A Comparative Study of Mathematical Modelling for GPS/GLONASS Real-Time

Kinematic (RTK)

Al-Shaery, A.1,2, Zhang, S. 3, Lim, S.1, and Rizos, C.1 1!School of Surveying and Spatial Information Systems, the University of New South Wales, Sydney, NSW, Australia

2 Civil Engineering Department, Umm Al-Qura University, Makkah, Saudi Arabia 3SPACE Research Centre, RMIT University, Melbourne, Australia

BIOGRAPHIES Ali Al-Shaery is currently a doctoral student in the School of Surveying and Spatial Information Systems at the University of New South Wales (UNSW), Australia. He obtained his B.Sc. degree in Civil Engineering from the University of Umm Al-Qura, Saudi Arabia and the M.Sc. degree from University College London, United Kingdom in 2003 and 2007, respectively. His current research interests are network-RTK algorithms and applications. Shaocheng Zhang is a research fellow from the SPACE Research Centre, RMIT University, Australia. He obtained his BSc, MSc. and Ph.D. degrees from Wuhan University in China, and his current research is focused on algorithms for GNSS positioning and radio occultation studies. Samsung Lim is an Associate Professor in the School of Surveying and Spatial Information Systems at UNSW. Samsung is a graduate of Seoul National University, South Korea, with a Bachelor's degree from the Department of Mathematics in 1988, and a Master's degree from the same department in 1990. Samsung obtained his PhD from the Department of Aerospace Engineering & Engineering Mechanics in the University of Texas at Austin in 1995. Chris Rizos is a graduate of the School of Surveying, UNSW, obtaining a Bachelor of Surveying in 1975, and a PhD in 1980. Chris has been researching the technology and high precision applications of GPS since 1985, and has published over 500 journal and conference papers. He is currently the President of the International Association of Geodesy and a member of the Governing Board of the International GNSS Service. ABSTRACT The benefit of combining GLONASS and GPS measurements is obvious when a limited number of satellites are available due to the sky view being partially blocked. However, adding GLONASS observations to

those of GPS is not a straightforward process. Due to the differences in signal structure between GPS and GLONASS, the techniques to eliminate relative receiver clock errors from GLONASS carrier phase observations are different from the GPS data processing algorithms. Several approaches have been suggested to address this challenge when processing combined GPS and GLONASS observations. These can be grouped into two main classes: receiver clock error estimation or receiver clock elimination. This paper reports on the issues related to the mathematical modelling of a combined GPS/GLONASS real-time kinematic (RTK) system. Two experiments, static RTK and kinematic RTK, were carried out to compare the two strategies for combining GPS and GLONASS observations. In the static positioning experiment, three short baselines of 24hour data set among five CORS stations in a network located in the Sydney region, Australia, were processed on an epoch-by-epoch basis. Several data quality measures were used to compare different processing strategies. The influence of system selection on ambiguity resolution was assessed using some of the commonly used measures for ambiguity validation such as the F-ratio (Frei and Beutler 1990), R-ratio (Euler and Schaffrin 1991) and W-ratio (Wang et al. 1998). The precision of baseline vector estimation was also analysed. The investigation found insignificance difference between both modelling strategies considering both baseline precision and ambiguity resolution. INTRODUCTION The performance, in terms of accuracy, availability and reliability, of GPS-RTK is a function of the number of satellites being tracked. Thus, the positioning function of GPS is degraded in ‘urban canyon’ environments or in deep open cut mines where the number of visible satellites is limited. Adding measurements from additional navigation satellites is one of the means of aiding position solutions.

Augmenting GPS satellite measurements with those from GLONASS benefits high precise positioning applications in both real-time and post-mission modes, especially in areas where a limited number of GPS satellites are visible. The inclusion of GLONASS observations in positioning solutions does increase the available number of satellites and thus positioning accuracy may improve as a result of enhanced overall satellite geometry. The GLONASS constellation reached Full-Operational-Capability (FOC) on 8 December 2011 with 24 satellites being set to ‘healthy’ (IAC 2011). The inclusion of GLONASS into positioning solutions is also encouraged by the availability of GLONASS final orbits from the International GNSS Service (IGS). Adding GLONASS observations to those of GPS is not a straightforward process. This is due to the fact that GLONASS satellites broadcast signals based on a frequency division multiplexing strategy (FDMA-Frequency Division Multiple Access) whereas GPS signals are based on a code division multiplexing strategy (CDMA-Code Division Multiple Access). Another difference between both systems is in the coordinate and timing reference frames. This difference can be easily dealt with using well-defined conversion and transformation parameters (El-Mowafy 2001). However, the former difference should be given special care. The main consequence of the difference in signal structure is negating the use of the double-differencing approach to eliminate relative receiver clock error from GLONASS carrier phase observations. Unlike GPS, the mathematical modelling of double-differenced GLONASS carrier phase observations is more complex (Leick 1998, Leick et al. 1998, Wang et al. 2001). The consequence of signal structure differences is more complicated when receivers of different brands are used. An attempt to mitigate this challenge has been made by several researchers (Al-Shaery et al. 2012, Wanninger 2012, Yamada et al. 2010, Zhang et al. 2011). This paper focuses on the case of using homogenous (i.e. same brand) receivers. Li and Wang (2011) compared mathematical modelling of GPS/GLONASS integration for short sessions and for static positioning. However, as far as the authors are aware, no comprehensive comparison between different approaches of mathematical modelling for combined GPS/GLONASS-RTK solutions has been published. This paper reports on a study to examine different approaches for mathematical modelling for a GPS+GLONASS RTK system. Static RTK for longer sessions and kinematic RTK tests were carried out to

assess the effect of different approaches on ambiguity resolution and coordinate accuracy. Mathematical modelling For centimetre-level accuracy GNSS positioning, carrier phase (CPH) observables must be used. The CPH observable of GPS or GLONASS is modelled as follows (Hofmann-Wellenhof et al. 2008, Leick 1998, Wang et al. 2001):

!!,!! = ! !!!

! !!! + !!,!! − !!!! !!! − !!!! − !!

!

! !!,!! +

!!!

! !!! + !!! + !!! + !!,!! (1)

The symbol r refers to a user receiver located at station r. Superscript P refers to GPS or GLONASS satellite, !!,!! is a CPH measurement in units of cycles from receiver r to satellite p, !!! indicates the frequency of the carrier signal transmitted by satellite p in this case the L1 frequency. In the case of GPS satellites the carrier frequency is the same for all satellites. For GLONASS satellites the carrier frequency of each satellite is slightly different because the FDMA signal structure is used to distinguish between the different satellite transmissions, !!! is the geometric distance in units of metres between receiver r and satellite p, !!,!! is the unknown integer ambiguity in units of cycles in the CPH measurement between receiver r and satellite p, !!! and !!! are satellite and receiver clock errors, respectively, !!,!! is the ionospheric delay in the CPH measurement with respect to the carrier frequency in units of metres, !!! is the tropospheric delay in units of metres, !!! refers to the inter-channel hardware bias due to modern receivers design (Hofmann-Wellenhof et al. 2008) which is ignored for either GPS or GLONASS satellites if homogenous receivers are employed, !!! is the initial phase, !!,!! denotes the measurement noise and residual biases (receiver noise, multipath, etc.). In differential GNSS positioning the unknown coordinates of receiver r are determined relative to a precisely known reference station using CPH observables. In order to obtain accurate estimates of the coordinates, the ambiguity parameters are required to be correctly fixed to their theoretical integer values. What makes the process difficult is that the unknown integer ambiguities are contaminated by random and systematic measurement errors, such as ionospheric, tropospheric, satellite clock and orbit, receiver clock errors, and measurement noise. Therefore, the reduction or mitigation of these errors is a pre-requisite for accurate coordinate estimation. This is traditionally achieved by using the data double-differencing approach.

Double-differencing is a very effective way of mitigating or eliminating the common errors affecting GNSS observations. The double-differenced (DD) observable is formed by differencing a single-differenced (SD) observable associated with satellite p from a SD observable associated with satellite q. It should be noted that both measurements are made at the same time. The GLONASS DD observable is different to that of GPS. This is due to the fact that GLONASS satellites transmit signals at different frequencies. Therefore receiver clock errors cannot be cancelled out. In contrast, receiver clock bias can be cancelled out in the case of GPS even for different types of receivers at both ends of a baseline. The initial phase bias is cancelled in both observables of GPS and GLONASS. Receiver clock bias The double-differencing operation is carried out in two steps: SD observables are firstly formed, and then the DD observables. The traditional way to form a SD observable is to difference measurements of the rover r from that of the master reference station m observed at the same time from the same satellite (Leick 1998, Rizos 1999). This is the between-receiver SD. It is assumed here that the distance between the two receivers is short enough to cancel out the distance-dependent errors (ionospheric, tropospheric and orbit errors) (Rizos 2002). The SD equation for either GPS or GLONASS CPH observable is expressed (in units of cycles) as:

!!",!! = ! !!!

! !!"! + !!",!! + !!!! !!! − !!!! +

!!!!!!!!!!!!!!!!!!"! !+!!",!! ! (2) In the case of GPS-only or GPS/GLONASS SD observables the atmospheric (ionosphere and troposphere) delay biases are significantly reduced for a short baseline which is often considered as one whose length is <20km. Orbit errors are largely mitigated for baselines up to 100km in length. Satellite clock errors are cancelled no matter what the baseline length. This reduces the errors which are required to be estimated or assumed to have been cancelled in order to reliably resolve the integer ambiguities. Inter-channel biases are cancelled assuming both receivers are from same manufacturer (or brand). However, receiver clock errors and the initial phase bias still exist. These errors will be accounted for in the DD observable:

!!",!!" = !!!

! !!"! − !!!

!

! !!"! + !!",!!" + !!! − !!!! !!!" +

!!!!!!!!!!!!!!!!!!",!!" (3) Although double-differencing GNSS observations is an effective way of mitigating or eliminating the common errors, it is not straightforward when GLONASS

observations are used. The GLONASS DD observable is different from that of GPS because the frequencies of the two GLONASS satellites involved in the DD are not the same. On the other hand, the initial phase bias can be assumed cancelled from both GPS and GLONASS observables. To overcome this problem the bias can be either estimated or eliminated. The first strategy is to estimate the receiver clock error and then correct the associated CPH observable, as suggested by Raby and Daly (1993), Pratt et al. (1997) and Leick (1998). Another approach was implemented by Wang et al. (2001) where the DD GPS and SD GLONASS pseudorange (PR) and the DD GPS and GLONASS CPH observables are used together to estimate relative receiver clock error, integer ambiguities and baseline components. The following model (System 1) represents this strategy: !!",!!"#,!" = !!"!" + !!",!!" !!",!!"#,! = !!"! + !"#!" + !!",!!"

!!",!!"#,!" = !!! !!"

!" + !!",!!" + !!",!!"

!!",!!"#,!" = !!!! !!"

! − !!!

! !!"! + (!!

!

! − !!!

! )!"#!" + !!",!!"

+ !!",!!" (4)

The second strategy requires scaling of the CPH observation into distance. This method successfully eliminates receiver clock errors from the DD observables. However the DD integer ambiguity becomes a non-integer number: !!",!!" = !!"!" + !

!!!!!",!! − !

!!!!!",!! + !!",!!" (5)

Leick (1998) proposed a solution to preserve the integer nature of ambiguities by introducing the common frequency by applying two integer factors to the DD CPH observation. However, this method produces micrometre-sized integer ambiguities which can not be easily estimated. Takasu and Yasuda (2009) addressed this problem by solving for baseline components and the SD integer ambiguities. Then, the SD integer ambiguities are transformed into the DD form using a transformation matrix. Adding PR observables ensures single-epoch fixed ambiguity solutions as seen in the following functional model, which represents the second strategy to overcome GLONASS DD challenges (System 2):

!!",!!"#,!" = !!"!" + !!",!!" !!",!!"#,!" = !!"! + !!",!!"

!!",!!"#,!" = !!"!" +!!!!!",!!" + !!",!!"

!!",!!"#,!" = !!"!" +!!!! !!",!

! − !!!

! !!",!! + +!!",!!"

(6) Dai et al. (1999) proposed a three-step approach, which is an extension of Wang’s model, by combining the estimation and the elimination approach. In the first step, relative receiver clock errors and baseline components are estimated using DD GPS and SD GLONASS PR. Then these values with their variance-covariance information are used to fix the integer ambiguities of DD GPS and GLONASS CPH observations. Then the fixed ambiguities are used in the third step to estimate the GLONASS SD integer ambiguities involving the reference satellite and baseline components using DD CPH expressed in units of metres. The third step was proposed to exclude the effect of receiver clock error on the estimation of baseline components. However, Li and Wang (2011) found that the performance of this approach is identical to the optimal model identified in (Wang et al. 2001). Experiments Two experiments, static RTK and kinematic RTK, were carried out in order to compare the two strategies (System 1 and System 2) of addressing the GPS/GLONASS mathematical challenges. Several quality measures were used to compare the strategies. The influence of system selection on ambiguity resolution (AR) was assessed using some of the commonly used measures for ambiguity validation (AV), such as F-ratio (Frei and Beutler 1990), R-ratio (Euler and Schaffrin 1991), and W-ratio (Wang et al. 1998). Baseline precision was also investigated. Static RTK Three short baselines between five continuously operating reference stations (CORS), CHIP, UNSW, PBOT, BATH and RGLN, of the CORSnet-NSW network located in the Sydney region were processed on an epoch-by-epoch basis. The CHIP CORS was used in two baselines as the reference station while UNSW and PBOT were assumed to be user receivers. BATH was used in the third baseline as a reference station with RGLN as the user receiver. The 24 hour data set used in this test was from 29 January 2012 for the first two baselines and 29 February 2012 for the third baseline. Table 1 summarises the data set information. Table 1 Summary of static RTK test parameters. CHUN029 CHPB029 BTRG060 Data length 24 hr 24hr 24hr Obs. Type L1+L2 L1+L2 L1+L2 Baseline Length 4.4km 9.7km 8.3km

Receiver Types Leica Leica Trimble

Elevation mask 15º 15º 15º

Interval 15s 15s 15s

Several discrimination tests were used to assess the performance of the two strategies in terms of AR. F-ratio with critical value of 2 (Landau and Euler 1992) was used, as well as R-ratio test with critical value of 3 (Verhang 2004) and W-ratio with critical value chosen based on the student’s t-distribution (Verhang 2004, Wang et al. 2000). Table 2 summarises the success rate for the two strategies and the baselines. In the case of the first baseline (CHUN), the System 2 achieved slightly higher success rate over all AV measures than that of the System 1 except for W-ratio test, where System 1 is better by an insignificant amount. In contrast, System 1 performs better than System 2 for all baselines for all validation tests except the W-ratio in CHPB029. A clear conclusion of which approach is better cannot be drawn from these results. Table 2 Success rate of tested baselines processed by both strategies. Baseline GPS-GLO

model Success rate (%)

F R W

CHUN029 System 1 98.53 97.54 94.71

System 2 99.09 97.57 94.64

CHPB029 System 1 93.48 87.53 90.81

System 2 92.57 86.81 90.70

BTRG060 System 1 70.85 76.98 80.17

System 2 67.66 76.90 81.55

With respect to the coordinate precision, both strategies achieved similar values for each baseline (Table 3). No significant differences between strategies can be noted (Figures 1-3). Figures 1-3 show fixed solutions based on the R-ratio test. The standard deviation values listed in the figures are from ambiguity fixed solutions only.

06:00 12:00 18:0010

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Figure 1 CHIP-UNSW (29/01/2012) baseline processing of Systems 1 and 2.

Figure 2 CHIP-PBOT (29/01/2012) baseline processing of Systems 1 and 2.

Figure 3 BATH-RGLN (29/02/2012) baseline processing of Systems 1 and 2. Table 3 Standard deviation summary for tested baselines. Baseline GPS-GLO

model Standard deviation (m)

E N U

CHUN029 System 1 0.006 0.005 0.013

System 2 0.006 0.005 0.013

BTRG060 System 1 0.007 0.010 0.021

System 2 0.007 0.010 0.021

CHPB029 System 1 0.008 0.006 0.019

System 2 0.008 0.006 0.019

Kinematic RTK

Two kinematic tests were carried out on 21/12/2011 using LEICA GX1230GG dual-frequency receiver mounted on a car, and forming a baseline with the CORS station (UNSW) located on the roof of the Electrical Engineering Building at UNSW. The first trajectory (Traj1) started from Maroubra Junction to the UNSW campus (Figure 4). Signal environments of the trajectory can be categorised as a moderate cut-off elevation angle which may reach 30 degrees at maximum. The second trajectory (Traj2) was generated by driving the car around the university campus (Figure 7). The sky view for some parts of the trajectory was almost blocked, especially the east and west sides of the campus. The number of fixed epochs according to R-ratio test with critical value 3 was used in this test to assess the performance of both systems.

Figure 4 Trajectory of kinematic test 1 (Traj1).

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Figure 5 Traj1 processed by System 1.

Figure 6 Traj1 processed by System 2.

Figure 7 Trajectory of kinematic test 2 (Traj2).

Figure 8 Traj2 processed by System 1.

Figure 9 Traj2 processed by System 2. From Figures 5-6, almost the same characteristics can be seen in Traj1 from both strategies (System 1 and System 2). However, System 2 produced a slightly larger number of fixed ambiguity epochs than that of System 1. The same holds for the second kinematic test (Figures 8-9). The signals from satellites in the second test were frequently obstructed by trees and tall buildings, whereas environment was relatively benign in the first test. The consequence of which can be seen in the number of gaps in both tests. The minor difference between the number of ambiguity fixed solutions between the strategies is of little significance. Concluding remarks Receiver clock error cancellation in the double-differenced GPS and GLONASS CPH observations is a challenge for combined measurement processing. Two main approaches for integrated GPS/GLONASS mathematical modelling to overcome such challenges were analysed. The first approach is to estimate the error, while the second one is to eliminate the error.

3.36 3.362 3.364 3.366 3.368 3.37 3.372 3.374

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Fix Epochs: 72.03 %

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Fix Epochs: 80.6962 %

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Fix Epochs: 82.2785 %

FixedFloat

There is similar performance of both approaches for dealing with receiver clock error. Field static and kinematic RTK experiments were carried out to assess the performance of the two approaches in terms of ambiguity resolution success rate and the coordinate accuracy. The static test indicates that there is an insignificant advantage of one approach over the other even though the estimation approach performs better than the elimination approach in two baselines of the three tested in terms of ambiguity validation results. However, similar results of coordinate accuracy were obtained from both approaches for the tested baselines. In contrast, the second approach appears to have a slightly higher performance than the first in the kinematic test by producing slightly larger number of fixed epochs. However the difference is not significant to support the conclusion that one approach is clearly superior to the other. Acknowledgments The first author would like to thank the NSW Land & Property Information, Department of Finance & Services, for the provision of the CORSnet-NSW data. He is also grateful to the scholarship provider, the Saudi Higher Education Ministry, and especially the University of Umm Al-Qura. References Al-Shaery, A., S. Lim & C. Rizos (2011) Assessment of

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ambiguity resolution issues, GPS Solutions, 5(1), 55-64.

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Ali, see edited version attached. I have real problems with your conclusions in Table 3 (which are not referenced in the text), and Figures 5-8. Where do you get the accuracies? Or are they precisions? Please address my comments/questions embedded in text. CR