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Many Beidou navigation satellite system (BDS) receivers or boards provide dual-frequency measurements to conduct precise positioning and navigation for low-power consumption. Cycle-slip processing is a primary work to guarantee consistent, precise positioning with the phase data. However, the cycle-slip processing of BDS dual-frequency phases still follows with those of existing GPS methods. For single-satellite data, cycle-slip detection (CSD) with the geometry-free phase (GF) is disturbed by severe ionospheric delay variations, while CSD or cycle-slip repair (CSR) with the Melbourne–Wubbena combination (MW) must face the risk of the tremendous disturbance from large pseudorange errors. To overcome the above limitations, a new cycle-slip repair method for BDS dual-frequency phases (BDCSR) is proposed: (1) An optimal model to minimize the variance of the cycle-slip calculation was established to the dual-frequency BDS, after correcting the ionospheric variation with a reasonable and feasible way. (2) Under the BDS dual-frequency condition, a discrimination function was built to exclude the adverse disturbance from the pseudorange errors on the CSR, according to the rankings of the absolute epoch-difference GFs calculated by the searched cycle-slip candidates after correcting the ionospheric variation. Subsequently, many compared CSR tests were implemented in conditions of low and medium elevations during strong geomagnetic storms. Comparisons from the results of different methods show that: (1) The variations of ionospheric delays are intolerable in the cycle-slip calculation during the geomagnetic storm, and the tremendous influence from the ionospheric variation should be corrected before calculating the cycle-slip combination with the BDS dual-frequency data. (2) Under the condition of real dual-frequency BDS data during the geomagnetic storm, the actual success rate of the conventional dual-frequency CSR (CDCSR) by employing the optimized combinations, but absenting from the discrimination function, is lower than that of BDCSR by about 2%; The actual success rate of the CSD with MW (MWCSD), is lower than that of BDCSR by about 2%. (3) After adding gross errors of 0.7 m to all real epoch-difference pseudoranges epoch-by-epoch, results of CDCSR and MWCSD showed many errors. However, BDCSR achieved a higher actual success rate than those of CDCSR and MWCSD, about 43% and 16%, respectively, and better performance of refraining the disturbance of large pseudorange error on the cycle-slip determination was achieved in the BDCSR methodology.

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The cycle slip detection (CSD) and cycle slip repair (CSR) are easily affected by ionospheric delay and observational noise. Aiming at mitigating the above disadvantage, a new BeiDou navigation satellite system (BDS) triple-frequency CSR method (BTCSR) is proposed for the undifferenced phase. BTCSR learns from the classic triple-frequency CSR (CTCSR), with combinations of phases and pseudoranges in correcting ionospheric delay and optimizing observational noise. Different from CTCSR, though, BTCSR has made the following improvements: (1) An optimal model of calculating cycle slip combination is established, which further takes into account the minimization of the effect of residual ionospheric error after the correction. The calculation of cycle slip combination is obtained with the root mean squared errors (0.0646, 0.1261, 0.1069) of cycles, resulting in CSR success rate of 99.9927%, and the wavelengths (4.8842,3.5738,8.1403) of m. (2) A discriminant function is added to guarantee the CSR correctness. This function utilizes epoch-difference value of the ionosphere-free and geometry-free phase to select the correct cycle slip value, which eliminates the interference of large pseudorange errors in determining the final cycle slip. Consequently, the performances of BTCSR and CTCSR have been compared. For the real BDS pseudorange observation with additional 1.5 m errors, which can cover situations of 99.96% pseudorange noise, results of CTCSR show failure, but results of BTCSR keep correct. Moreover, BTCSR has made the following improvements relative to the geometry-free cycle slip detection method (GFCSD) and Melboune–Wubbena cycle slip combination detection method (MWCSD): (1) During a moderate magnetic storm of level 6, CSR testing, with the BDS monitoring station in a low latitude region, showed that some failures occur in GFCSD because of severe ionospheric variation, but BTCSR could correctly identify and fix cycle slips. (2) For the BDS observation data with an additional 1.5 m error on the actual pseudoranges, MWCSD exhibited failures, but the repair results of BTCSR were correct and reliable. (3) For the special slips of (0,59,62) cycles, and equal slips of (1,1,1) cycles on (B1,B2,B3), that are hard to detect by GFCSD and MWCSD, respectively, BTCSR could repair these correctly. Finally, BTCSR obtains reliable repair results under large pseudorange errors and severe ionospheric variations, and the cut-off elevation larger than 10 degrees is the suggested background.

In this paper, a single-frequency real-time kinematic positioning (RTK) robust adaptive Kalman filtering algorithm is proposed in order to realize real-time dynamic high-precision positioning of smartphone global navigation satellite systems (GNSSs). A robust model is established by using the quartile method to dynamically determine the threshold value and eliminate the gross error of observation. The Institute of Geodesy and Geophysics Ⅲ (IGG Ⅲ) weight function is used to construct the position and speed classification adaptive factors to weaken the impact of state mutation errors. Based on the analysis of the measured data of Xiaomi 8 and Huawei P40 smartphones, simulated dynamic tests show that the overall accuracy of the Xiaomi 8 is improved by more than 85% with the proposed robust RTK algorithm, and the overall positioning error is less than 0.5 m in both open and sheltered environments. The overall accuracy of the Huawei P40 is improved by more than 25%. Furthermore, the overall positioning accuracy is better than 0.3 m in open environments, and about 0.8 m in blocked situations. Dynamic experiments show that the use of the robust adaptive RTK algorithm improves the full-time solution planar positioning accuracy of the Xiaomi 8 by more than 15%. In addition, the planar positioning accuracy under open and occluded conditions is 0.8 m and 1.5 m, respectively, and the overall positioning accuracy of key nodes whose movement state exhibits major changes improves by more than 20%.