The Chinese BeiDou system (BDS), having different types of satellites, is an important addition to the ever growing system of Global Navigation Satellite Systems (GNSS). It consists of Geostationary Earth Orbit (GEO) satellites, Inclined Geosynchronous Satellite Orbit (IGSO) satellites and Medium Earth Orbit (MEO) satellites. This paper investigates the receiver-dependent bias between these satellite types, for which we coined the name “inter-satellite-type bias” (ISTB), and its impact on mixed receiver attitude determination. Assuming different receiver types may have different delays/biases for different satellite types, we model the differential ISTBs among three BeiDou satellite types and investigate their existence and their impact on mixed receiver attitude determination. Our analyses using the real data sets from Curtin’s GNSS array consisting of different types of BeiDou enabled receivers and series of zero-baseline experiments with BeiDou-enabled receivers reveal the existence of non-zero ISTBs between different BeiDou satellite types. We then analyse the impact of these biases on BeiDou-only attitude determination using the constrained (C-)LAMBDA method, which exploits the knowledge of baseline length. Results demonstrate that these biases could seriously affect the integer ambiguity resolution for attitude determination using mixed receiver types and that a priori correction of these biases will dramatically improve the success rate.
References
[1]
CSNO. BeiDou Navigation Satellite System Signal in Space Interface Control Document: Open Service Signal B1I. Version 1.0; China Satellite Navigation Office: Beijing, China, 2012.
[2]
Cao, C.; Jing, G.; Luo, M. COMPASS Satellite Navigation System Development. PNT Challenges and Opportunities Symposium, Stanford, CA, USA, 5–6 November 2008.
[3]
Grelier, T.; Ghion, A.; Dantepal, J.; Ries, L.; de Latour, A.; Issler, J.L.; Avila-Rodriguez, J.; Wallner, S.; GW, H. COMPASS signal structure and first measurements. ION GNSS 2007, 2007, 3015–3024.
[4]
Chen, H.; Huang, Y.; Chiang, K.; Yang, M.; Rau, R. The performance comparison between GPs and BeiDou-2/COMPASS: A perspective from Asia. J. Chin. Inst. Eng. 2009, 32, 679–689, doi:10.1080/02533839.2009.9671550.
[5]
Yang, Y.; Li, J.; Xu, J.; Tang, J.; Guo, H.; He, H. Contribution of the Compass satellite navigation system to global PNT users. Chin. Sci. Bull. 2011, 56, 2813–2819, doi:10.1007/s11434-011-4627-4.
[6]
Zhang, S.; Guo, J.; Li, B.; Rizos, C. An Analysis of Satellite Visibility and Relative Positioning Precision of COMPASS. Proceedings of Symposium for Chinese Professionals in GPS, Shanghai, China, 18–20 August 2011; pp. 41–46.
[7]
Verhagen, S.; Teunissen, P.J. Ambiguity resolution performance with GPS and BeiDou for LEO formation flying. Adv. Space Res. 2013, doi:10.1016/j.asr.2013.03.007.
[8]
Montenbruck, O.; Hauschild, A.; Steigenberger, P.; Hugentobler, U.; Teunissen, P.J.G.; Nakamura, S. Initial assessment of the COMPASS/BeiDou-2 regional navigation satellite system. GPS Solut. 2013, 17, 211–222, doi:10.1007/s10291-012-0272-x.
[9]
Montenbruck, O.; Hauschild, A.; Steigenberger, P.; Hugentobler, U.; Riley, S. A COMPASS for Asia: First Experience with the BeiDou-2 Regional Navigation System. IGS Workshop, Olsztyn, Poland, 23–27 July 2012.
[10]
Shi, C.; Zhao, Q.; Li, M.; Tang, W.; Hu, Z.; Lou, Y.; Zhang, H.; Niu, X.; Liu, J. Precise orbit determination of BeiDou Satellites with precise positioning. Sci. China Earth Sci. 2012, 55, 1079–1086, doi:10.1007/s11430-012-4446-8.
[11]
Steigenberger, P.; Hauschild, A.; Montenbruck, O.; Hugentobler, U. Performance Analysis of COMPASS Orbit and Clock Determination and COMPASS-only PPP. IGS Workshop, Olsztyn, Poland, 23–27 July 2012.
[12]
Odolinski, R.; Teunissen, P.J.G.; Odijk, D. An Analysis of Combined COMPASS/BeiDou-2 and GPS Single- and Multiple-Frequency RTK Positioning. Proceedings of the Institute of Navigation Pacific PNT 2013, Honolulu, HI, USA, 22–25 April 2013; pp. 69–90.
[13]
Cai, C.; Gao, Y.; Pan, L.; Dai, W. An analysis on combined GPS/COMPASS data quality and its effect on single point positioning accuracy under different observing conditions. Adv. Space Res. 2013, doi:10.1016/j.asr.2013.02.019.
[14]
Li, W.; Teunissen, P.J.G.; Zhang, B.; Verhagen, S. Precise Point Positioning Using GPS and Compass Observations. Proceedings of the 4th China Satellite Navigation Conference (CSNC), Wuhan, China, 15–17 May 2013.
[15]
Zhao, Q.; Guo, J.; Li, M.; Qu, L.; Hu, Z.; Shi, C.; Liu, J. Initial results of precise orbit and clock determination for COMPASS navigation satellite system. J. Geod. 2013, 87, 475–486, doi:10.1007/s00190-013-0622-7.
[16]
Nadarajah, N.; Teunissen, P.J.G.; Buist, P.; Steigenberger, P. First Results of Instantaneous GPS/Galileo/COMPASS Attitude Determination. Proceedings of the 6th ESA Workshop on Satellite Navigation User Equipment Technologies, Noordwijk, The Netherlands, 5–7 December 2012; p. 8.
[17]
Nadarajah, N.; Teunissen, P.J.G. Instantaneous GPS/BeiDou/Galileo Attitude Determination: A Single-Frequencyr Robustness Analysis under Constrained Environments. Proceedings of The Institute of Navigation Pacific PNT, Honolulu, HI, USA, 22–25 April, 2013; pp. 1088–1103.
[18]
Nadarajah, N.; Teunissen, P.J.G.; Raziq, N. Instantaneous COMPASS-GPS attitude determination: A robustness analysis. Adv. Space Res 2013. submitted.
[19]
Shi, C.; Zhao, Q.; Hu, Z.; Liu, J. Precise relative positioning using real tracking data from COMPASS GEO and IGSO satellites. GPS Solut. 2013, 17, 103–119, doi:10.1007/s10291-012-0264-x.
[20]
Steigenberger, P.; Hauschild, A.; Montenbruck, O.; Rodriguez-Solano, C.; Hugentobler, U. Orbit and clock determination of QZS-1 based on the CONGO networkION-ITM-2012. 2012.
[21]
Steigenberger, P.; Hugentobler, U.; Hauschild, A.; Montenbruck, O. Orbit and clock analysis of Compass GEO and IGSO satellites. J. Geod. 2013, 87, 515–525, doi:10.1007/s00190-013-0625-4.
[22]
Cohen, C. Attitude Determination Using GPS. Ph.D. Thesis, Stanford University, Stanford, CA, USA, 1992.
[23]
Lu, G. Development of A GPS Multi-Antenna System for Attitude Determination. Ph.D. Thesis, University of Calgary, Calgary, Alberta, Canada, 1995.
[24]
Crassidis, J.L.; Markley, F.L. New algorithm for attitude determination using Global Positioning System signals. J. Guidance Control Dyn. 1997, 20, 891–896, doi:10.2514/2.4162.
[25]
Li, Y.; Zhang, K.; Roberts, C.; Murata, M. On-the-fly GPS-based attitude determination using single- and double-differenced carrier phase measurements. GPS Solut. 2004, 8, 93–102.
[26]
Lin, D.; Voon, L.; Nagarajan, N. Real-time Attitude Determination for Microsatellite by LAMBDA Method Combined with Kalman Filtering. Proceedings of the 22nd AIAA International Communications Satellite Systems Conference and Exhibit, Monterey, California, USA, 9–12 May 2004.
[27]
Madsen, J.; Lightsey, E.G. Robust spacecraft attitude determination using global positioning system receivers. J. Spacecr. Rocket. 2004, 41, 635–643, doi:10.2514/1.1324.
[28]
Psiaki, M.L. Batch algorithm for global-positioning-system attitude determination and integer ambiguity resolution. J. Guidance Control Dyn. 2006, 29, 1070–1079, doi:10.2514/1.18351.
[29]
Teunissen, P.J.G. The least-squares ambiguity decorrelation adjustment: A method for fast GPS integer ambiguity estimation. J. Geod. 1995, 70, 65–82, doi:10.1007/BF00863419.
[30]
Boon, F.; Ambrosius, B. Results of Real-time Applications of the LAMBDA Method in GPS Based Aircraft Landings. Proceedings of the International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Banff, AB, Canada, 3–6 June 1997; pp. 339–345.
[31]
Cox, D.B.; Brading, J.D.W. Integration of LAMBDA Ambiguity Resolution with Kalman Filter for Relative Navigation of Spacecraft. Proceedings of the 55th Annual Meeting of The Institute of Navigation, Cambridge, MA, USA, 27–30 June, 1999; pp. 739–745.
[32]
Ji, S.; Chen, W.; Zhao, C.; Ding, X.; Chen, Y. Single epoch ambiguity resolution for Galileo with the CAR and LAMBDA methods. GPS Solut. 2007, 11, 259–268, doi:10.1007/s10291-007-0057-9.
[33]
Huang, S.; Wang, J.; Wang, X.; Chen, J. The application of the LAMBDA method in the estimation of the GPS slant wet vapour. Acta Astron. Sinica 2009, 50, 60–68.
[34]
Kroes, R.; Montenbruck, O.; Bertiger, W.; Visser, P. Precise GRACE baseline determination using GPS. GPS Solutions 2005, 9, 21–31, doi:10.1007/s10291-004-0123-5.
[35]
Jin, S.; Luo, O.; Ren, C. Effects of physical correlations on long-distance GPS positioning and zenith tropospheric delay estimates. Adv. Space Res. 2010, 46, 190–195, doi:10.1016/j.asr.2010.01.017.
[36]
Jin, S.; Wang, J.; Park, P.H. An improvement of GPS height estimations-Stochastic modeling. Earth Planets Space 2005, 57, 253–259.
[37]
Park, S.Y. Thermally induced attitude disturbance control for spacecraft with a flexible boom. J. Spacecr. Rocket. 2002, 39, 325–328, doi:10.2514/2.3816.
[38]
Teunissen, P.J.G. An optimality property of the integer least-squares estimator. J. Geod. 1999, 73, 587–593, doi:10.1007/s001900050269.
[39]
Teunissen, P.J.G.; De Jonge, P.; Tiberius, C. Performance of the LAMBDA method for fast GPS ambiguity resolution. Navigation 1997, 44, 373–383.
[40]
Verhagen, S.; Teunissen, P.J.G. New global navigation satellite system ambiguity resolution method compared to existing approaches. J. Guidance Control Dyn. 2006, 29, 981–991, doi:10.2514/1.15905.
[41]
Park, C.; Teunissen, P.J.G. A New Carrier Phase Ambiguity Estimation for GNSS Attitude Determination Systems. Proceedings of International Symposium on GPS/GNSS, Tokyo, Japan, 15–18 November 2003; pp. 283–290.
[42]
Teunissen, P.J.G. The LAMBDA method for the GNSS compass. Artif. Satell. 2006, 41, 89–103.
[43]
Buist, P.J. The Baseline Constrained LAMBDA Method for Single Epoch, Single Frequency Attitude Determination Applications. Proceedings of the 20th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS 2007), Fort Worth, TX, USA, 25–28 September 2007; pp. 2962–2973.
[44]
Park, C.; Teunissen, P.J.G. Integer least squares with quadratic equality constraints and its application to GNSS attitude determination systems. Int. J. Control Autom. Syst. 2009, 7, 566–576, doi:10.1007/s12555-009-0408-0.
[45]
Giorgi, G.; Teunissen, P.J.G.; Buist, P.J. A Search and Shrink Approach for the Baseline Constrained LAMBDA Method: Experimental Results. Proceedings of International Symposium on GPS/GNSS, Tokyo, Japan, 11–14 November 2008; pp. 797–806.
[46]
Giorgi, G.; Buist, P. Single-epoch, Single-frequency, Standalone Full Attitude Determination: Experimental Results. Proceedings of the 4th ESA Workshop on Satellite Navigation User Equipment Technologies, NAVITEC, Noordwijk, The Netherlands, 10–12 December; p. 8.
[47]
Teunissen, P.J.G. Integer least-squares theory for the GNSS compass. J. Geod. 2010, 84, 433–447, doi:10.1007/s00190-010-0380-8.
[48]
Giorgi, G.; Teunissen, P.J.G.; Verhagen, S.; Buist, P.J. Testing a new multivariate GNSS carrier phase attitude determination method for remote sensing platforms. Adv. Space Res. 2010, 46, 118–129, doi:10.1016/j.asr.2010.02.023.
[49]
Teunissen, P.J.G.; Giorgi, G.; Buist, P.J. Testing of a new single-frequency GNSS carrier phase attitude determination method: Land, ship and aircraft experiments. GPS Solut. 2011, 15, 15–28, doi:10.1007/s10291-010-0164-x.
[50]
Teunissen, P.J.G.; Kleusberg, A. GPS for Geodesy, 2nd ed. ed.; Springer: Berlin/Heidelberg, Germany, 1998.
[51]
Harville, D.A. Matrix Algebra From A Statistician's Perspective; Springer: New York, NY, USA, 1997.
[52]
Magnus, J.R.; Neudecker, H. Matrix Differential Calculus with Applications in Statistics and Econometrics; Wiley: New York, NY, USA, 1995.
[53]
Teunissen, P.J.G. Zero Order Design : Generalized Inverses, Adjustment, the Datum Problem and S-transformations. In Optimization and Design of Geodetic Networks; Grafarend, E., Sans, F., Eds.; Springer: Berlin/Heidelberg, Germany, 1985; pp. 11–55.
[54]
Teunissen, P.J.G.; Odijk, D.; Zhang, B. PPP-RTK: Results of CORS network-based PPP with integer ambiguity resolution. J. Aeronaut. Astronaut. Aviat. Series A 2010, 42, 223–230.
[55]
Odijk, D.; Teunissen, P.J.G.; Zhang, B. Single-frequency integer ambiguity resolution enabled precise point positioning. J. Surv. Eng. 2012, 138, 193–202, doi:10.1061/(ASCE)SU.1943-5428.0000085.
[56]
Euler, H.J.; Goad, C. On optimal filtering of GPS dual frequency observations without using orbit information. J. Geod. 1991, 65, 130–143.
[57]
Geng, J.; Meng, X.; Dodson, A.; Teferle, F. Integer ambiguity resolution in precise point positioning: Method comparison. J. Geod. 2010, 84, 569–581, doi:10.1007/s00190-010-0399-x.
[58]
Zhang, B.; Teunissen, P.J.G.; Odijk, D. A novel un-differenced PPP-RTK concept. J. Navig. 2011, 64, S180–S191, doi:10.1017/S0373463311000361.
