Adaptive Kalman filtering for integration of GPS with GLONASS and INS

J. Wang, M. P. Stewart and M. Tsakiri School of Spatial Sciences Curtin University of Technology GPO Box U 1987; Perth, WA 6845, Australia

Abstract. In an integrated kinematic system, the Kalman filter is commonly used to integrate the data from different sensors (such as GPS/GLONASS and INS) for precise positioning. Reliable Kalman filtering results rely heavily on the correct definition of both the mathematical and stochastic models used in the filtering process. Whilst the mathematical models for various positioning measurements are (sufficiently) known and well documented in the current literature, stochastic modelling is not trivial, in particular for real-time applications.

In this paper, a newly developed adaptive Kalman filter algorithm is introduced to directly estimate the variance and covariance components for the measurements. Example applications of the proposed algorithm in GPS/GLONASS kinematic positioning and GPSIINS integration are discussed using test data sets. Test results show that the proposed algorithm can improve the performance of the filtering process.

Keywords. Kalman filtering, stochastic modelling, GPS, GLONASS, INS.

1 Introduction

Kinematic positioning as a geodetic tool has been widely used in a variety of areas, such as real-time mapping and precise navigation. During the last decade, GPS has been the major technology for kinematic positioning techniques. A recent development in kinematic GPS positioning is the technique of using pseudo-range and carrier phase measurements to resolve the carrier phase ambiguities almost instantaneously. Kinematic techniques improve productivity in surveying applications, and also apply well to navigation. Reliable kinematic positioning results depend on the availability of a large number of GPS satellites. GPS kinematic positioning techniques, therefore, may not be feasible for some situations, such as the

Intematiooal Association of Geodesy Symposia, Vol. 121

positioning in built up areas, where the number of visible satellites is limited. One possible scheme to increase the availability of satellites is to combine GPS and GLONASS, which is the Russian counterpart to GPS.

Satellite-based kinematic positioning techniques can offer consistent precision during the period of positioning. However, such systems require line-of­sight between the orbiting satellites and receiver antennas. In some situations, in highway tunnels and under trees or bridges, GPS/GLONASS receivers cannot track satellite ranging signals, and thus will fail to provide positioning results. Integration of GPS with a self-contained inertial navigation system (INS) can overcome this disadvantage of the satellite-based systems. On the other hand, precise GPS positioning results are ideally suited for continuous INS calibration. GPSIINS integration has been successfully used in modern real-time mapping systems (e.g. Grejner­Brezinska, 1997; Schwarz, 1998).

In an integrated kinematic positioning system, the Kalman filter is commonly used as a data fusion tool because a kinematic system will include some time-dependent unknown states, such as coordinates of a moving platform. The Kalman filter, as a set of mathematical equations, can provide an efficient computational (recursive) solution of the least­squares estimation problem. Therefore, the Kalman filter can produce optimal estimation results in real­time if the mathematical and stochastic models used in the filtering process are both correctly defined. Whilst the mathematical models for major positioning systems (such as GPS, GLONASS and INS) are sufficiently known and well documented, the stochastic modelling for kinematic positioning is a critical issue to be investigated.

For example, in the commonly used procedures for constructing the covariance matrices of the differenced GPS and GLONASS measurements, all the one-way code or carrier phase measurements are assumed to be independent and to have the same accuracy. In fact, these assumptions do not fit

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reality (e.g. Gianniou and Groten, 1996; Wang et ai, 1997). First, the code or carrier phase observations are correlated (the single- and double-difference methods are based explicitly on this fact). Secondly, measurements obtained from different satellites cannot have the same variance because of various noise sources. This is particularly true when the measurements come from different systems in combined GPS and GLONASS positioning. Any deficiency in the stochastic models for the differenced GPS and GLONASS measurements in data processing will inevitably

-result in unreliable statistics for ambiguity resolution and bi~ed positioning results (e.g. Cannon and Lachapelle, 1995; Wang et ai, 1997).

In the Kalman filtering process, online stochastic modelling can be implemented using the so-called adaptive filtering techniques (Chin, 1979; Mehra, 1972). Applications of adaptive Kalman filtering techniques in real-time GPS kinematic positioning (Wang et ai, 1997) and GPSIINS integration (Mohamed and Schwarz, 1999) have shown encouraging results. This paper will introduce a simplified explanation for the algorithms used in the adaptive Kalman filtering procedure. Latest test results in GPS/GLONASS and GPSIINS integration are discussed in detail.

2 Adaptive Kalman filter procedure

Since the true values of the· model errors (measurement noise or process noise) are unknown, stochastic modelling has to be based on the filtering residuals of the measurements and state corrections, which are generated in the process of parameter estimation. A problem here is that the parameter estimation process itself relies on the estimated measurement covariance matrix R and process covariance matrix Q. To tackle this problem, an

adaptive procedure can be used. The basic idea of the adaptive filtering procedure

is that the residuals collected from the previous segment of positioning results are used to estimate the covariance matrices of the measurement noise and process noise for the current epoch. (Preset default covariance matrices are needed to seed the adaptive estimation process.) Within a segment, the covariance matrices for each epoch is assumed to be the same. Therefore, the formulation of an adaptive stochastic modelling method includes two critical steps, namely: (a) to derive suitable formulae for use in estimating the covariance matrices, and (b) to determine the optimal segment (or window size), which is application-dependent and will be

discussed in Section 3. The formulae for constructing R and Q can be derived using the so-

called 'covariance matching' method (Mehra, 1972).

2.1 Estimating the measurement covariance matrix

Suppose the measurement filtering residuals are:

(1)

where Zk is the measurement vector and Hk,is the

measurement design matrix. Equation (1), obviously, is the optimal estimator of the measurement noise level because the estimated values Xk (not the predicted values xk ) of the state

parameters are used in their computations. In order to obtain the covariance matrix of the measurement filtering residuals, equation (1) is modified

326

where Gk is the gain matrix and dk is the

innovation vector. By applying the error propagation law to equation (2), after extensive computations, one obtains

(3)

In equation (3), if the covariance matrix Qv is Zk

computed using the measurement filtering residuals from the previous m epochs, the covariance matrix Rk can be estimated as (Wang, 1998)

Rk = QVZk + HkQxkHI

1 m-l T T =;; i~O vZk-ivZk_i + HkQxkHk

(4)

which can be used in the computation of epoch k + 1. In equation (4), m is called the width of moving windows. It is noted that the covariance

matrix Rk estimated with equation (4) is always

positive definite because it is the sum of the two positive definite matrices. Equation (4) requires some extra computations for both VZt and

HkQxl; Hi, which are not generated by the

standard Kalman filtering process. Fortunately, the amount of these additional computations is small

and leads to no significant time delay in data processing.

2.2 Estimating the process covariance matrix

Similar to the measurement covariance matrix, the process noise covariance matrix can be derived using the state corrections as follows

(5)

By applying the error propagation law to equation (5), one obtains

QVXk = GkOtik GI

= Qxk -Qxk

= cI>k,k-1Qxk_lQl.k-l +Qk -Qxk

(6)

In equation (6), if the covariance matrix Qv is Xk

computed using the state corrections from the previous m epochs, the covariance matrix Ok can

be estimated as

Qk = QVXk + QXk - cI> k,k-1Qxk_l cI>r.k-l

1 m-l =- L Vx .vT

m i=O k-l Xk-i

+ QXk - cI> k,k-1Qxk-l cI>r.k-l

(7)

which can be used in the computation of epoch k + 1. Unlike equation (4), however, with equation (7), it cannot be guaranteed that the estimated process noise matrix will be positive definite.

In many physical situations, therefore, it. is very difficult to model the dynamic behaviour of a moving platform with the accuracy that GPS/GLONASS measurements can give. In order to avoid the effect of errors in the process noise covariance matrix on the state estimation, the filter can be set up to operate only on the measurement noise. The basic concept regarding this filter operating mode has been discussed extensively in the literature (e.g. Hwang and Brown, 1990; Lachapelle et at, 1992). In the following

327

experiment, the process covariance matrix will be constructed using default values.

It is noted that the formulae derived above,

equations (4) and (7), can also be derived using a maximum likelihood criterion (Mohamed and Schwarz, 1999).

3 Experiments for GPS/GLONASS and GPSnNS integration

3.1 GPS/GLONASS positioning

The test data set was collected on a 1.2km (static) baseline, on February 16, 1998, in Perth, Australia, using two Ashtech GG24 GPS/GLONASS receivers. The data span was 25 minutes with a data interval of 1 second. During the whole session of observation, 7 GPS and 5 GLONASS satellites were tracked.

The realistic estimation of measurement covariance matrices provides reliable statistics for ambiguity resolution. To demonstrate this more clearly, both the solutions with the preset and estimated covariance matrix were generated. The ambiguity dilution of precision (ADOP) (Teunissen and Odijk, 1997) is illustrated in Figure 1. It indicates that, with the estimated covariance matrix, the accuracy of the estimated ambiguities was greatly improved. As a consequence of this, the ambiguity search volume was significantly reduced and faster search process could be expected (Landau and Euler, 1992; Teunissen and Odijk, 1997).

The improved float ambiguity estimates are of great importance for the ambiguity validation test, which is one of the critical steps in ambiguity resolution. The ambiguity validation results, using the ambiguity validation test statistic W s, see description in Wang et al (1998), are graphed in Figure 2. In all cases, the ambiguity validation test statistics with the estimated measurement covariance matrices are much larger than those with the preset measurement covariance matrices. The averaged confidence levels of the ambiguity resolution for the preset and estimated stochastic models are 0.83 and 1.0, respectively. This improvement in confidence level is statistically significant.

0.2

~ 0.15 GI

---PresetR - - - - - -Estimated R

~ 2. 0.1 0-o Q

<C 0.05

o 200 400 600 800 1000 1200 1400 1600

Epoch Number

Fig. 1: Ambiguity dilution of precision (ADOP. after Teunissen and Odijk. 1997)

30

---PresetR - - - - - - Estimated R 1/1 20 ~ 1.1 ;: 1/1

~ 10

0

0 200 400 600 800 1000 1200 1400 1600

Epoch Number

Fig. 2: Ambiguity discrimination test statistic Ws (After Wang et aI.1998)

Table 1: Impacts of the width of moving windows on ambiguity resolution

Width 6 7 8

Averaged F-ratio 3.5 5.2 13.3

AveragedWs 2.4 3.5

Success Rate % 62.7 96.9

A realistic stochastic model can also improve the accuracy of the ambiguity fixed solutions. For instance, at epoch 8 when the filtering process was using the preset measurement covariance matrix, the standard deviations of (x,y,z) coordinated were 0.008m, 0.016m and 0.012m, respectively. At epoch 9 when the filtering process began to utilise the estimated measurement covariance matrix, the standard deviations of the coordinates were down to 0.006m, 0.013m and 0.009m, respectively. The

6.6

100

328

9 10 30 120

12.0 11.1 6.0 4.1

6.3 6.2 4.3 3.3

100 100 99.5 94.2

differences in the coordinates between the two solutions are around 0.005 m.

To determine the optimal width of the moving window, some further tests for various widths have been conducted. The averaged ambiguity validation test statistics and success rate of ambiguity resolution are listed in Table 1, which show that for the tested GPS/GLONASS data set, the optimal value for the width of the moving window is 8.

3.2 GPSIINS integration

In a combined GPSIINS positioning system, GPS measurements can be used to calibrate INS instrument and alignment errors. Alignment is the process whereby the orientation of the axes of an inertial navigation system are determined. Whilst the initial position and velocity can be directly obtained using GPS measurements, the orientation parameters may be determined using the so-called (INS and GPS) velocity matching method (e.g. Titterton and Weston, 1997). This method may be implemented using a Kalman filter procedure, in which the measurements are differences of the GPS and INS velocity.

A GPS/INS data set was used to test the performance of the adaptive Kalman filtering procedure in the GPS-INS velocity match alignment. The data set was collected using a Litton IMU and Trimble 4000SSE GPS receivers.

In the filter, the orientation parameters (pitch, roll and yaw) are of direct interest. Therefore, the major concern is the precision of the estimated orientation parameters. The test results show that the precision of the yaw parameter can be significantly reduced (see Figure 3). However, only a slight improvement in the precision of the pitch and roll parameters was observed. The possible reason for this is that the pitch and roll precision may rely heavily on the geometry of the system instead of the measurement covariance matrix. This, however, needs further investigation.

For this data set, the different window sizes (ranging from 2 to 10) did not influence significantly the filtering results.

GI .s:. -... .2

4 Concluding remarks

In an integrated GPS/GLONASS kinematic positioning system, ambiguity resolution and positioning results rely on the correctness of the measurement noise covariance matrix. The common practice of assuming that the one-way (code or ~arrier phase) measurements are statistically IOdependent, and have the same accuracy, is certainly not realistic, and thus inevitably leads to an unsuitable measurement covariance matrix.

With the proposed adaptive Kalman filtering procedure, the measurement noise covariance matrix can be applied to real-time data processing. By using an estimated covariance matrix, the reliability of ambiguity resolution and the accuracy of kinematic positioning can be significantly improved.

The proposed adaptive Kalman filtering procedure can also be used in other integrated kinematic positioning systems. Test results from an integrated GPSIINS data set indicate that by using the estimated measurement covariance matrix, the precision of the orientation parameters can be improved, particularly that of the estimated yaw parameter in the velocity matching alignment process.

It is noted that the optimal window size used in the adaptive filtering procedure may depend on the geometry of the solution. In practical applications, the optimal window size may need to be determined in real-time. On the other hand, the adaptive estimation of the filtering process noise covariance matrix needs further investigation.

CIl-

.§ ~ 25 iii e >~

---PresetR .. .. .. Estimated R

~ :I: 'E~ ca 'D C ~ 15 L-__ --'-___ '--__ ......... __ ---'-_ __ -'--__ ..J

o 30 60 90 120 150 180

Time (seconds)

Fig. 3: Standard deviation for the yaw parameter

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Acknowledgment. The authors thank Dr. Dorota A. Grejner-Brzezinska (The Ohio State University Center for Mapping) for kindly providing us GPSIINS data.

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