GLONASS Broadcast Orbit Computation MIKE STEWART and MARIA TSAKIRI

School of Spatial Sciences, Cuptin University of Technology, SPO Box U 1387, Perth 6845 WA, Australia

The potential 48-satellite constellation offered by the combination of observations from both the GPS and GLONASS positioning systems has created considerable interest amongst existing GPS users. In the published literature, a considerable amount of work has been devoted to the theoretical issue of algorithm design for combined GPS/GLONASS positioning solutions. Little work has been published, however, on the practical conversion of existing GPS software to include GLONASS observations. This paper considers the computation issues pertaining to the GLONASS broadcast ephemeris for inclusion of GLONASS observations into existing GPS software. The format of the GLONASS broadcast ephemeris is discussed and theory of satellite orbits and their stepwise numerical integration is reviewed. Finally, a strategy for GLONASS broadcast ephemeris computation is proposed to facilitate combination of GPS and GLONASS observations. © 1998 John Wiley & Sons, Inc.

HffltODOCTSOW he two Global Satellite Navigation Systems, GPS and GLONASS, have attracted considerable interest amongst the geodetic and navigation community. While GPS has been fully operational for several years, considerable doubt remains over the future of GLONASS. At the time of writing, only 13 satellites of the full 24 GLONASS constellation are operational. However, the potential addition of a fully operational GLONASS system to GPS, creating a 48 satellite constellation if GPS and GLONASS observations can be inte-

GPS Solutions, Vol. 2, No. 2, pp. 16-27 (1998) © 1998 John Wiley & Sons, Inc. CCC 1080-5370/98/020016-12

grated directly, has many attractions, such as improved reliability for real time positioning techniques and increased coverage in areas of limited satellite visibility. Recent developments in the hardware and algorithm design for the processing of the combined observations from both systems (e.g., Beser et al, 1995; Gourevitch et al, 1996; Heinrichs et al., 1996; Walsh et al., 1996) have demonstrated the advantages of the combined system. The success of GPS has, ironically, served to highlight its deficiencies. Many research groups around the world are now concerned with pushing the boundaries of satellite positioning into areas where GPS cannot reliably operate as a stand-alone system. A common strategy, particularly among the navigation community, is to supplement GPS observations with measurements from other sensors, such as dead reckoning or inertial systems (e.g., Chen, 1992; Tsakiri, 1995). GLONASS observations offer an additional source of measurements with which to supplement GPS. Therefore, any research group working in the field of satellite positioning will wish, at some stage, to modify existing GPS software to incorporate GLONASS observations. The GPS and GLONASS systems are very similar in terms of design. Raw GPS and GLONASS code and carrier phase measurements represent ranges from a receiver to a satellite. To use these measurements for position computation, the locations of the GPS and GLONASS satellites must be known. The locations of the satellites are known from the GPS or GLONASS broadcast ephemerides. Despite many similarities, GPS and GLONASS are autonomous systems, each with its own reference system for time and space. GPS and GLONASS signal structure is very similar as both employ codes at the L-band frequencies. However, GPS satellites are distinguished by the code division multiple access characteristic;

GLONASS uses the method of frequency division multiple access. This means that GPS satellites transmit on the same frequency using different codes to distinguish between satellites whilst GLONASS satellites use the same code sequence but transmit on different frequencies. Furthermore, the internal structure of GLONASS signals is different to GPS. For example, while GLONASS code or carrier phase observations are very similar to GPS observations, the broadcast ephemeris format of the GLONASS navigation message is different to GPS. The aim of this paper is to illustrate a procedure for computing satellite positions from the GLONASS broadcast ephemeris that can be directly used to process GLONASS data in conjunction with GPS data. The paper is essentially a review paper, in that no new theories or models are proposed. Instead, the paper will concentrate on the practical issues of implementing the GLONASS broadcast ephemeris into existing GPS processing software. First, the structure of the GLONASS broadcast ephemeris will be reviewed, followed by a discussion of the orbital force model as recommended by the GLONASS Interface Control Document (ICD-95). The algorithms for the numerical integration of the ICD-95 force model are quoted explicitly. Some comments and tests related to the accuracy of the integrator are also given. The effect of differences between GLONASS and GPS system times and reference frames are discussed and some final comments regarding further issues in GPS/GLONASS system integration are made.

ilOSMSS BROADCAST EPMERIS The GLONASS broadcast ephemeris, while essentially containing similar information, is structured somewhat differently to the GPS broadcast ephemeris. The fundamental information in the GPS broadcast ephemeris is given in terms of the Kepler elements described in standard GPS textbooks (e.g., Leick, 1995, p. 38-41). The GPS broadcast ephemeris is updated hourly and is nominally valid for two hours either side of its specified reference time. The GLONASS almanac, containing information not necessary for computation of instantaneous satellite positions, is also transmitted in terms of Kepler elements. However, the GLONASS broadcast ephemeris information is typically transmitted as a half-hourly satellite state vector, expressed in PZ90 geocentric cartesian coordinates. The elements of the GLONASS broadcast ephemeris relevant to satellite orbital computation are summarised in Table 1.

A full description of the format and information contained in the GLONASS navigation message can be found in ICD-95. The ephemeris parameters are calculated by the GLONASS ground control segment and uploaded to the GLONASS satellites, typically at 30-min intervals. In Table 1, the reference time tb represents the time within the current day that the following information is referred to. This time is transmitted in the GLONASS system time frame. tb is always given at discrete 15-min intervals relative to the top of the hour and is equivalent to the reference time of a broadcast ephemeris block for an individual GLONASS satellite. Additional flags in the ephemeris can indicate changes to the 30-min update interval. For further information, the reader is again referred to ICD-95. The mean square values of errors in the GLONASS broadcast satellite position and velocity vectors are quoted by ICD-95 as 20 m and 0.05 cm/s in the along track component, 10 m and 0.1 cm/s in the cross track component, and 5 m and 0.3 cm/s in the radial component. It may be pointed out that the reliability of GLONASS orbits can be somewhat variable. Occasions when either some or all terms in a satellite navigation message are set to zero happen sporadically. It is therefore possible for the receiver to measure a full set of code and carrier phase data for an individual satellite but have little or no ephemeris information for that satellite. For further analysis of the reliability of the GLONASS system, the reader is referred to Holmes and Trethewey (1997), GPS/GLONASS Working Group (1996), Misra et al. (1996b), and Daly (1995).

ORBITAL FORCE MODEL Position computation using raw GPS/GLONASS code or carrier phase measurements requires the position of each GPS/GLONASS satellite at a time corresponding to each observation time. Given a GLONASS satellite state vector at time tb (or the equivalent GPS set of Keplerian elements with reference time tb), the orbit must be calculated for time tt, the time at which the raw GPS/ GLONASS observations have been made. Note that tf actually represents the time the signal left the satellite for any particular observation (transmission time), rather than the time at the receiver (reception time). It is also noted that tb and tt are representative of the respective GPS and GLONASS system time frames. For GPS, the computation of a satellite position in WGS84 cartesian coordinates (x, y, z) at time tl from Keplerian elements with reference time tb is well documented (see, e.g., Leick, 1995; Kaplan, 1996; Parkinson et al, 1996). For GLONASS, the computation of a satellite position at time tt from a satellite state vector with reference time tb requires the numerical integration of an orbital force model. Here, an important difference between the representation of GPS and GLONASS broadcast ephemerides becomes apparent. While GPS ephemerides are updated hourly, any single GPS ephemeris is valid for a 4-h time span. GLONASS ephemerides are updated half-hourly but are only valid for as long as the force model is valid. In principle, with half-hourly ephemeris updates, the longest integration time period required should be 15 min forward and backward from the current state vector. Therefore, a relatively simple orbital force model is required. Such an orbital force model is given in ICD-95 and is developed in the subsequent section. However, in situations where halfhourly ephemeris updates are not available, possibly due to problems in the navigation message or where integration is required beyond the first or last recorded ephemerides, a simplified orbital force model may not be valid. More complex models may be used, at the cost of decreased computational efficiency. The derivation of the simplified force model recommended in ICD-95 is given below, primarily to illustrate the assumptions incorporated in the model and the limitations imposed by the model on the GLONASS broadcast ephemeris.

Derivation of SCD-S5 simplified orbital force mode! The orbit of a satellite around the Earth is governed by the forces acting on that satellite. The primary force is

that due to the Earth's gravity field, the potential of which can be written as:

where [7 is the part of the potential of the Earth's field departing from radiality, r is the distance from the Earth's geocenter to the center of mass of the satellite, G is the universal gravitational constant, and M is the mass of the Earth. Expanding U into spherical harmonics gives (Heiskenan and Moritz, 1967):

is the Earth's equatorial radius; are Earth-fixed polar coordinates (geocentric radius, longitude and colatitude); are the degree and order of the spherical harmonic expansion; are associated Legendre functions; are spherical harmonic coefficients. The zonal harmonics (those independent of latitude, 6) are much more significant to satellite orbits than the tesseral harmonics (those dependent on longitude, X). The first assumption to be made is that the contribution of the tesseral harmonics over a short time period is insignificant. Therefore, Equation (2) can be rewritten as (ibid):

which depends on the zonal harmonics /„ and is a function of r and 0 only. Other perturbing forces act on a satellite, such as the gravitational effect of the sun, moon and planets, and solar radiation pressure. Table 2, taken from Parkinson et al. (1996), gives an indication of the relative magnitude of these perturbing forces. It can be seen that relative to /2 term, all other perturbing forces are small, although their effects do cumulate with time. The effects of lunar and solar gravity

(8)

It is seen from the previous derivation that the ICD95 orbital force model neglects minor forces such as nonzonal components of the Earth's gravity field and solar radiation pressure. Furthermore, Equation (9) models only the radial component of satellite motion, neglecting the colatitude dependent component which, in itself, is not insignificant with time. Justifiably, the ICD-95 force model is recommended for only a time period of 15 min either side of the broadcast ephemeris reference time. However, it will subsequently be shown that the integration time period can be extended beyond 15 min with a relatively small degradation in satellite coordinate quality.

The equations of satellite motion given by the ICD-95 model are second-order differential equations, given as a function of time, position, and velocity:

where Rt is any one of the components of the satellite position vector (x, y, z] and Rg are the corresponding acceleration components, r, X, 9 are as defined in Equation (2) above with:

The acceleration vector is integrated twice, via Equations (lla) and (lib), to give satellite position. However, prior to integration, a further issue regarding the force model given in Equation (9) must be addressed. In integrating Equations (lla) and (lib) in PZ90, a critical assumption is being made that PZ90 is an inertial (i.e., nonrotating, nonaccelerating) reference system. This is because Newton's laws of motion, which are represented in Equation (9), are valid only in an inertial reference frame. This is clearly not true as PZ90 rotates at the same rate as the daily rotation of the Earth. Therefore, a rigorous integration would first have to transform the satellite state vectors into an inertial reference frame, perform the integration, then transform the solution back from the inertial reference frame to the conventional reference frame. The transformation from earth-fixed to inertial reference frame is given below (after Agrotis, 1984):

which is the final form of the force model given in ICD95. Here, to represents the rotation rate of the Earth (radians/second). The concept of the integration of the GLONASS ephemerides is illustrated in Figure 1. State vectors given at half-hourly reference times are integrated backwards and forwards for 15 min each, giving a continuous representation of the orbit over the complete time scale. If necessary, the first and last state vectors can be extrapolated backwards and forwards beyond the recommended 15-min intervals, although, as discussed above, extended extrapolation will degrade the quality of the computed satellite coordinates.

The Runge-Kutta method of numerical integration ICD-95 recommends the numerical integration be carried out by the fourth-order Runge-Kutta method. This is one of the most popular and accurate numerical procedures capable of obtaining approximate solutions to differential equations and is documented in many standard mathematical texts (e.g., Zill and Cullen, 1992). Single step methods, such as the Runge-Kutta method, employ the current value of the desired quantity to predict the value of that quantity at the following epoch. The Runge-Kutta method has the form: backward integration

Over relatively short time periods, only the rotation of the Earth will cause an ECEF reference frame to differ from an inertial reference frame, as the other factors represent long period variations. Therefore, equation (9) must be modified to account for the rotation of the Earth (assumed to be about the cartesian z-axis) over the integration time period. Equation (9) becomes (setting Pr, N, and P to identity matrices) :

where kt are evaluated from /(t, f, r) and a, b, c, d are constants. These coefficients are evaluated from a Taylor's expansion of the components of fn+l around fn. In the following, we follow the notation of Agrotis (1984). Applying the fourth Runge-Kutta method to the ICD-95 orbital force model is relatively straightforward. To compute the components of acceleration, velocity, and position vectors at time ti+l, we use the satellite state vector components at time tt such that: The coefficients Knm depend on the functions flt f2, . . . f e , from Equations (17a) and (17b), and on the integration step length h, which is given in seconds. Appendix 1 explicitly states the equations for the computation of the coefficients Knm.

Error in Runge-Kutla numerical integration

The full 4th order Runge-Kutta formulae for the double integration of the 2nd order differential Equations (17a) and (17b), written in the form of Equation (14), are:

The main user-defined variable in Runge-Kutta numerical integration is the integration step length, h. Two factors influence the accuracy of the numerical integration. First, is the validity of the force model over the integration time period. Second, is the magnitude of the integration interval itself, which will influence the size of the neglected higher order terms in the Taylor expansion discussed in conjunction with Equation (14). In practice, it is difficult to separate these errors. Both factors will cause errors in the computed satellite position and velocity, which, because of the stepwise nature of the integration process, will lead to greater error in the computed positions and velocities of later integration step. The error is cumulative, so it is logical to assume that errors will be minimized by integrating with a small step length over a relatively short time period. Fourth-order numerical integration can be somewhat numerically intensive. Four function evaluations must be made for each of the six components of the satellite state vector at each step. Obviously, these computations must also be made for every visible satellite. Furthermore, for positioning, processing software requires the satellite position at the time the signal left the satellite, rather than the time the observation was made by the receiver. The times at which the satellite coordinates are required are, therefore, not necessarily at regular intervals. Additionally, for differencing techniques, satellite coordinates must be computed separately for each individual receiver, as the time the signal

left the satellite is slightly different for measurements made simultaneously at each receiver. The obvious approach is to integrate in equal steps up to the observation time, then make an extra RungeKutta integration for the final fractional step. However, a valid step length must be chosen to minimize integration errors and maintain numerical efficiency. For example, for a 30-s data interval, it would be more efficient, though less accurate, to integrate using a 30-s step length rather than a 1-s step length. Figure 2 shows the error between the 15-min forward and backward integration of adjacent 30-min GLONASS broadcast ephemerides (cf Figure 1) for different integration step lengths. Step lengths in Figure 2 have been chosen as multiples of 15 min, from a step length of 0.1 s up to a maximum of 60 s, which is the slowest data sampling rate in common use. The comparison between the forward and backward integrations has been made exactly at the midpoint between the two adjacent 30-min ephemerides. Figure 2 illustrates the error in x, y, and z components. For a 0.1-s integration interval, the differences between the integrated forward and backward solutions at the 15-min midpoint are -0.5 m, 1.2 m, and 1.0 m, respectively. A step length of 30 s yields errors in x, y, and z of - 0.7 m, 4.0 m, and 2.5 m, and a step length of 60 s has errors of - 1.0 m, 6.8 m, and 4.0 m, respectively. As expected, the integrated solution degrades with increasing step length. As integration error is minimized at short step lengths, the magnitude of the error given above for a 0.1-s integration interval can be said to rep-

resent the approximations made in the orbital force model over a 15-min integration period. Figure 3 illustrates how the degradation of the integrated orbits with time can be estimated using a sequence of 30-min GLONASS broadcast ephemeris state vectors. Figure 4 illustrates the magnitude of orbital error, for an individual satellite in the x, y, and z components, from the ICD-95 force model over several hours. Orbital positions have been computed at an integration step length of 1 s and compared with half-hourly broadcast ephemeris updates. In Figure 4, after 30 min the error in the integrated orbit in this example is - 4.7 m, 2.5 m, and - 0.9 m in x, y, and z, respectively. It is clear from Figure 4 that the approximations of the ICD-95 force model cause an accumulation of orbital error as the time interval of the numerical integration increases. This could be compounded by the choice of integration step length. While extending the integration period of the ICD-95 model beyond the ± 15 min recommended in the ICD-95 document can yield errors of less than 20 m over 1 h, a more detailed force model, such as including changes in the solar or lunar gravitational forces, would give a more precise orbit. From Figure 2, it is clear that integration step lengths of similar time spans to the data update rate will yield errors of only a few meters, even for data interval of 60 s or greater. For relative positioning over short baselines, where the majority of orbital errors are differenced away, such an error is acceptable. It must be stressed that errors caused by force model approxima-

tions and the approximations inherent in the RungeKutta method are additional to the errors in the broadcast state vectors themselves. Over longer baselines, it will be desirable to minimize integration errors by using as short a step length as possible, as orbital errors will become more significant in the final baseline solution.

ments), while GPS "follows" UTC time of the U.S. Naval Observatory (USNO) but without leap seconds adjustments. Currently, the GLONASS data message transmits a parameter which relates GLONASS system time to UTC(SU) and is updated to the satellites approximately once a day. To equate the two time systems, GLONASS measurements at UTC(SU) time need to be corrected for the offset between UTC(USNO) and UTC(SU) followed by the leap second correction between UTC(USNO) and GPS system time. Furthermore, GLONASS time is 3 h ahead of UTC(SU). The relationship between GLONASS time and GPS time is monitored and regularly published by the Bureau Internationale des Poids et Mesures in Paris (BIPM). An example of BIPM time offsets for February 1998 is given in Table 3. Using the most recent updates, it is possible to estimate the difference between the two time systems. Neglecting satellite clock error and relativistic effects, the reference time for GLONASS broadcast ephemeris state vectors can be converted to GPS time using (with reference to Table 3):

GLONASS ORBIT COMPUTATION FOR 6PS/GLONASS DATA PROCESSING GLONASS ephemeris information is available every 30 min. However, similarly to GPS observations, GLONASS phase and code observations are usually logged at intervals of between 1 s and 30 s. For data processing, it is necessary to compute GLONASS satellite positions at times equating to the observations made by the receiver. As a GLONASS receiver is likely to be simultaneously making GPS measurements, it is logical to compute GLONASS orbits as accurately as possible in the WGS84 reference frame and the GPS time system. By computing GLONASS orbits in this way, the problems of direct integration of GPS and GLONASS observations are simplified. The primary issues are therefore to: a) convert the GLONASS reference time for that state vector tb from UTC(SU) to GPS time, b) rotate the GLONASS satellite position from PZ90 to WGS84.

Time corrections GLONASS space vehicles use GLONASS system time which is referenced to Universal Coordinated Time of Soviet Union UTC(SU) (including leap second adjust-

where ?6(GLONAss)ls the broadcast reference time of the GLONASS broadcast ephemeris and taps is the equivalent GPS time. Correction of r&CGLONASS) to jfGPS for the broadcast ephemeris state vector allows the orbital integration to be performed in GPS time. This has obvious advantages in that the time offset between systems can be applied directly as part of the GLONASS broadcast orbit computation. The satellite clock offset is linearly interpolated from the broadcast ephemerides and could also be applied at this stage. Although the Russian authorities have proposed including differences between the two time and reference systems in the navigation message (Langley, 1997), to date this information is not available in real-time. Until this is the case, some uncertainty will always exist in the difference between time systems when integrating GPS and GLONASS. The time scale issue can also be resolved without other external information, by using a measurement to estimate the instantaneous bias between the two scales. One approach in relative positioning is to only differentiate observations between similar systems, essentially using a GPS reference satellite and a GLONASS reference satellite. Other approaches esti-

mate the offset as part of the least squares adjustment procedure. Further discussion of the timing issue can be found in, for example, Walsh and Daly (1996) and Kozlov and Tkachenko (1997).

FZ90 - WSS84 coordinate transformation Given the PZ90 coordinates of a GLONASS satellite, a simple transformation is required to convert these coordinates to WGS84 and hence make them compatible with GPS. However, the values of the transformation parameters between PZ90 and WGS84 are still relatively uncertain, primarily due to a lack of known points worldwide in the PZ90 reference frame. To date, two estimates of transformation parameters have been published, by Misra et al. (1996a) and Rossbach et al. (1996).

Both solutions reveal only small rotations about the z axis, with the solution of Mirsa et al. suggesting a 2.5 m origin shift along the y-axis. The PZ90 - WGS84 transformation can be expressed as:

G! = 1.9 X 10~ 6 radians (Misra et al, 1996a); 1.6 X 10~ 6 radians (Rossbach et al., 1996). In practice, for short baselines the effect of neglecting the difference in coordinate reference frame will be small. For longer baselines, however, the effect will become significant. It may be hoped that the forthcoming IGEX98 campaign (Willis et al., 1998) will go some way to resolving PZ90 - WGS84 transformation parameters with more certainty. While error may be small on short baselines, it may be argued that applying either of the above transformation parameter sets is more rigorous than simply ignoring the problem when integrating GPS and GLONASS.

SUMMARY AND FINAL REMARKS GLONASS broadcast ephemerides are broadcast in a different format to GPS. They are transmitted as satellite state vectors every 30 min and must be numerically integrated to give satellite positions for raw code and carrier phase observations. The GLONASS Interface Control Document (ICD-95) gives a simplified orbital force model which is accurate to around 1 m in the x, y and z components over a ± 15-min interval. This model has the advantage that it is more numerically efficient than more complex models (such as models which take the relative motion of the sun and the moon into account), although errors can accumulate beyond the specified ± 15-min interval. In the example given, extending the total integration period to 30 min caused the error to increase from 1 m to 4 m in the y component. Situations where the total integration period is required to be greater than the requisite ±15 min may arise if GLONASS broadcast ephemeris outages occur. In such cases, users must be aware of the degrading quality of the integrated orbit if the ICD-95 force model is being used. The fourth-order Runge-Kutta method of numerical integration inherently involves approximations and adds further errors to GLONASS satellite orbit determination. Choice of integration step length should be as short as possible to minimize integration errors. It is suggested that a step length of between 1 and 30 s is adequate for short baselines. For long, high precision baselines where orbital errors become more significant, a 1 s step length is recommended. For combination of GLONASS with GPS observations, it is possible to create GLONASS orbits directly compatible with GPS by performing the GLONASS broadcast orbit integration in the GPS time system and

applying the PZ90 to WGS84 coordinate transformaton to the integrated GLONASS positions. The offset between GPS and GLONASS time systems is not currently available in real time so this approach is suitable only for post-processed solutions. However, the prospect of the relevant information being broadcast in the GLONASS navigation message sometime in the future means that this approach could become applicable for high accuracy real time systems, as opposed to, for example, sacrificing an observation to resolve the time offset. Once the reference system and time offsets between GLONASS and GPS have been resolved, GLONASS code data processing can be combined directly with GPS code processing, with GLONASS satellite coordinates and clock errors treated as if they were GPS. The practical issues of combining GPS and GLONASS carrier phase data are somewhat more complicated and have been covered by Leick (1998). •

APPENDIX 1

Rynge-Kutta method parameters for numerical Integration of satellite orbits (after Agrotss, 1884)

Kozlov, D., and M. Tkachenko. 1997. Instant RTK cm with low cost GPS and GLONASS C/A receivers, Proceedings of the IONGPS-97, September 16-19, Kansas City, MO, USA, 1559-1570. Landau, H., and U. Vollath. 1996. Carrier phase ambiguity resolution using GPS and GLONASS signals, Proceedings of the ION-GPS-96, September 17-20, Kansas City, MO, USA, 917923. Langley, R.B. 1997. GLONASS: Review and update, GPS World, 8(7), 46-51. Leick, A. 1995. GPS Satellite Surveying, Wiley, New York. Leick, A. 1998. GLONASS satellite surveying, Journal of Surveying Engineering, May, 91-99.

REFERENCES Agrotis, L.G. 1984. Determination of satellite orbits and the global positioning system, Ph.D. thesis, The University of Nottingham, UK. Beser, J., A. Balendra, E. Erpelding, and S. Kim. 1995. Differential GLONASS, differential GPS and integrated differential GLONASS/GPS—Initial results, Proceedings oftheION-GPS-95, September 12-15, Palm Springs, CA, USA, 507-515.

Misra, P., R. Abbot, and E. Gaposchkin. 1996a. Integrated use of GPS and GLONASS: Transformation between WGS84 and PZ90, Proceedings of the ION-GPS-96, September 17-20, Kansas City, MO, USA, 307-314. Misra, P., M. Pratt, R. Muchnik, B. Burke, and T. Hall. 1996b. GLONASS performance: Measurement data quality and system upkeep, Proceedings of the ION-GPS-96, September 17-20, Kansas City, MO, USA, 261-270. Parkinson, B. and J. Spilker (Eds). 1996. Global Positioning System: Theory and Applications Volume I. Progress in Astronautics and Aeronautics, Volume 163, Washington, DC, USA.

Daly, P. 1995. GLONASS approaches full operational capability, Proceedings of the ION-GPS-95, September 12-15, Palm Springs, CA, USA, 1021-1025.

Remondi, B. 1989. Extending the National Geodetic Survey Standard GPS Orbit Formats, NOAA Technical Report NOS 133 NGS 46, Rockville, MD, November.

Chen, W. 1992. Integration of GPS and INS for precise surveying applications, Ph.D. thesis, The University of Newcastle upon Tyne, UK.

Pratt, M., B. Burke, and P. Misra. 1997. Single-epoch integer ambiguity resolution with GPS-GLONASS LI data, Proceedings of the 53rd Annual Meeting of the Institute of Navigation, June 30-July 2, Albuquerque, NM, 691-699.

Gourevitch, S., S. Sila-Novitsky, and F. van Diggelen. 1996. The GG24 combined GPS + GLONASS receiver, Proceedings of the ION-GPS-96, September 17-20, Kansas City, MO, USA, 141145.

Rapoport, L. 1997. General purpose kinematic/static. GPS/ GLONASS postprocessing engine, Proceedings of the ION-GPS97, September 16-19, Kansas City, MO, USA, 1757-1772.

GPS/GLONASS Working Group. 1996. A report on the first meeting of the GPS/GLONASS Interoperability Working Group, September 17-20, Kansas City, MO, USA. http://www.ion.org/ workgroupmin.html.

Rossbach, U., H. Habrich, and B. Zarraoa. 1996. Transformation parameters between PZ90 and WGS84, Proceedings of the ION-GPS-96, September 17-20, Kansas City, MO, USA, 279285.

Heinrichs, G. and S. Gotz. 1996. The GNSS-200: Features and performance of the fully integrated single frequency, 12 Channel, C/A code GPS/GLONASS receiver, Proceedings of the ION-GPS-96, September 17-20, Kansas City, MO, USA, 125130.

Tsakiri, M. 1995. GPS and DR for land vehicle navigation, Ph.D. thesis, The University of Nottingham, UK.

Heiskenan, W. and H. Moritz. 1967. Physical Geodesy, WH Freeman, San Francisco, CA. Holmes, D. and M. Trethewey. 1997. NAGUs: An analysis of GLONASS availability statistics, Proceedings oftheION-GPS-97, September 16-19, Kansas City, MO, USA, 1521-1525. ICD-95 (GLONASS Interface Control Document). 1995. International Civil Aviation Organisation (ICAO), RTCA Paper No. 639-95/SC159-685, GNSSP/2-WP/66.

Tsakiri, M., M. Stewart, T. Forward, D. Sandison, and J. Walker. 1998. Urban fleet monitoring with GPS and GLONASS, Journal of Navigation, UK (in press). Walsh, D., S. Riley, J. Cooper, and P. Daly. 1996. Precise positioning using GPS/GLONASS carrier phase and code phase observables, Proceedings of the ION-GPS-96, September 17-20, Kansas City, MO, USA, 499-506. Walsh, D., and P. Daly. 1996. GPS and GLONASS carrier phase ambuigity resolution, Proceedings of the ION-GPS-96, September 17-20, Kansas City, MO, USA, 899-907.

Kealy, A., M. Stewart, M. Tsakiri, and L. Woolhouse. 1998. An evaluation of GLONASS for maritime positioning, The Hydrographic Journal, 88, 19-22.

Willis, P., G. Beutler, W. Gurtner, G. Hein, R. Neilan, and J. Slater. 1998. The international GLONASS experiment IGEX-98, September 20-December 20, 1998. Final Version (March 6). http://igscb.jpl.nasa.gov/igscb/mail/. May 18, 1998.

Kaplan, E. 1996. Understanding GPS Principles and Applications. Artech House Publishers, Norwood, MA.

Zill, D.G., and M.R. Cullen. 1992. Advanced Engineering Mathematics, PWS Publishing, Boston, MA.

BIOGRAPHIES Mike Stewart has resided in Australia since 1994 and is a senior lecturer in GPS and Geodesy at Curtin University. He holds a degree in geophysics from the University of Liverpool (UK) and was awarded a Ph.D. from the University of Edinburgh (UK) in 1990. He has been active in GPS software and algorithm development since commencing postdoctoral work at the University of Nottingham in 1990.

Maria Tsakiri was appointed as a lecturer at Curtin University following the completion of her Ph.D. research in 1995 at the Institute of Engineering Surveying and Space Geodesy (IESSG) in the University of Nottingham, UK. She holds a degree in Engineering Surveying (1992) from the National Technical University of Athens, Greece. Her principal research area is in kinematic GPS/GLONASS and navigation.

GLONASS Broadcast Orbit Computation

27