| Asian GPS ---> Proceeding ---> 2002---> Keynote Session |
Using GPS Velocity Information in Enhancement of GPS Position Accuracy
Dr. K. Mohammadi, Mohammad Reza Zamani
College of Electrical Engineering
Iran University of Science and Technology
Tehran, Iran
mohammadi@iust.ac.ir , 79708704@iust.ac.ir
Dr. K. Mohammadi, Mohammad Reza Zamani
College of Electrical Engineering
Iran University of Science and Technology
Tehran, Iran
mohammadi@iust.ac.ir , 79708704@iust.ac.ir
Abstract
Position information obtained from standard GPS receivers is known to be corrupted with coloured (time-correlated) noise. To make effective use of GPS information in a navigation system it is essential to model this coloured noise and to incorporate additional information to de-correlate and eliminate its effect. In this paper frequency domain, techniques are employed to generate a model for GPS noise sources. This model shows clearly, what type additional information such as velocity information is necessary to separate GPS errors and for improving the GPS position accuracy in navigation tasks.
Introduction.
GPS provides worldwide positioning with acceptable accuracy, if four or more satellites are in view. Various dynamic models for GPS positioning have been proposed over the years, differing in their complexity . However, there are problems inherent in the system such as correlated errors on the satellite signal, which mean that GPS alone does not meet the requirements for such a system . State estimators, such as the Kalman filter, which are often used as the main navigation algorithm, implicitly reflect these models and act as low-pass filters for the observation information. Therefore, true high frequency information, associated with vehicle maneuvers such as turning, is lost. To overcome this, inertial information is often fed-forwarded through the state estimator as a prediction to be corrected by absolute information. The net effect of this is that while absolute information is subjected to a low-pass filter, prediction information is subjected to a complimentary high-pass filter. Together these two sources of information span the complete frequency spectrum in a complimentary manner . The approach adopted in this paper is to develop and exploit frequency domain error and system models to specify and compare different GPS velocity information. Section 2 in this paper introduces an experimentally derived frequency domain error model for standard GPS position estimates. This model is obtained directly from , data provided by a commercial GPS receiver. Section 3 describes how this model is exploited in a state estimator (Kalman filter), extended with an appropriate shaping filter to take into account the coloured noise and incorporating an additional information to de-correlate the position state from the shaping state. Section 4 shows how the filter is tuned and how the quality of aiding information is studied in the frequency domain.
GPS Error model
State estimation methods such as the Kalman filter assume that measurement noise is white or non-correlated. If this is not the case then a shaping filter must be constructed whose input is white noise and whose output is the observed coloured noise . This filter can be obtained by obtaining a power spectral density for the measurement errors on the assumption that the true states are known. In the work described in this paper, we exploit this knowledge and construct only shaping models for and errors by using real GPS position information.
Figure 1 shows the and errors in meters obtained with a GPS unit working without differential correction. It is clear from the figure that the error is time correlated, showing a characteristic oscillatory behavior with amplitude between 10m and 20m.
To generate an error model the auto-correlation and corresponding power spectral density (PSD) of the error signal must be estimated. The PSD of and errors is shown in Figure 2 and 3.
Figure 2. PSD of X Error
Figure 3. PSD of Y Error
The PSD has a gain of approximate 26db below 70mHz, continuing with a sharp roll-off of approximate 40db/decade up to 1000mHz where the magnitude starts to decrease at approximate 20db/decade.
A transfer function may be fitted to these PSD estimates in the form
Where parameters are as shown in following table.
This model may be described by a shaping filter with process model as:
Where the driving noise w (t) is now white noise with unit variance.
Filter Implementation
It is well known that if the process can be approximated with a linear model plus white noise with known statistics, then an optimal (minimum mean squared error) Kalman Gain can be evaluated. In this application, the observation is composed of the GPS position measurement that is corrupted with coloured and white noise. In order to de-correlate the GPS position information from the coloured noise, it is necessary to obtain other information free of coloured noise in the frequency range of interest. The additional information used in this example is the GPS velocity information corrupted by white noise only. We will show that the quality of the additional information is of fundamental importance to de-correlate the position information from the non-white noise.
In this application, we consider a constant velocity model. A full implementation for , and estimation will require 12 states to account for the coloured noise. The results presented in this work correspond to the estimation of information.
The constant velocity model and error model to estimate expressed as
The measurements available are GPS position corrupted by coloured noise and white noise and GPS velocity information corrupted by white noise only. The white noise present in both measurement is modeled with a variance of and respectively.
Filter Tuning and Simulation Results
A key issue to be resolved is what quality (noise variance) is required of the GPS velocity information to successfully de-correlate GPS position information from the coloured noise. To begin with, it is assumed that the GPS velocity information is high, with a standard deviation of 35.4mm/sec. The Bode plot describing the transfer functions between GPS position, and the output estimates corresponding to position and shaping state is shown in Figure 4.
Figure 4. Bode plot of Filter Transfer function with a standard deviation of 35.4mm/sec in GPS velocity information
Figure 5 shows the time-domain performance of the system with respect to the GPS position error Errx, GPS velocity error Errvx, shaping state estimation xs and position estimate x.
Figure 5. State estimations and GPS position and velocity errors
The standard deviation in the GPS velocity information is now reduced to 10.6mm/sec. The resulting transfer function for the new system is shown in the bode diagrams of Figure 6.
Figure 6. Bode plot of Filter Transfer function with a standard deviation of 10.6mm/sec in GPS velocity information
The figure show that the position estimate transfer function gain has decreased but the shaping state estimation transfer function has not changed. Figure 7 shows the time-domain performance of the new system.
Figure 7. State estimations and GPS position and velocity errors
The time response in figure 7 shows that the position estimated error has decreased but the shaping state has not changed.
Conclusions
This report has described the frequency domain techniques in providing an effective way of designing navigation system. Frequency domain methods are of particular value in showing how additional information may be used to improve overall system performance. The results showing how GPS position data may be improved using GPS velocity information to de-correlate coloured noise errors are presented to demonstrate the proposed methods.
The frequency domain methods described in this paper provide a powerful method of describing and analyzing the systems such as GPS and radar.
References
Position information obtained from standard GPS receivers is known to be corrupted with coloured (time-correlated) noise. To make effective use of GPS information in a navigation system it is essential to model this coloured noise and to incorporate additional information to de-correlate and eliminate its effect. In this paper frequency domain, techniques are employed to generate a model for GPS noise sources. This model shows clearly, what type additional information such as velocity information is necessary to separate GPS errors and for improving the GPS position accuracy in navigation tasks.
Introduction.
GPS provides worldwide positioning with acceptable accuracy, if four or more satellites are in view. Various dynamic models for GPS positioning have been proposed over the years, differing in their complexity . However, there are problems inherent in the system such as correlated errors on the satellite signal, which mean that GPS alone does not meet the requirements for such a system . State estimators, such as the Kalman filter, which are often used as the main navigation algorithm, implicitly reflect these models and act as low-pass filters for the observation information. Therefore, true high frequency information, associated with vehicle maneuvers such as turning, is lost. To overcome this, inertial information is often fed-forwarded through the state estimator as a prediction to be corrected by absolute information. The net effect of this is that while absolute information is subjected to a low-pass filter, prediction information is subjected to a complimentary high-pass filter. Together these two sources of information span the complete frequency spectrum in a complimentary manner . The approach adopted in this paper is to develop and exploit frequency domain error and system models to specify and compare different GPS velocity information. Section 2 in this paper introduces an experimentally derived frequency domain error model for standard GPS position estimates. This model is obtained directly from , data provided by a commercial GPS receiver. Section 3 describes how this model is exploited in a state estimator (Kalman filter), extended with an appropriate shaping filter to take into account the coloured noise and incorporating an additional information to de-correlate the position state from the shaping state. Section 4 shows how the filter is tuned and how the quality of aiding information is studied in the frequency domain.
GPS Error model
State estimation methods such as the Kalman filter assume that measurement noise is white or non-correlated. If this is not the case then a shaping filter must be constructed whose input is white noise and whose output is the observed coloured noise . This filter can be obtained by obtaining a power spectral density for the measurement errors on the assumption that the true states are known. In the work described in this paper, we exploit this knowledge and construct only shaping models for and errors by using real GPS position information.
Figure 1 shows the and errors in meters obtained with a GPS unit working without differential correction. It is clear from the figure that the error is time correlated, showing a characteristic oscillatory behavior with amplitude between 10m and 20m.
To generate an error model the auto-correlation and corresponding power spectral density (PSD) of the error signal must be estimated. The PSD of and errors is shown in Figure 2 and 3.
Figure 2. PSD of X Error
Figure 3. PSD of Y Error
The PSD has a gain of approximate 26db below 70mHz, continuing with a sharp roll-off of approximate 40db/decade up to 1000mHz where the magnitude starts to decrease at approximate 20db/decade.
A transfer function may be fitted to these PSD estimates in the form
Where parameters are as shown in following table.
This model may be described by a shaping filter with process model as:
Where the driving noise w (t) is now white noise with unit variance.
Filter Implementation
It is well known that if the process can be approximated with a linear model plus white noise with known statistics, then an optimal (minimum mean squared error) Kalman Gain can be evaluated. In this application, the observation is composed of the GPS position measurement that is corrupted with coloured and white noise. In order to de-correlate the GPS position information from the coloured noise, it is necessary to obtain other information free of coloured noise in the frequency range of interest. The additional information used in this example is the GPS velocity information corrupted by white noise only. We will show that the quality of the additional information is of fundamental importance to de-correlate the position information from the non-white noise.
In this application, we consider a constant velocity model. A full implementation for , and estimation will require 12 states to account for the coloured noise. The results presented in this work correspond to the estimation of information.
The constant velocity model and error model to estimate expressed as
The measurements available are GPS position corrupted by coloured noise and white noise and GPS velocity information corrupted by white noise only. The white noise present in both measurement is modeled with a variance of and respectively.
Filter Tuning and Simulation Results
A key issue to be resolved is what quality (noise variance) is required of the GPS velocity information to successfully de-correlate GPS position information from the coloured noise. To begin with, it is assumed that the GPS velocity information is high, with a standard deviation of 35.4mm/sec. The Bode plot describing the transfer functions between GPS position, and the output estimates corresponding to position and shaping state is shown in Figure 4.
Figure 4. Bode plot of Filter Transfer function with a standard deviation of 35.4mm/sec in GPS velocity information
Figure 5 shows the time-domain performance of the system with respect to the GPS position error Errx, GPS velocity error Errvx, shaping state estimation xs and position estimate x.
Figure 5. State estimations and GPS position and velocity errors
The standard deviation in the GPS velocity information is now reduced to 10.6mm/sec. The resulting transfer function for the new system is shown in the bode diagrams of Figure 6.
Figure 6. Bode plot of Filter Transfer function with a standard deviation of 10.6mm/sec in GPS velocity information
The figure show that the position estimate transfer function gain has decreased but the shaping state estimation transfer function has not changed. Figure 7 shows the time-domain performance of the new system.
Figure 7. State estimations and GPS position and velocity errors
The time response in figure 7 shows that the position estimated error has decreased but the shaping state has not changed.
Conclusions
This report has described the frequency domain techniques in providing an effective way of designing navigation system. Frequency domain methods are of particular value in showing how additional information may be used to improve overall system performance. The results showing how GPS position data may be improved using GPS velocity information to de-correlate coloured noise errors are presented to demonstrate the proposed methods.
The frequency domain methods described in this paper provide a powerful method of describing and analyzing the systems such as GPS and radar.
References
- Robert Grover Brown, Patrick Y.C. Hwang. " Introduction to random signals and applied Kalman filtering: with MATLAB exercises and solutions " 1997.
- Nebot, E.M.; Whyte, H.D.; Sheding, S, "Kalman filtering design techniques for aided GPS land navigation applications", Data Fusion Symposium, 1996.ADFS`96.First Australian, pp.83-88, 1996.
- Dr.K.Mohammadi, Dr.M.H.Refan, " A new method for improving the time accuracy of GPS Receivers using Kalman Filter ", International Journal of Engineering Science, IUST, Vol.13, No.1, pp.11-24, 2002.
- Cooper, S.; Durrant -Whyte, H, " A Kalman filter model for GPS navigation of land vehicles ", Intelligent robots and Systems ' 94.' Advanced Robotics Systems and the Real World', IROS' 94. Proceedings of the IEEE/RSJ/GI International Conference on, pp. 157-163 Vol.1, 1994.