS ---> Proceeding ---> 2002---> Trends in GPS data processing |
Time varying ARMA processing on low cost GPS receiver data to improve the position accuracy
M. R. Mosavi
Department of Computer Engineering
Shahid Rajaee Teachers Training University
Lavizan, Tehran, Iran
mailto:M.Mosavi@srttu.edu
M. H. Refan
Department of Computer Engineering
Shahid Rajaee Teachers Training University
Lavizan, Tehran, Iran
M.Mosavi@srttu.edu
K. Mohammadi
Department of Electrical Engineering
Iran University of Science and Technology
Narmak, Tehran, Iran
Mohammadi@iust.ac.ir
M. R. Mosavi
Department of Computer Engineering
Shahid Rajaee Teachers Training University
Lavizan, Tehran, Iran
mailto:M.Mosavi@srttu.edu
M. H. Refan
Department of Computer Engineering
Shahid Rajaee Teachers Training University
Lavizan, Tehran, Iran
M.Mosavi@srttu.edu
K. Mohammadi
Department of Electrical Engineering
Iran University of Science and Technology
Narmak, Tehran, Iran
Mohammadi@iust.ac.ir
Abstract
This paper presents a method to determine an accurate position by using a low cost GPS receiver and proposes a new time varying ARMA model for better accuracy in GPS positioning. At first the GPS systems errors are defined. Then calculating the position component errors, real and dynamic patterns of the errors will be created and applied in to time varying ARMA model. The parameter matrix is obtained with such real data to predict the later seconds errors. The steps of design and implementation of the time varying ARMA model are presented and the experimental results of the tests are stated with real data. They show the position components errors decrease by using of time varying ARMA processing.
Introduction
Global Positioning System (GPS) has replaced prior positioning systems. It can cover all the earth by satellites to measure accurate time, altitude, longitude and latitude in every desirable point [1,2]. Positioning began from 1950s and improved in 1970s. In 1980s, GPS became an operational positioning system. At first it was designed and used for military purposes. But later its commercial applications have been increased. Nowadays commercial receivers take a great part in its market [1,2].
There is some error sources that cause the position and time measuring from GPS receivers to be inaccurate. Significant error sources are signal delays from ionospheric and tropospheric effects, satellite clock drift, satellite orbital position errors, signal multi path and noise generated within the receiver itself. Table1 shows the common errors of GPS system in meter and nanoseconds.
Because of above mentioned error sources, all GPS receivers have a certain number of errors. This means that received data from GPS receiver will not reflect the real location. Therefore, users who wish to increase the accuracy of their GPS receiver must take steps to minimize the errors. In this paper, a new modeling to decrease the positioning measurement errors in low cost GPS receiver is described. The theoretical background for better accuracy is based on time varying ARMA model.
GPS Receiver
To achieve information of position and implementing an operational system, MicroTracker Low Power (MLP) as a low cost GPS engine manufactured by Rockwell Company was used. This miniature receiver with a small volume is appropriate for a vast range of Original Equipment Manufacturer (OEM) products. OEM receiver provides the possibility of improving software by presenting raw data [5,6].
This receiver has 5 parallel channels. It can track up to 9 satellites simultaneously. This receiver supports approved and improved NMEA-0183 protocol. It can receive differential RTCM messages to improve the accuracy of positioning in differential mode. Its serial port can receive and transmit NMEA or Binary data with the rate of 4800 or 9600 bit per second. The Binary protocol provides more detailed information compare with NMEA protocol [5,6].
Data Collection
To study the function of receiver, the GPS receiver was installed and set up in a fixed position. There are several binary messages provided by MLP. One famous and general purpose of these messages is message No.103, which is available on the first receiver output port as default, when we configure the receiver in binary mode. The message 103 is contain of very useful detail information of position and time [5,6].
In order to setup the receiver, connecting to PC and data collection, a hardware designed and implemented. Fig.1 shows the hardware structure. The output data was collected for a few months. A Pentium III computer with 450 MHZ clock speed saved it.
Fig.1. Hardware structure
Position Components Errors
Since MLP is a low cost nonmilitary receiver, its measurement errors are not neglected (188 meters RMS 3D, when S/A was on and 60 meters RMS 3D, when S/A is off) [5,6]. To study the receiver data and achieving of the errors, the data of position were studied in World Geodetic System-1984 (WGS-84). Therefore the x,y and z magnitude in the No.103 Binary message were collected and saved in separate files every 1 second period.
We focus on variation of x,y and z components in studying ARMA model [7].A software was developed for this purpose. By calculating the average of each quantity in file length, the software provides difference of the instantaneous magnitude of each point with its corresponding quantity average according to equation (1) to (6) and saves them in other files [7].
Where xi,yi,zi are instantaneous magnitude of x,y,z and Ax, Ay, Az are the average magnitude of x,y,z and xi,yi,zi are instantaneous error magnitude of x,y,z respectively. n is number of samples.
Linear Correlation between Position Components
We collected position data for a fix point at 17 days and saved them to 17 files. Based on the equations (7) to (9), linear correlation between x,y and y,z and z,x for 17 data files were calculated. The obtained results of correlation coefficient were recorded in Table 2.
Where,rxy, rxz and ryz are linear correlation between x,y and y,z and z,x , respectively. The results of these statistical studying show that the average correlation coefficients are: rxy=0.58,rxz=0.42 and zi=0.51 Hence, the result shows linear correlation between x,y and y,z and x,z.
Table 2. x,y,z correlation coefficients for 17 days
Model Description
ARMA (Auto-Regressive Moving Average) is a famous model, which is used a discrete-time stochastic process. A time varying ARMA model of order (p,q) which is conventionally noted as ARMA(p,q), is mathematically described by [8]:
Where x(k-j) and y(k-i) are the input and output of the model and e(k) is the noise value at time k for k=1,2,... .ai(k) for i=1,2,...,p and for bj(k)=1,2,...,q are the sets of parameters which describes the model structure. Without loss of generality, it is always assumed that ao(k) 1[9].
Proposed Time Varying ARMA Model
Because of linear correlation between x,y and y,z and also x,z , the ARMA model should be of such kinds that their input variables include patterns of positioning errors conjointly. Structure of this time varying ARMA model is as:
For simplicity, we assume in this research P1=P2=P3=P, q1=q2=q3=q and r1=r2=r3=r.
The Calculation Process
We start from dx component. Equation (11) can restate as:
If we rewrite equation (14) in matrix format, We get:
The parameter matrix q may be estimated using the Least-Squares (LS) method. The LS estimation of the parameter matrix q can be obtained by [9]:
Once the model parameters a1i(n),a1j(n) and a1k(n) are known, the calculation of dx(n) for an arbitrary n can be accomplished by [9]:
Modeling for dy and dz components are similar to equations (14) to (23).
Experimental Results
To study the efficiency of the model, we used 5000 position test data, which were collected on the building of Computer Control and Fuzzy Logic Research Lab in the Iran University of Science and Technology. Statistical significance results from tests are shown in Table3. Prediction errors averages are obtained as following equations:
Where exi,eyi and ezi are errors of prediction dx, dy and dz, respectively. M is the number of test samples. Also, Prediction errors norms are obtained as following equations:
Table3. Average, Variance, Standard Deviation and Norm of positioning error components with (p=q=r=3)
Conclusion
This paper has described how the positioning accuracy of a low cost GPS receiver can be greatly improved with a time varying ARMA model. The Least-Squares method was used to determine the parameter matrix according to the measurement or available information. The result is a highly effective estimation technique for accurate positioning. The validity of the proposed time varying ARMA model was confirmed by experimental results on implemented unit in this paper (Fig.2). So, the average errors in prediction of x, y and z are less than 1 meter.
Fig.2. Implemented unit in this research
References
This paper presents a method to determine an accurate position by using a low cost GPS receiver and proposes a new time varying ARMA model for better accuracy in GPS positioning. At first the GPS systems errors are defined. Then calculating the position component errors, real and dynamic patterns of the errors will be created and applied in to time varying ARMA model. The parameter matrix is obtained with such real data to predict the later seconds errors. The steps of design and implementation of the time varying ARMA model are presented and the experimental results of the tests are stated with real data. They show the position components errors decrease by using of time varying ARMA processing.
Introduction
Global Positioning System (GPS) has replaced prior positioning systems. It can cover all the earth by satellites to measure accurate time, altitude, longitude and latitude in every desirable point [1,2]. Positioning began from 1950s and improved in 1970s. In 1980s, GPS became an operational positioning system. At first it was designed and used for military purposes. But later its commercial applications have been increased. Nowadays commercial receivers take a great part in its market [1,2].
There is some error sources that cause the position and time measuring from GPS receivers to be inaccurate. Significant error sources are signal delays from ionospheric and tropospheric effects, satellite clock drift, satellite orbital position errors, signal multi path and noise generated within the receiver itself. Table1 shows the common errors of GPS system in meter and nanoseconds.
Table1. The common errors of GPS system in meter and nanoseconds [3,4]
| Error Component |
Standard Deviation (m) |
Standard Deviation (ns) |
Time Constant |
| Satellite Parameters of Broadcast | 30 | 100 | More than 1 Hour |
| S/A | 30 | 100 | 2 Minute |
| Ionosphere | 5 | 17 | More than 1 Hour |
| Multi path | 5 | 17 | 0.5 to 10 Minute |
| Troposphere | 2 | 7 | More than 1 Hour |
Because of above mentioned error sources, all GPS receivers have a certain number of errors. This means that received data from GPS receiver will not reflect the real location. Therefore, users who wish to increase the accuracy of their GPS receiver must take steps to minimize the errors. In this paper, a new modeling to decrease the positioning measurement errors in low cost GPS receiver is described. The theoretical background for better accuracy is based on time varying ARMA model.
GPS Receiver
To achieve information of position and implementing an operational system, MicroTracker Low Power (MLP) as a low cost GPS engine manufactured by Rockwell Company was used. This miniature receiver with a small volume is appropriate for a vast range of Original Equipment Manufacturer (OEM) products. OEM receiver provides the possibility of improving software by presenting raw data [5,6].
This receiver has 5 parallel channels. It can track up to 9 satellites simultaneously. This receiver supports approved and improved NMEA-0183 protocol. It can receive differential RTCM messages to improve the accuracy of positioning in differential mode. Its serial port can receive and transmit NMEA or Binary data with the rate of 4800 or 9600 bit per second. The Binary protocol provides more detailed information compare with NMEA protocol [5,6].
Data Collection
To study the function of receiver, the GPS receiver was installed and set up in a fixed position. There are several binary messages provided by MLP. One famous and general purpose of these messages is message No.103, which is available on the first receiver output port as default, when we configure the receiver in binary mode. The message 103 is contain of very useful detail information of position and time [5,6].
In order to setup the receiver, connecting to PC and data collection, a hardware designed and implemented. Fig.1 shows the hardware structure. The output data was collected for a few months. A Pentium III computer with 450 MHZ clock speed saved it.
Fig.1. Hardware structure
Position Components Errors
Since MLP is a low cost nonmilitary receiver, its measurement errors are not neglected (188 meters RMS 3D, when S/A was on and 60 meters RMS 3D, when S/A is off) [5,6]. To study the receiver data and achieving of the errors, the data of position were studied in World Geodetic System-1984 (WGS-84). Therefore the x,y and z magnitude in the No.103 Binary message were collected and saved in separate files every 1 second period.
We focus on variation of x,y and z components in studying ARMA model [7].A software was developed for this purpose. By calculating the average of each quantity in file length, the software provides difference of the instantaneous magnitude of each point with its corresponding quantity average according to equation (1) to (6) and saves them in other files [7].
Where xi,yi,zi are instantaneous magnitude of x,y,z and Ax, Ay, Az are the average magnitude of x,y,z and xi,yi,zi are instantaneous error magnitude of x,y,z respectively. n is number of samples.
Linear Correlation between Position Components
We collected position data for a fix point at 17 days and saved them to 17 files. Based on the equations (7) to (9), linear correlation between x,y and y,z and z,x for 17 data files were calculated. The obtained results of correlation coefficient were recorded in Table 2.
Where,rxy, rxz and ryz are linear correlation between x,y and y,z and z,x , respectively. The results of these statistical studying show that the average correlation coefficients are: rxy=0.58,rxz=0.42 and zi=0.51 Hence, the result shows linear correlation between x,y and y,z and x,z.
ARMA (Auto-Regressive Moving Average) is a famous model, which is used a discrete-time stochastic process. A time varying ARMA model of order (p,q) which is conventionally noted as ARMA(p,q), is mathematically described by [8]:
Where x(k-j) and y(k-i) are the input and output of the model and e(k) is the noise value at time k for k=1,2,... .ai(k) for i=1,2,...,p and for bj(k)=1,2,...,q are the sets of parameters which describes the model structure. Without loss of generality, it is always assumed that ao(k) 1[9].
Proposed Time Varying ARMA Model
Because of linear correlation between x,y and y,z and also x,z , the ARMA model should be of such kinds that their input variables include patterns of positioning errors conjointly. Structure of this time varying ARMA model is as:
For simplicity, we assume in this research P1=P2=P3=P, q1=q2=q3=q and r1=r2=r3=r.
The Calculation Process
We start from dx component. Equation (11) can restate as:
If we rewrite equation (14) in matrix format, We get:
The parameter matrix q may be estimated using the Least-Squares (LS) method. The LS estimation of the parameter matrix q can be obtained by [9]:
Once the model parameters a1i(n),a1j(n) and a1k(n) are known, the calculation of dx(n) for an arbitrary n can be accomplished by [9]:
Modeling for dy and dz components are similar to equations (14) to (23).
Experimental Results
To study the efficiency of the model, we used 5000 position test data, which were collected on the building of Computer Control and Fuzzy Logic Research Lab in the Iran University of Science and Technology. Statistical significance results from tests are shown in Table3. Prediction errors averages are obtained as following equations:
Where exi,eyi and ezi are errors of prediction dx, dy and dz, respectively. M is the number of test samples. Also, Prediction errors norms are obtained as following equations:
Table3. Average, Variance, Standard Deviation and Norm of positioning error components with (p=q=r=3)
| Statistical Significance |
Position Component x |
Position Component y |
Position Component z |
| Error Average | -0.1426 | -0.2303 | 0.1601 |
| Error Variance | 0.1452 | 0.1038 | 0.0804 |
| Error Standard eviation |
0.3810 | 0.3222 | 0.2836 |
| Error Norm | 3.5537 | 3.0173 | 2.6559 |
Conclusion
This paper has described how the positioning accuracy of a low cost GPS receiver can be greatly improved with a time varying ARMA model. The Least-Squares method was used to determine the parameter matrix according to the measurement or available information. The result is a highly effective estimation technique for accurate positioning. The validity of the proposed time varying ARMA model was confirmed by experimental results on implemented unit in this paper (Fig.2). So, the average errors in prediction of x, y and z are less than 1 meter.
Fig.2. Implemented unit in this research
References
- Paul Zarchan," Global Positioning System: Theory and Applications ", Volume I, American Institute of Aeronautics and Astronautics, 1996.
- B.Hofmann-Wellenhof, H.Lichtenegger and J.Collins, " Global Positioning System: Theory and Practice ", Third Revised Edition, Springer-Verlag Wien New York, April 1994.
- Sien-Chong Wu and William G. Melbourne, " An Optimal GPS Data Processing Technique for Precise Positioning ", IEEE Transactions on Geoscience and Remote Sensing, Vol. 31, No. 1, Jan. 1993, pp.146-152.
- M.H.Refan and K.Mohammadi, " Point Averaging of the Position Components, before and after S/A IS Turned off ". www.gisdevelopment.net/technology/gps/techgp0014. htm, 2001
- " Micro tracker LP Operations Manual ", Rockwell International Corporation, GPS-16, January 10, 1994.
- " Micro tracker LP Designer's Guide ", Rockwell International Corporation, GPS-22, January 1 , 1995.
- Xian-Jun Gao and Yi-Song Dai Ke Wang, " A Study on the Self-Difference GPS Positioning by Dynamic and Fictitious Datum Station ", IEEE Conference on Vehicle Electronics,Vol.1, 1999, pp.16-18.
- Simon Haykin, " Adaptive Filter Theory ", Third Edition, Prentice Hall, 1996.
- Yi Bian, " GPS Signal Selective Availability Modeling and Simulation for FAA WAAS IV & V ", ION GPS-95, September 1995, pp.761-769.