# GPS + Galileo tightly combined RTK positioning for medium-to-long baselines based on partial ambiguity resolution

- Guangcai Li
^{1}Email author, - Jianghui Geng
^{1, 2}, - Jiang Guo
^{1}, - Sen Zhou
^{3}and - Shuang Lin
^{4}

**16**:3

https://doi.org/10.1186/s41445-018-0011-x

© The Author(s) 2018

**Received: **20 September 2017

**Accepted: **23 January 2018

**Published: **2 March 2018

## Abstract

With the modernization of the GNSS, the techniques of multi-GNSS navigation and positioning are becoming increasingly important. For multi-GNSS double-difference data processing, a tight combination (TC) strategy can provide more observations and higher reliability, which emploies a single reference satellite for all observations from different GNSS. However, multi-GNSS will bring some challenges to the high-dimension ambiguity resolution (AR). In this contribution, a GPS + Galileo tightly combined real-time kinematic (RTK) positioning strategy is proposed, which introduces the partial ambiguity resolution (PAR) method. A set of baselines ranging from about 22 to 110 km are used to test the positioning performance of this strategy. Experimental results demonstrate that the TC strategy can improve the success rate, but it can’t increase the ambiguity ratio values. Using the PAR method can reduce convergence times and improve the ambiguity fixing rate. Combining the TC strategy with the PAR method can provide better positioning performance, especially for long baselines.

## Keywords

## Introduction

With the modernization of Global Navigation Satellite System (GNSS), multi-GNSS navigation and positioning techniques are becoming increasingly important. Combining observations from various GNSS constellations significantly increases the number of observations and improves the positioning accuracy and reliability, especially in difficult environments (Li et al., 2016a). Multi-GNSS double difference combination strategies include loose combination (LC) in which each of the systems uses its own reference satellite and no double differences are formed across systems (Zhang et al. 2003), and tight combination (TC) in which two systems use the same reference satellite and permitting double differences across different systems (Julien et al. 2003). Therefore, the TC strategy can provides more observations than the LC strategy. However, as the ambiguity dimension increases sharply, the success rate of integer ambiguity resolution is reduced (Teunissen et al., 1999), while the key requirement for real-time kinematic (RTK) is to quickly and correctly fix the ambiguities of carrier phase measurements. For multi-GNSS data processing, it is often impossible to fix all ambiguities simultaneously due to the large number of observations, which is even deteriorated in case of medium-to-long baselines (more than 20 km) when various residual errors cannot be mitigated completely (Li et al., 2016, b).

To solve this problem, the idea of partial ambiguity resolution (PAR), which means to resolve a subset of the candidate ambiguities, was suggested to maintain a sufficiently high success rate (Teunissen et al., 1999). The selection of an ambiguity subset could be based on pre-defined subset sizes (Mowlam and Collier, 2004), ambiguity variances (Wang and Feng, 2012), satellite elevations (Li et al., 2014), satellite variances (Li and Teunissen, 2014), combined phase observation wavelengths (Li et al., 2015, b) and composite methods that combine such strategies (Gao et al., 2017). In addition, the algorithm of satellite selection algorithm for PAR (Wang and Feng, 2013) and the method of EWL/WL as well as NL PAR for triple-frequency GNSS signals (Li et al., 2015a) are studied systematically.

Many studies have applied the PAR method to the LC strategy and have achieved significant results. For example, the reliability characteristics of PAR solutions were verified by Wang and Feng (2012), and the PAR method was applied to the LC RTK positioning with the GPS constellation and virtual Galileo constellation to demonstrate the advantages of the proposed PAR method. Hou and Verhagen (2014) proposed a model and data driven PAR (MD-PAR) strategy and evaluated the performance of MD-PAR for GPS + BDS LC RTK positioning using simulated GPS and BDS observations. Li et al. (2015, b) presented the multi-carrier fast PAR (MCFPAR) strategy to solve multi-system multi-frequency high-dimensional AR problems, and its validity was demonstrated with BDS + GPS LC RTK positioning using real dual- and triple-frequency observations. Gao et al. (2015, 2015) quoted the partial wide-lane ambiguity resolution strategy to GPS + BDS LC RTK positioning and GPS+ GLONASS + BDS LC RTK positioning and validated with real observations.

However, there are few publications applying the PAR method to the TC strategy, although this strategy provides more observations. The PAR method was introduced into the GPS + Galileo TC RTK positioning by Cao et al. (2007), and verify the reliability performance of this strategy in short baseline RTK. However, the simulation data is used and the inter-system bias is ignored.

In this paper, a GPS + Galileo TC RTK positioning strategy with PAR method is proposed. A set of real baseline observations ranging from about 22 to 110 km are used to test the performance of this strategy, including success rate, convergence time and ratio values. The experimental results are provided to demonstrate the benefits of introducing the PAR method into the TC strategy for multi-GNSS, which is finally followed by the summary and conclusions of this study.

## Methods

### Multi-GNSS observation models

*P*and Φ are pseudorange and carrier phase measurements, respectively;

*ρ*is the distance between the receiver and the satellite; \( \mathrm{ucd} \) and

*upd*are receiver uncalibrated code and phase delays, respectively; These two quantities are related to the initial phase and the hardware phase delays (Gu, 2013); The symbol

*I*denotes the ionospheric delay;

*T*is the tropospheric delay;

*λ*is the wavelength;

*N*is the integer phase ambiguity; \( {\varepsilon}_P \) and

*ε*

_{Φ}are the mixture of measurement noise and multipath error for pseudorange and carrier phase observations, respectively. Note that all variables are expressed in meters, except the ambiguity which is expressed in cycles. Furthermore, the reference receiver is denoted with subscript \( {r}_1 \), the rover receiver is denoted using subscript \( {r}_2 \), the reference satellite is denoted using superscript \( {s}_1 \) and its system is labeled using superscript \( {A}_1 \), the non-reference satellite is denoted using superscript \( {s}_2 \) and its system is labeled using superscript \( {A}_2 \), and superscript

*i*and

*j*refer to carrier frequencies.

The above observation equations for multi-GNSS DD operations can be generalized to inter-system mixed DD which can be further categorized into those between the same frequencies or the diverse frequencies of observations (Li et al., 2017).

Because the frequencies are the same, the ambiguities \( {N}_{r_1{r}_2,i}^{s_1{s}_2} \) still have integer characteristics. However, the receiver UPDs which are related to the initial phase and hardware delay are consequently contained in the inter-system bias (ISB) and therefore cannot be eliminated, i.e. \( {ucd}_{r_{\boldsymbol{1}}{r}_{\boldsymbol{2}},i}^{A_{\boldsymbol{1}}{A}_{\boldsymbol{2}}}\ne 0 \) and \( {upd}_{r_1{r}_2,i}^{A_1{A}_2}\ne 0 \).

### Ambiguity resolution in the DD model

**y**is the vector of ‘observed minus computed’ DD observations; x is the vector of incremental baseline coordinates, the residual tropospheric zenith delay, and the DD ionospheric slant delays for each measurement epoch;

**N**is the vector of carrier-phase integer ambiguities;

**ε**is the vector of unmodeled effects and measurement noise. The matrices A and B are the corresponding design matrices of x and

**N**, respectively.

**X**and variance-covariance matrix

**Q**from a least-squares estimation can be expressed as

*R*, defined as the ratio of the weighted sum of the squared residuals by the second best solution \( {\overset{\smile }{\mathbf{N}}}_2 \) to the best \( \overset{\smile }{\mathbf{N}} \) is used to check the reliability of AR. In general, validation threshold

*R*

_{ thres }can be 1.5 to 3.0 (Wang and Feng, 2012), and we used 3.0 for this study.

If the validation fails, the current epoch will keep the ambiguities float instead.

### Partial ambiguity resolution strategy

Then the LAMBDA method is used to fix \( {\tilde{\boldsymbol{N}}}_b \), and if \( {\tilde{\boldsymbol{N}}}_b \) can be fixed, \( {\overset{\smile }{\mathbf{N}}}_a \) and \( {\tilde{\boldsymbol{N}}}_b \) are used to update \( \overset{\smile }{\mathbf{x}} \) and \( {\mathbf{Q}}_{\overset{\smile }{\mathbf{x}}} \). Otherwise, only \( {\overset{\smile }{\mathbf{N}}}_a \) is used to update \( \overset{\smile }{\mathbf{x}} \) and \( {\mathbf{Q}}_{\overset{\smile }{\mathbf{x}}} \).

First, the PAR process starts with the decorrelation of the ambiguities and the diagonal elements of the decorrelated matrix are sorted in the ascending order. We can get the diagonal element set \( D=\left\{{d}_1,{d}_2,\cdots {d}_i,\cdots, {d}_n\left|{d}_1<{d}_2<\cdots {d}_i<\cdots <{d}_n\right.\right\} \) after sorting by conditional variance. Then, by updating the traversal of *i* = *n* to the minimum threshold *i* = *n*_{0} and pick the subset \( D=\left\{{d}_1,{d}_2,\cdots {d}_i\right\} \) and the corresponding ambiguity subset \( {\widehat{\boldsymbol{N}}}_a\left({D}_i\right) \) and variance-covariance matrices \( {\mathrm{Q}}_{{\widehat{\boldsymbol{N}}}_a}\left({D}_i\right) \). The minimum threshold *n*_{0} is typically 6 to ensure the selected satellites are still sufficient to get reliable positioning results. Then, the LAMBDA method is applied in the ambiguity search process. If \( {P}_S\ge {P}_{S0} \) and \( R>{R}_{\mathtt{thres}} \), the fixed ambiguities can be considered to pass the acceptance test and be used into the following position calculation. Otherwise, we will update the subset and repeat the ambiguity search and test. If the number of selected ambiguities is less than \( {n}_0 \), the iteration will stop and only the float solutions are made available.

## Results and Discussion

Medium and long-baseline data processing strategy settings

Item | Models |
---|---|

Satellites | GPS + Galileo |

Observations | Phase and code observations |

Signal selection | GPS: L1, Galileo: E1 |

Cutoff elevation | 15° |

Sampling rate | 30s |

Observation weight | Elevation dependent weight |

Phase-windup effect | Corrected |

Satellite Antenna PCO and PCV | DD elimination or weakening |

Receiver antenna PCO and PCV | DD elimination or weakening |

Earth tides | Corrected |

Relativity correction | Corrected |

Satellite orbit | Broadcast/Precise ephemerides |

Satellite clock | DD elimination or weakening |

Receiver clock | DD elimination or weakening |

Station coordinate | estimated as parameters |

Tropospheric delay | estimated as parameters |

Ionospheric delay | estimated as parameters |

Phase ambiguities | LAMBDA/MLAMBDA |

AR success rate threshold | 0.999 |

Ratio threshold | 3 |

Minimum threshold | 6 |

### Carrier phase and code ISB estimation

Zero-Baselines used in the experiment of ISB estimates

Number | Station name | Receiver type |
---|---|---|

1 | CUT0-CUT1 | Trimble NETR9-Septentrio POLARX4 |

2 | CUT0-CUT2 | Trimble NETR9-Trimble NETR9 |

3 | CUT1-CUT2 | Septentrio POLARX4-Trimble NETR9 |

We selected the ISB corrections from the experiment in 2017 to correct the ISBs in the long-baseline experiment because the receiver’s brands and firmware versions used in this experiment were the same as that used in the long-baseline experiment. The receiver brands and firmware versions of the long-baseline experiment are shown in Fig. 2.

### Results of ambiguity resolution

Figure 5 shows that the AR success rate and number of ambiguities of the three baselines are similar. However, the number of AR changes more frequently as the baseline length increases. When there is a newly-rising satellite, the success rate will drop dramatically. Fortunately, the TC strategy can provide more satellite observations, at least one more than the LC strategy. In this way, the TC strategy can be faster to reach the success rate at 99%, and has the ability to eliminate the sudden change in the number of satellites.

It is worth noting that the ratio values of the medium-long-baseline and the long-baseline are very different. The ratio of the medium-long-baseline ratio values greater than 3 is about 90% and the long-baseline is just about 5%. This reflects the fact that the long-baseline is affected by the atmospheric delay error and is difficult to satisfy the condition of correct AR. In addition, TC did not improve the ratio values, but deteriorate. For example, the ratio of the NNOR-CUT0 long-baseline to the ratio values greater than 3 under the three different combinations of LC, TC, and tight combination with ISB corrections (TC + ISB) is 4.83, 4.16 and 4.34%, respectively. This indicates that more observations do not increase the ambiguity ratio values, but rather deteriorate.

Figure 6 shows that the application of the PAR method can significantly shorten the convergence time. The three baselines under different combinations can all reach the success rate of 99% within two or three epochs.

For medium-long-baselines, the number of resolved ambiguity is relatively stable and the problem of newly-rising or falling satellites is effectively suppressed. Because the ambiguity subset selection process can remove these satellites according to the variance of their ambiguity, and the ambiguity variance of such satellites is generally large. In the following epochs, their ambiguity precision improves, and their corresponding ratio values have also been improved, the ratio values also drops a lot only in the number of ambiguity changes epoch, but it still meets the threshold.

However, for long-baselines, the number of ambiguities to be fixed changes frequently. Because of the influence of atmospheric delay and so on, the ambiguity subset can’t be fixed, even when the number of ambiguity reaches the threshold. Moreover, the frequent occurrence of falling satellites also affects the ambiguity subset fixed. Since the ambiguity subset can’t be fixed, only the float solutions are available and then the ambiguity subset selection is tried again in the next epoch. However, the PAR method is still able to greatly improve the ambiguity fixed rate. For example, the ratio of the NNOR-CUT0 long-baseline to the ratio values greater than 3 under the three different combinations of LC, TC, and TC + ISB for the PAR method is 27.36, 27.23 and 27.30%, respectively. Compared to the FAR strategy, it was increased by 22.53, 23.07 and 22.96%, respectively.

### Results of positioning

Figure 7 shows that the medium-long-baseline error (1.5 cm) is smaller than that of the long-baseline (10 cm) static relative positioning for the FAR strategy. Affected by the residual atmospheric errors, such as residual tropospheric and ionospheric delays, the positioning results reflect some systematic biases especially for the longer baseline. The positioning accuracy based on the TC strategy is improved, especially for the medium length baseline. The accuracy of positioning based on TC + ISB is equivalent to that of TC, because the ISBs of this experiment are so small that it can be ignored. The positioning accuracy based on PAR method has improved, especially for the long-baseline. Because of after fixing the partial ambiguities by the PAR method, the fixed integer ambiguities can be back tracked into the observations to update the troposphere and the ionospheric parameters, a more accurate atmospheric delay correction is obtained, and the positioning parameters are updated by fixing the remaining ambiguities. The proportions of fixed solution of the NNOR-CUT0 long-baseline static relative positioning under the three different combinations of LC, TC, and TC + ISB for the FAR method is 11.98, 17.33 and 17.40%, respectively. For the PAR strategy, that is 75.80, 76.70 and 76.96%, respectively.

## Conclusions

A GPS + Galileo tightly combined RTK positioning strategy is proposed for medium-to-long baselines, which introduces the PAR method to the strategy. The method has been verified to be effective for faster and more reliable AR. Tests on middle-long and long-baselines demonstrate that TC strategy can provide more observations, which can improve the success rate. However, TC strategy does not increase the ambiguity ratio values, but rather deteriorate. The reason may be that the TC strategy increases the number of observations and increases the ambiguity dimension. Using the PAR method not only can make initialization time within three epochs, but also improve the ambiguity fixed rate. LC and TC strategies can get centimeter level positioning accuracy, but PAR of the TC strategy can provide better performance, especially for long-baselines. The selection of ambiguity subsets and the elimination of atmospheric delay are the areas that require further research in the future.

## Declarations

### Acknowledgments

This work is funded by National Science Foundation of China (No.41674033) and State Key Research and Development Program (2016YFB0501802). We thank Curtin University for the baseline observations. We are also grateful for the high performance computing facility at Wuhan University which support all the computational work of this study.

### Authors’ contributions

GL developed the algorithm. GL, JGeng and JGuo carried out most of the analyses and drafted the manuscript. SZ and SL participated in the design of the study and helped algorithm development. All authors have read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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