Abstract
Complex urban environments limit the reception of global navigation satellite system (GNSS) signals, increasing the difficulty of ambiguity resolution (AR) for GNSS attitude determination based on carrier-phase observations. To constrain the Least-squares Ambiguity Decorrelation Adjustment (LAMBDA) and improve the success rate and accuracy of AR, a fixed length between the antennas is proposed. Heading angle- and length-constrained LAMBDA (HLC-LAMBDA) and converted baseline vector-constrained LAMBDA (VC-LAMBDA) were proposed because baseline length-constrained LAMBDA (BC-LAMBDA) cannot guarantee three-dimensional accuracy. Because BC-LAMBDA, HLC-LAMBDA, and VC-LAMBDA are based on multi-antenna attitude determination, this study introduces heading angle-constrained LAMBDA (HC-LAMBDA), which can also be applied to single-antenna GNSS attitude determination scenarios. To identify the optimal ambiguity candidate in BC-LAMBDA, HLC-LAMBDA, HC-LAMBDA, and VC-LAMBDA, search and shrink strategy were performed based on the boundary function rather than an exhaustive search. Additionally, a simple validation was implemented to ensure the reliability of the ambiguous candidates during the search process. The car experiment was conducted in an urban environment using a low-cost Ublox-F9p. The test of different maximum candidate numbers indicated that the accuracy of the baseline solution did not continue to increase as the maximum candidate number increased in the search space. The constrained LAMBDAs (C-LAMBDAs) with the boundary function had a higher fix rate and smaller bias than those of the exhaustive search; however, the consumed time by the boundary function was larger owing to the Modified Search (M-Search) in the shrink process. The results also show that different constraints have different effects; HC-LAMBDA has a similar performance to HLC-LAMBDA in terms of heading angle, which should be considered when the baseline length is unknown. The baseline vector of VC-LAMBDA performed better in three dimensions if there was an accurate constraint on the baseline vector.
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The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable requests.
Abbreviations
- AC-LAMBDA:
-
Affine-constrained LAMBDA
- AFM:
-
Ambiguity function method
- AR:
-
Ambiguity resolution
- BC-LAMBDA:
-
Baseline length-constrained LAMBDA
- C-LAMBDA:
-
Constrained LAMBDA
- CPU:
-
Central processing unit
- DD:
-
Double-difference
- E:
-
East
- GNSS:
-
Global navigation satellite system
- HC-LAMBDA:
-
Heading angle-constrained LAMBDA
- HLC-LAMBDA:
-
Heading angle- and length-constrained LAMBDA
- IAR:
-
Integer ambiguity resolution
- ILS:
-
Integer least-squares
- IMU:
-
Inertial measurement unit
- INSs:
-
Inertial navigation systems
- LAMBDA:
-
Least-squares ambiguity decorrelation adjustment
- MC-LAMBDA:
-
Multivariate-constrained LAMBDA
- N:
-
North
- NSAT:
-
Number of satellites
- PDOP:
-
Position dilution of precision
- RMSE:
-
Root-mean-square error
- U:
-
Up
- VC-LAMBDA:
-
Vector-constrained LAMBDA
References
Buist P (2007). The baseline constrained LAMBDA method for single epoch, single frequency attitude determination applications. In: Proc ION GNSS 2007, institute of navigation, Fort Worth, TX, USA,September 25–28, 2962–2973
Chang XW, Yang X, Zhou T (2005) MLAMBDA: a modified LAMBDA method for integer least-squares estimation. J Geodesy 79:552–565. https://doi.org/10.1007/s00190-005-0004-x
Cohen CE (1996) Attitude determination, global positioning system: theory and applications. In: Parkinson BW, Spilker JJ (eds) Reston. AIAA, VA, pp 519–538
Cohen CE, Parkinson BW (1992) Integer ambiguity resolution of the GPS carrier for spacecraft attitude determination. Adv Astronaut Sci 78(8):91–118
Counselman C, Gourevitch S (1981) Miniature interferometer terminals for earth surveying: ambiguity and multipath with global positioning system. IEEE Trans Geosci Remote 19(4):244–252. https://doi.org/10.1109/tgrs.1981.350379
Crassidis JL, Markley FL, Lightsey EG (1999) Global positioning system integer ambiguity resolution without attitude knowledge. J Guid Control Dyn 22:212–218. https://doi.org/10.2514/2.4395
Eling C, Zeimetz P, Kuhlmann H (2013) Development of an instantaneous GNSS/MEMS attitude determination system. GPS Solut 17:129–138. https://doi.org/10.1007/s10291-012-0266-8
Gabriele G, Teunissen PJG (2013) Low-Complexity instantaneous ambiguity resolution with the affine-constrained GNSS attitude model. IEEE Trans Aerosp Electron Syst 49:1745–1759. https://doi.org/10.1109/taes.2013.6558017
Giorgi G, Teunissen PJG, Verhagen S, Buist PJ (2010) Testing a new multivariate GNSS carrier phase attitude determination method for remote sensing platforms. Adv Space Res 46:118–129. https://doi.org/10.1016/j.asr.2010.02.023
Giorgi G, Teunissen PJG, Buist P (2008) A search and shrink approach for the baseline constrained LAMBDA method: experimental results. In: Proceedings of the international GPS/GNSS symposium, Tokyo, Japan,2008,pp 797-806
Groves PD (2013) Principles of GNSS, inertial, and multisensor integrated navigation systems. Artech House, Boston
Han S, Rizos C (1999) Single-epoch ambiguity resolution for real-time GPS attitude determination with the aid of one-dimensional optical fiber gyro. GPS Solut 3:5–12. https://doi.org/10.1007/pl00012779
Jin W, Gong W, Hou T, Sun X, Ma H (2023) A dual-antenna heading determination method for single-frequency GNSS antennas with large phase pattern variations. GPS Solut. https://doi.org/10.1007/s10291-023-01429-1
Kuylen L V, Boon F, Simsky A (2005) Attitude determination methods used in the polarx2@ multi-antenna GPS receiver. In: Proc. ION GNSS 2005, institute of navigation, Long Beach, CA, USA, September 13–16, 125–135
Li Q, Zhang L, Wu J, Wang D, Dong Y (2017) A novel constrained ambiguity resolution approach for beidou attitude determination. Adv Space Res 60:2423–2436. https://doi.org/10.1016/j.asr.2017.09.027
Liu X, Ballal T, Ahmed M, Al-Naffouri TY (2023) Instantaneous GNSS ambiguity resolution and attitude determination via riemannian manifold optimization. IEEE Trans Aerosp Electron Syst 59:3296–3312. https://doi.org/10.1109/taes.2022.3223330
Ma L, Lu L, Zhu F, Liu W, Lou Y (2021) Baseline length constraint approaches for enhancing GNSS ambiguity resolution: comparative study. GPS Solut. https://doi.org/10.1007/s10291-020-01071-1
Ma L, Zhu F, Liu W, Lu L, Lou Y, Zhang X (2022) VC-LAMBDA: a baseline vector constrained LAMBDA method for integer least-squares estimation. J Geod. https://doi.org/10.1007/s00190-022-01644-7
Monikes R, Wendel J, Trommer GF (2005) A modified LAMBDA method for ambiguity resolution in the presence of position domain constraints. Proc. ION GNSS 2005, Institute of Navigation, Long Beach, CA, USA, September 13–16, 81–87
Nadarajah N, Teunissen PJG (2014) Instantaneous GPS/Galileo/QZSS/SBAS attitude determination: a single-frequency (L1/E1) robustness analysis under constrained environments. Navigation 61(1):65–75. https://doi.org/10.1002/navi.51
Nadarajah N, Teunissen PJG, Raziq N (2013) Instantaneous GPS–Galileo attitude determination: single-frequency performance in satellite-deprived environments. IEEE Trans Veh Technol 62:2963–2976. https://doi.org/10.1109/tvt.2013.2256153
Nadarajah N, Teunissen PJG, Giorgi G (2014) GNSS attitude determination for remote sensing: on the bounding of the multivariate ambiguity objective function. In: Proceedings of IAG 25th general assembly of the IUGG, Melbourne, Australia, 2014,139, pp 503–509
Nie Z, Wang Z, Ou J, Ji S (2015) On the efect of linearization and approximation of nonlinear baseline length constraint for ambiguity resolution. Acta Geod Cartogr Sin 44(2):168–173. https://doi.org/10.11947/j.AGCS.2015.20130491
Park C (2009) Teunissen PJG (2009) Integer least squares with quadratic equality constraints and its application to GNSS attitude determination systems. Proc Int J Control Autom Syst 7(4):566–576
Park C, Teunissen P (2003) A new carrier phase ambiguity estimation for GNSS attitude determination systems. In: Proceedings of international GPS/GNSS symposium, Tokyo, 2003, 8
Remondi BW (1991) Pseudo-kinematic GPS results using the ambiguity function method. Navigation 38(1):17–36. https://doi.org/10.1002/j.2161-4296.1991.tb01712.x
Safoora Z, Peter T, Nandakumaran N (2017) IRNSS/NavIC L5 attitude determination. Sensors. https://doi.org/10.3390/s17020274
Takasu T, Yasuda A (2009) Development of the low-cost RTK-GPS receiver with an open source program package RTKLIB. In: Proceedings of the international symposium on GPS/GNSS, Jeju, Korea. pp 4–6
Teunissen PJG (2006) The lambda method for the GNSS compass. Artif Satell. https://doi.org/10.2478/v10018-007-0009-1
Teunissen PJG (2007) A general multivariate formulation of the multi-antenna gnss attitude determination problem. Artif Satell 42(2):97–111. https://doi.org/10.2478/v10018-008-0002-3
Teunissen PJG (2010) Integer least-squares theory for the GNSS compass. J Geodesy 84:433–447. https://doi.org/10.1007/s00190-010-0380-8
Teunissen PJG (2012) The affine constrained GNSS attitude model and its multivariate integer least-squares solution. J Geodesy 86:547–563. https://doi.org/10.1007/s00190-011-0538-z
Teunissen PJG, Giorgi G, Buist PJ (2011) Testing of a new single-frequency GNSS carrier phase attitude determination method: land, ship and aircraft experiments. GPS Solut 15:15–28. https://doi.org/10.1007/s10291-010-0164-x
Teunissen PJG (1993) Least-squares estimation of the integer GPS ambiguities. Invited lecture, Section IV Theory and Methodology, IAG General Meeting, Beijing, China, August, also in Delft Geodetic Computing Centre LGR series, No 6, pp 16
Wang B, Miao L, Wang S, Shen J (2009) A constrained LAMBDA method for GPS attitude determination. GPS Solut 13:97–107. https://doi.org/10.1007/s10291-008-0103-2
Wang Y, Zhao X, Pang C, Wang X, Wu S, Zhang C (2019a) Improved pitch-constrained ambiguity function method for integer ambiguity resolution in BDS/MIMU-integrated attitude determination. J Geodesy 93:561–572. https://doi.org/10.1007/s00190-018-1182-7
Wang Z et al (2019b) Multipath mitigation based on trend surface analysis applied to dual-antenna receiver with common clock. GPS Solut. https://doi.org/10.1007/s10291-019-0897-0
Acknowledgements
We would like to thank the students in the laboratory at the Tokyo University of Marine Science and Technology for their help throughout the collection of the experimental data. This work was sponsored by the National Natural Science Foundation of China (No. 41771475), Social Development Project of Science and Technology Innovation Action Plan of Shanghai (No. 20dz1207107), and the authors gratefully acknowledge the financial support from China Scholarship Council. This work was also supported by JSPS KAKENHI Grant Number JP19H02355.
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CZ did the experiment and wrote the manuscript. DD helped solve the key problem, gave suggestions, and modified the manuscript. NK provided the idea, gave suggestions, and modified the manuscript. KK collected the data and gave suggestions on the program. NK and WC provided the funding support. JW gave suggestions on the writing of this manuscript and the support of experiment.
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Zhang, C., Dong, D., Kubo, N. et al. Evaluation of different constrained LAMBDAs for low-cost GNSS attitude determination in an urban environment. GPS Solut 28, 42 (2024). https://doi.org/10.1007/s10291-023-01584-5
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DOI: https://doi.org/10.1007/s10291-023-01584-5