- Original article
- Open Access

# Accuracy analysis of GPS/BDS relative positioning using zero-baseline measurements

- Gethin Wyn Roberts
^{1}Email author, - Xu Tang
^{2}and - Xiufeng He
^{3}

**16**:7

https://doi.org/10.1186/s41445-018-0015-6

© The Author(s) 2018

**Received:**22 December 2017**Accepted:**13 April 2018**Published:**25 April 2018

## Abstract

This paper focuses on assessing the precision of carrier phase relative positioning using GPS-only, BDS-only and GPS/BDS measurements. A zero baseline is used in order to achieve this. Software for GPS and BDS processing has been developed, allowing static and kinematic data processing, as well as the combined GPS and BDS processing. Ionospheric and tropospheric delays are significantly reduced by double differencing between satellites and receivers, but the Multipath signals are still a major source of error for the various general GNSS baseline applications. In this paper, two Multi-GNSS receivers are connected to one antenna by an antenna splitter. This strategy results in all the delays or errors being mitigated, leaving only the random measurement noises resulted from the double difference processing. The time series of the final baseline error reveal that both GPS and BDS can achieve a precision of millimetres, but GPS performs better than BDS. Results from the combined processing of GPS and BDS demonstrate that the integration of GPS and BDS can significantly improve the precision, compared with the GPS-only and BDS-only results.

## Keywords

- BDS satellite navigation system
- Integration of GPS and BDS
- Kinematic positioning error

## Introduction

Current BDS Operational Satellites (October 2014)

PRN | Type | Launch Time | Longitude | Latitude | Approximate Height |
---|---|---|---|---|---|

C01 | GEO | 2010/01/07 | 140.07° | ~ 0° | ~ 36,000 km |

C02 | GEO | 2012/10/25 | 80.22° | ~ 0° | ~ 36,000 km |

C03 | GEO | 2010/06/02 | 110.56° | ~ 0° | ~ 36,000 km |

C04 | GEO | 2010/11/01 | 160.00° | ~ 0° | ~ 36,000 km |

C05 | GEO | 2012/02/25 | 58.65° | ~ 0° | ~ 36,000 km |

C06 | IGSO | 2010/08/01 | 104.63°E - 136.05°E | 54:61°S - 54:61°N | ~ 36,000 km |

C07 | IGSO | 2010/12/18 | 102.60°E - 134.13°E | 54:81°S - 54:81°N | ~ 36,000 km |

C08 | IGSO | 2011/04/10 | 100.49°E - 133.82°E | 56:02°S - 56:02°N | ~ 36,000 km |

C09 | IGSO | 2011/07/27 | 80.12°E - 111.79°E | 54:93°S - 54:93°N | ~ 36,000 km |

C10 | IGSO | 2011/12/02 | 78.66°E - 110.33°E | 54:93°S - 54:93°N | ~ 36,000 km |

C11 | MEO | 2012/04/30 | 180°E - 180°W | 55:31°S - 54:61°N | ~ 21,500 km |

C12 | MEO | 2012/04/30 | 180°E - 180°W | 55:25°S - 54:81°N | ~ 21,500 km |

C13 | MEO | 2012/09/19 | 180°E - 180°W | 54:99°S - 56:02°N | ~ 21,500 km |

C14 | MEO | 2012/09/19 | 180°E - 180°W | 55:10°S - 59:93°N | ~ 21,500 km |

The first test satellite, called M1, was launched in 2007. The BDS code structure is analyzed by using high gain parabolic antennas at two stations located at (Grelier et al. 2007) CNES (Toulouse, France) and Leeheim (Germany). Gao et al. decoded the M1 satellite’s codes in the E2, E5b and E6 bands, extracted the code bits, and derived the code generators (Gao et al. 2009). All GPS satellites’ ephemerides can employ a uniform parameter, as the GPS constellation consisted of only MEO satellites. The BDS constellation is a mixture of GEO, IGSO and MEO satellites. A method has been presented to design optimal parameter sets for all these three types of satellites (Fu and Wu 2011). The clock offset was estimated by processing the BDS M1 observations based on the orbit solution from laser ranging measurements (Hauschild et al. 2012). Unexpectedly high dynamics in the clock results were found, which affect both the pseudo-range and carrier phase measurements.

Since the increase of BDS satellite numbers, many researches have focused on assessing BDS position precision, orbital determination and integration of BDS with other satellite navigation systems. *(*Shi et al. 2012*)* carried out early research, using one week of BDS GEO and IGSO satellites’ observations to assess the preliminary positioning performance. Their work revealed that BDS code measurements’ noise was higher than GPS’s. Both data were collected using the same GPS/BDS receiver. The precise relative positioning using code and carrier phase measurements revealed that BDS precision is better than 2 cm in the North-South component and 4 cm at the vertical component. The standard deviation of the East-West component is smaller than 1 cm. The BDS code measurements using 5 GEO, 5 IGSO and 4 MEO satellites’ carrier phase measurements were assessed in both static and kinematic positioning (Tang et al. 2014; Tang 2014). Researchers have combined BDS with other satellite navigation systems in order to assess the short baseline ambiguity resolution reliability, and the availability in high cut-off elevation situation (He et al. 2014; Deng et al. 2013; Odolinski et al. 2014; Teunissen et al. 2014). Multipath delay is one of the main error sources, which is a limitation of GNSS positioning accuracies improving (Wang et al. 2014). The BDS GEO satellites’ multipath are even more difficult to mitigate due to the relatively stationary geometry when compared to MEO satellites in particular.

This paper assesses the precision of GPS and BDS carrier phase positioning using the current constellations. The results of this study reveal that by integrating both GPS and BDS, it is possible to improve the precision compared to GPS-only and BDS-only results. This paper is organized as follows. In section 2, the zero-baseline experiment is briefly described. In section 3, the GPS and BDS satellites’ availability is detailed. In section 4, the GPS-only, BDS-only and combined GPS and BDS kinematic results are presented.

## Results and discussion

### Zero baseline data collection

#### Field experiments

### Zero baseline error analysis

*i*and

*j*, and the two tracked satellites’ names as

*p*and

*q*. Considering the tropospheric delay, ionospheric delay, multipath error, satellite and receiver clock errors, as well as the random receiver noise errors, these two receiver observation equations to satellite

*p*can be written as:

*ϕ*is the carrier phase measurement;

*ρ*is the geometrical range between satellite

*p*and receivers

*i*and

*j*;

*c*is the speed of light in a vacuum.

*δt*

^{ p }is the satellite

*p*clock error;

*δt*

_{ i }and

*δt*

_{ j }are the receiver

*i*and receiver

*j*clock errors, respectively; \( {N}_i^p \)and \( {N}_j^p \) are the integer ambiguities of receivers

*i*and

*j*from satellite

*p*carrier phase measurements, respectively. \( {T}_i^p \) and \( {T}_j^p \) are the tropospheric delays of receivers

*i*and

*j*from satellite

*p*, respectively; \( {I}_i^p \) and \( {I}_j^p \) are the ionospheric delays from satellite

*p*to receivers

*i*and

*j*, respectively; \( {M}_i^p \) and \( {M}_j^p \) are the multipath noise of receivers

*i*and

*j*from satellite

*p*. Receivers

*i*and

*j*have the same tropospheric and ionospheric delay when they track the same satellite over a zero baseline. Choke ring antennas have the capacity of resisting most of the multipath. The remaining multipath impacts equally onto receivers

*i*and

*j*in this zero-baseline scenario. The single difference equation between the two receivers can be written as:

*q*, a second single difference equation can be written as:

*g*

_{i, j}=

*g*

_{ j }−

*g*

_{ i }. The tropospheric, ionospheric and multipath errors could be totally eliminated by differencing between the receivers’ carrier phase data. The random errors which reflect the quality of the signals, single difference integer ambiguity and the single difference receivers’ clock error remain. Through calculating the difference between eq. 4 minus eq. 3, the double difference observation is derived as:

*δb*is the error of the baseline,

*δs*is the satellite orbital error,

*ρ*is the distance between the tracked satellite to the receiver

*b*is the length of the baseline. In the zero-baseline scenario, the error of the satellite orbit can be eliminated through differencing (Eq. 6).

\( \left[{l}_{i,j}^{p,q}\kern0.5em {m}_{i,j}^{p,q}\kern0.5em {n}_{i,j}^{p,q}\right] \) is the double difference distance from the satellites to the receivers, which could be calculated by the receivers’ approximate coordinates and the tracked satellites’ coordinates. [*δx δy δz*] is the correction of the baseline’s 3D distance, \( {N}_{i,j}^{p,q} \) is the double difference integer ambiguity, \( {\xi}_{i,j}^{p,q} \) is the double difference random noise. The precision of the baseline correction is not only determined by the observation noise, but also the coefficient \( \left[{l}_{i,j}^{p,q}\kern0.5em {m}_{i,j}^{p,q}\kern0.5em {n}_{i,j}^{p,q}\right] \), after the double difference integer ambiguity \( {N}_{i,j}^{p,q} \) is correctly fixed. We use variance of unit weight during the least squares adjustment. The coefficients reflect the efficiency of different satellite geometry. In this paper, the final positional error is a combination of the observation random noise and the satellites’ geometrical impact. Distinguishing between these error sources is the topic of ongoing work. This paper pays more attention to the difference of GPS-only and BDS-only position precision and the improvement through the integration of GPS and BDS data.

## Satellites’ visibility

^{∘}52

^{′}

*N*and 121

^{∘}33

^{′}

*E*). IGSO satellites can be tracked for the most part of their orbital period by the station, but they have the same feature as with MEO satellites in that IGSO satellites can drop out of sight due to their orbit. There is a specific area in the sky where the satellites are never present. This area is located in the north of the sky plot to the northern hemisphere users.

^{∘}. It demonstrates that there are only 4 GPS satellites in sight during some epochs, but BDS has at least 6 satellites that could be tracked during the 24 h period. In some instances, there are more BDS satellites than GPS being tracked. There are at least 12 navigation satellites being tracked when GPS and BDS are combined for positioning. The integration of GPS and BDS can be very useful when the user is located in a non-GNSS friendly environment.

## GPS-only, BDS-only and GPS + BDS kinematic results

*mm*most of the time in both the North-South and the East-West components. BDS position error is no more than ±3

*mm*in the east-west component. The BDS precision in the north-south and height components is not as stable as GPS. Errors during 02:00–04:00, 10:00–13:00 and some epoches around 17:00 are observed as being bigger than other times in the north-south component, this also happens in the height component in the BDS positional error time series. Comparing the GPS-only and BDS-only resolutions in Figs. 5, 6 and 7, the GPS positional precision currently performs better than BDS in the three components. This is partly due to the fact that the BDS constellation is still incomplete. Figure 6 reveals that BDS position noise is much more obvious than GPS during 10:30 to 12:30. The GPS-only and BDS-only sky plots, Figs. 8 and 9 respectively, reveal that most of the BDS satellites are located to the south of the station. The GPS satellites’ orbital tracks are longer than BDS GEO and IGSO satellites during the 2 h period. The GPS satellites coverage are also more spread out than BDS during this 2 h period. The overall conclusion is that the incomplete BDS constellation has a poorer geometrical spread than GPS. However, the integrated GPS/BDS solution has the best spread of satellites. Both GPS and BDS have the same feature in that North-South and East-West position errors are much smaller than the height component.

GPS-only, BDS-only and GPS + BDS double difference position error Root Mean Squared value (RMS) in the different position components

components | North (mm) | East (mm) | Height (mm) |
---|---|---|---|

Model | |||

BDS-only | 1.22 | 1.03 | 3.70 |

GPS-only | 0.81 | 0.67 | 1.94 |

GPS + BDS | 0.69 | 0.62 | 1.71 |

## Conclusion

In this paper, zero baseline observations were gathered in order to assess the accuracy of GPS-only, BDS-only and the integration of GPS and BDS relative positioning. There are on average 8 GPS and 8 BDS satellites that could be tracked during the whole BDS orbital period at the University of Nottingham Ningbo, China. Sometimes the number of GPS satellites being tracked above an elevation cut-off angle of 15^{∘} can drop down to only four during the experiments. The minimum number of BDS tracked during the experiments were six using the same elevation mask of 15^{∘}. The results demonstrate that BDS relative positioning accuracy could achieve millimetre level, but is still weaker than that of GPS. The integration of GPS and BDS could improve not only BDS-only, but also GPS-only position precisions.

From the derivation of observation equation, the positional error is caused by the precision of the observation, as well as the coefficient of observation equation. We will study this aspect by simulating more MEO satellites with the help of a SPIRENT GNSS simulator in the future.

## Declarations

### Acknowledgements

The work in this paper is supported by Young Scientist programme of Natural Science Foundation of China (NSFC) with a project code 41704024, as well as Ningbo Science and Technology Bureau – China as Part of the Project: Structural Health Monitoring of Infrastructure in the Logistics Cycle (2014A35008).

### Availability of data and materials

Please contact the authors for data requests.

### Authors’ contributions

GWR obtained the Ningbo Science and Technology bureau funding that supported the research, and XT obtained the NSFC funding. XT carried out the field work and developed the algorithms. GWR and XT interpreted the results and co-wrote the paper. All authors read and approved the final manuscript.

### Authors’ information

Dr. Gethin Wyn Roberts is an Associate Professor in Geospatial Engineering at the University of the Faroe Islands. He was previously at the University of Nottingham as a staff member for 24 years, both in the UK and on the Ningbo, China campus.

Dr. Xu Tang is a research fellow at the University of Nottingham Ningbo, China.

Prof Xiufeng He is a professor at HoHai University in Nanjing China.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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