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Emerging technologies such as smart cities and unmanned vehicles all need Global Navigation Satellite Systems (GNSS) to provide high-precision positioning and navigation services. Fast and reliable carrier phase ambiguity resolution (AR) is a prerequisite for high-precision positioning. The poor satellite geometry and severe multipath effect caused by Beidou Navigation Satellite System (BDS) signal occlusion and reflection in complex environments will degrade the AR performance. In this contribution, a fast triple-frequency AR method combining Microelectromechanical System-Inertial Measurement Unit (MEMS-IMU) and BDS is proposed. First, the Extra-Wide Lane (EWL) ambiguity is fixed with the positioning parameters of MEMS-IMU instead of the pseudorange. Then, the phase noise variance of Narrow Lane (NL) observation is obtained from ambiguity-fixed EWL observation to reduce the total noise level of NL observation, and the NL ambiguity can be reliably fixed, and the BDS positioning result is obtained. Finally, the BDS positioning result is used as the posterior measurement of the extended Kalman filter to update the MEMS-IMU positioning parameters to form the coupling loop of MEMS-IMU and BDS. The data of urban road vehicle experiments were collected to verify the feasibility and effectiveness of the proposed algorithm. Results show that MEMS-IMU can speed up AR, and reduction of total noise level can significantly improve the reliability of AR.

With the indepth development of information technology, next-generation information technologies such as the Internet of Things (IOT), smart city, and unmanned vehicle all need high- accuracy time and space service support. The Global Navigation Positioning System (GNSS) can provide reliable high-precision positioning and timing services. The key to realizing high-precision positioning for GNSS is to quickly and reliably resolve integer carrier phase ambiguity [

The Chinese Beidou Navigation Satellite System (BDS) constellation transmits observations at three frequencies (B1, B2, and B3), including pseudorange and carrier phase observations [

However, the EWL ambiguity is fixed by pseudorange, and the multipath effect caused by the complex environment will reduce the AR performance of the traditional TCAR and its derivative methods. Obviously, completely eliminating code multipath error will improve AR performance. Feng-Yu Chu et al. proposed a method to resolve ambiguities using only triple-frequency carrier phase observations, completely eliminating the effects of code multipath, and pointed out that the positioning performance is still affected by the carrier phase multipath [

In fact, the original carrier wavelength of GNSS is about 20cm, while the prediction error of some low-cost Microelectromechanical System-Inertial Measurement Unit (MEMS-IMU) in within 1 epoch (1s) of GNSS for geometric distance is more than half of the original carrier wavelength [

We investigated the causes of AR performance degradation in complex environments and improved the first and third steps of the traditional TCAR method. The high quality positioning parameters of the MEMS-IMU are used to assist in fixing the EWL ambiguity, and the NL ambiguity is reliably fixed after the total noise variance is reduced. The MEMS-IMU and BDS are loosely coupled at the centimeter level to achieve high-precision positioning by Extended Kalman Filter (EKF).

The rest of this paper is organized as follows. Section

In the open sky environment, the traditional TCAR method can efficiently resolve the triple-frequency ambiguity due to high-precision observations. However, in a complex environment where the BDS signal occluded and reflected, the severe multipath effect makes it difficult for the traditional TCAR method to reliably and instantaneously resolve the triple-frequency ambiguity.

The observation formula of the BDS carrier phase and pseudorange can be expressed as

For short baselines (baseline length < 20km), the spatiotemporal correlation between observations is strong. After intersatellite difference and interreceiver difference, the satellite clock error, the receiver clock error, and the hardware delay are completely eliminated, the tropospheric error, the ionospheric error, and the satellite orbit error are also greatly weakened and can be ignored. Therefore, after the double differenced (DD) operation, the observation formula is

The triple-frequency DD observations can form virtual observations with various favorable characteristics for AR. The linear combination of triple-frequency DD observations can be expressed as [

The combined observations of DD pseudorange and carrier phase can be expressed as

The carrier phase combination observations are only mathematically combined and their noise originates from the original carrier noise. Assuming that the noise at the three frequencies of the BDS is equal and uncorrelated, equal to

The BDS triple-frequency linear combination can form a lot of virtual observations. Table

Commonly used BDS triple-frequency combinations and related parameters.

| | | |
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(1,0,0) | 0.192 | 1 | 1 |

(0,1,0) | 0.248 | 1.672 | 1 |

(0,0,1) | 0.236 | 1.515 | 1 |

(0,-1,1) | 4.884 | -1.592 | 28.529 |

(1,-1,0) | 0.847 | -1.293 | 5.575 |

(4,0,-3) | 0.123 | 0.1972 | 2.9984 |

The traditional TCAR method is a method that is independent of the satellite geometry, namely the GF- (geometry-free) TCAR method. The three combined observations are fixed by the rounding method.

EWL ambiguity

WL ambiguity:

NL ambiguity:

The traditional TCAR and its derivative methods use pseudorange to fix EWL or WL ambiguity. As can be seen from formula (

In the open sky environment, the accuracy of pseudorange and carrier phase is relatively high, and the accuracy of virtual observations is also relatively high. The BDS triple-frequency ambiguity can be resolved instantaneously and reliably by the traditional TCAR method.

For short baselines, the main error source of BDS positioning in complex environment is the multipath error, especially code multipath error. The multipath effect reduces the accuracy of the pseudorange and increases the convergence time of the pseudorange fixed EWL ambiguity and will bring the error to the fixed WL ambiguity.

The multipath effect will not only increase the code error but also increase the carrier phase error. Under the condition that the reflection coefficient is 1, the carrier phase multipath error is at most 0.25 wavelengths [

The fixed speed of ambiguity integer solution depends on the size of the float ambiguity search space, and the ambiguity search space depends on the float ambiguity variance covariance matrix [

Among the three factors affecting the float ambiguity search space,

In this study, the loosely coupled form of MEMS-IMU and BDS is realized by extended Kalman filter. Compared with the tightly coupled form, the loosely coupled form has a simple structure and a small amount of computation, which helps to improve the real-time of AR. A navigation system state error model based on the MEMS-IMU attitude angle is established [

Take into account the state incremental error, the error of the lever arm a total of 24 error terms characterization of the system state:

After the AR is completed, the carrier phase observation can be converted to a high-accuracy BDS positioning result. Considering the deviation between the BDS antenna phase center and the MEMS-IMU reference center, in the e-system, the antenna phase center can be expressed as

Because the integrated navigation system is loosely coupled, the BDS corrects the accumulated error of the MEMS-IMU over time at the positioning result level. The

The geometrical distance can be calculated from the positioning parameters output from the MEMS-IMU and the satellite coordinates obtained from the BDS precision satellite orbit products:

Obviously, the accuracy of the EWL float ambiguity is greatly improved, so that the EWL ambiguity can be fixed by rounding. Due to the high accuracy of

The fixation of NL ambiguity is the final step of the TCAR method and the key to achieving high-precision positioning. Based on the PNF relationship between EWL and NL observations, the phase noise variance of the NL observation is obtained from the ambiguity-fixed EWL observation to reduce the total noise of the NL observation, and the accuracy of float ambiguity and the ambiguity fixed success rate are improved.

The error sources of combined observations mainly include ionospheric delay error, tropospheric delay error, and other errors (mainly phase multipath error) [

Substituting the obtained

It should be noted that considering AR performance of NL ambiguity is sensitive to various noises, under the assumption that the DD eliminates the ionosphere delay, the ionosphere delay actually exists although it is small. It is necessary to select the NL combination with a small ISF as much as possible to reduce the negative effects of ionospheric delay. For example, the (4.0.-3) in Table

The main performance parameters of the two INS.

| | | | | |
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POS310 | Tactical | 0.5 | 25 | 200 | |

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POS1100 | MEMS | 10.0 | 2000 | 200 | |

In order to improve the real-time performance of the integrated system, the proposed algorithm adopt the loosely-coupled form with a small amount of computation. In the BDS triple-frequency AR process, PNF is used to reduce the total noise level of NL observations to improve the ambiguity fixed reliability.

Figure

Overview of BDS/INS loosely-coupled integration system.

In order to evaluate the performance of the proposed MEMS-IMU assisted BDS triple-frequency AR method, the field vehicular experiment of the integrated navigation system was conducted in Taiyuan, China, on June 6, 2018, from 9:30 to 10:30 am. The MEMS-grade INS and BDS rover receivers are installed on the top of the experimental vehicle to collect data on the test road with a sampling interval 1s. The entire test road can be divided into three types according to the characteristics of the road environment: open sky, urban canyon and roadside trees. The most serious occlusion and reflection of satellite signals is the urban canyon section, and the signal blockage of the roadside tree section is not serious, and the signal of the open sky road section is completely unobstructed. The speed of the experimental vehicle is maintained at about 5m/s, the baseline length of the entire experimental road is within 5km, and the data collection time is 1 hour. Figure

Experimental area and vehicle trajectory.

The POS1100 MEMS-grade INS was used in the experiment and included three MEMS gyroscopes and three quartz accelerometers. The coordinate parameters of the vehicle trajectory in the map are measured in advance using a tactical-grade INS POS310. Both of the two IMUs are provided by Wuhan MaiPu Space Time Technology Company (Wuhan, China). Their main performance specifications are shown in Table

The BDS base station receiver UR380 installed on the roof of the library, and the BDS Rover receiver UB4B0 installed on experimental vehicle. Both of the BDS receivers that can receive and process triple-frequency observations are provided by Unicore Communications, Inc. (Beijing, China).

In order to analyze the influence of different road environments on the observations, the entire test data is divided into 10 periods on average according to the length of time, each for 6 minutes (360 epochs). The 2th period (9:36-9:42) belongs to the open sky road section without any occlusion, and the 6th period (10:00-10:06) belongs to the urban canyon road section with severe occlusion. Only the data of these two periods of the extreme environment are selected for comparative analysis. Figure

Received BDS satellites. (a) The 2th period and (b) the 6th period.

The BDS includes geostationary earth orbit (GEO), Inclined Geosynchronous Satellite Orbit (IGSO), and Medium Earth Orbit (MEO) satellites. Tables

Relevant parameters of visible satellites during the 2th period.

| | | | | | |

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PRN | C01 | C02 | C03 | C04 | C05 | C06 |

| ||||||

Elev. (°) | 43 | 37 | 48 | 24 | 16 | 9 |

| ||||||

| | | | | | |

| ||||||

PRN | C07 | C08 | C09 | C10 | C13 | C14 |

| ||||||

Elev. (°) | 70 | 33 | 18 | 64 | 16 | 4 |

Related parameters of visible satellites during the 6th period.

| | | | | | |

| ||||||

PRN | C01 | C04 | C07 | C08 | C09 | C10 |

| ||||||

Elev. (°) | 40 | 24 | 68 | 33 | 16 | 62 |

Comparing Tables

The occlusion or reflection of the BDS signal not only leads to a reduction in the number of visible satellites, but also a multipath error that reduces the accuracy of the observations, especially pseudoranges. Figure

Code multipath error at three frequency in the 2th period (top) and the 6th period (bottom).

For short baselines, the DD eliminates most of the errors, and code multipath error becomes the main error affecting the AR performance. By comparing the pictures in the top and bottom rows in Figure

Further observation shows that the code multipath error at B3 is generally less than that at B1 and B2, This is because the B3 has a chip rate of 10.23 Mcps and the accuracy of chip is relatively high. In addition, the code multipath error of the C09 satellite is significantly higher than that of other satellites. It can be seen from the comparison that the elevation angle of the C09 satellite is relatively low, indicating that the satellite elevation angle is directly related to the code multipath error.

In order to fully prove that MEMS-IMU can improve the AR performance of EWL ambiguity, three different TCAR methods are compared with the proposed method. The data of 10 periods are processed by four TCAR methods. The AR performance indicator uses the fixed success rate and the time to first fix (TTFF). The four methods for fixing the EWL ambiguity are as follows:

Scheme A: GF-TCAR, which is described in Section

Scheme B: GB-TCAR, integer least squares model based on geometry.

Scheme C: GIF-TCAR, in addition to the GIF condition, the triple-frequency pseudorange observations together fix the EWL ambiguity.

Scheme D: INS-TCAR, the proposed method.

The cut-off elevation angle is set to 10°, and C07 satellite is used as the pivot satellite throughout the experiment.

The ambiguity fixed success rate is one of the commonly used indicators for measuring AR performance. Figure

EWL ambiguity fixed success rate of Scheme A (blue), Scheme B (green), Scheme C (yellow), and Scheme D (red) in 10 periods.

As can be seen from Figure

TTFF refers to the time it takes for the ambiguity to be correctly fixed for the first time. It can reflect the real-time performance of ambiguity fixation and positioning. Figure

TTFF of the four TCAR in 10 periods.

It can be seen from Figure

In order to prove that the proposed method can solve the problem that NL ambiguity is difficult to be fixed in a complex environment, we use scheme D to process and analyze the data of the 6th period with the largest multipath error. Only the ambiguities of EWL, WL, and NL are all fixed correctly to obtain BDS high-precision positioning results. Failure to fix the ambiguity not only reduces the positioning accuracy of the BDS but also does not correct the cumulative error of the MEMS-IMU. Figure

Ambiguity fixed success rate before the total noise level of the NL observation is reduced.

As can be seen from Figure

NL float ambiguity before (a) and after (b) the total noise level is reduced.

From the first and the third rows of Figure

It can be seen from the second and fourth rows of Figure

Fixed success rate of NL ambiguity before and after the total noise level is reduced.

It can be seen from Figure

In the open sky environment, the traditional TCAR and its derivative algorithms have fast and reliable AR performance due to the high quality BDS triple-frequency observations. However, in complex environments, satellite signals are blocked or reflected by obstacles causing severe multipath errors in pseudorange and carrier phase observations. To solve this problem, it is proposed to use the MEMS-IMU corresponding geometric distance instead of pseudorange to fix EWL ambiguity. Then, the noise variance of the NL observation is obtained from the EWL observation to reduce the total noise variance and improve the reliability of the NL ambiguity fixed. MEMS-IMU can output high-quality positioning parameters at high frequency in a short time; it not only improves the BDS AR success rate, but also improves the stability and continuity of BDS positioning, especially in the tunnel. The BDS positioning results can then be used to correct the cumulative error of the MEMS-IMU; both are loosely coupled by EKF.

The field vehicular experiment experiments were conducted on urban streets to verify the feasibility of the proposed algorithm. The result proves that the proposed algorithm not only speed up the fixation of the ambiguity through the combination of MEMS-IMU and BDS triple-frequency but also improves the reliability of the fixed ambiguity.

The proposed method is mainly to improve the AR performance for the short baseline in complex environments. For the medium-to-long baseline, the temporal and spatial correlation of observations is reduced, and ionospheric delays cannot be eliminated by DD. It is necessary to analyze the characteristics of various error sources and propose an effective algorithm.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

Junbing Cheng and Dengao Li conceived and designed the experiments; Junbing Cheng performed the experiments; Junbing Cheng and Jumin Zhao analyzed the data; Junbing Cheng wrote the paper; Dengao Li critically reviewed the paper.

This work is partially supported by the National High Technology Research and Development Program (“863” Program) of China with Grant no. 2015AA016901, partially sponsored by the General Object of National Natural Science Foundation with two Grant nos. 61772358 and 61572347, partially by a project funded by the International Cooperation Project of Shanxi Province with Grant no. 201603D421012.