^{1}

^{1}

^{2}

^{2}

^{1}

^{2}

Receiver Autonomous Integrity Monitoring (RAIM) method is an effective means to provide integrity monitoring for users in time. In order to solve the misjudgment caused by the interference of gross error to the least squares algorithm, this paper proposes a RAIM method based on M-estimation for multiconstellation GNSS. Based on five programs, BDS, GPS/BDS, and GPS/BDS/GLONASS at the current stage, the future Beidou Global Navigation Satellite System, and the future GPS/BDS/GLONASS/Galileo system, the new RAIM method is compared with the traditional least squares method by simulation. The simulation results show that, with the increase of constellations, RAIM availability, fault detection probability, and fault identification probability will be improved. Under the same simulation conditions, the fault detection and identification probabilities based on M-estimation are higher than those based on least squares estimation, and M-estimation is more sensitive to minor deviation than least squares estimation.

Receiver Autonomous Integrity Monitoring (RAIM) is an effective method of integrity monitoring [

Suppose that the receiver pseudorange observation equation is

And the least square positioning solution is

Pseudorange residual vector:

Make

The error in the a posteriori unit weight of the pseudorange residual vector is

Therefore, the unit weight error of the pseudorange residual vector

The detection threshold

Then the false alarm probability

According to the above equation, the detection threshold can be obtained. If

Fault detection is based on the test of pseudorange residuals sum of squares, and fault identification is based on the test of pseudorange residual element. And the basic idea is based on Baarda’s data snooping method [

From (

When there is no fault,

Where

From the above formula we can calculate the identification threshold

M-estimation used in this paper is an iterative weighted least squares estimator. Different weights are applied to different points according to the pseudorange residual vector; that is, the points with small residuals are given a larger weight, while those with larger residuals are given a smaller weight. And weighted least squares estimation is then established, repeatedly iterating to improve the weight coefficient.

Different from the least squares making pseudorange residual sum as the extreme function, the extreme function of M-estimation is

Make

Thus the M estimated value of the robustness of the parameter vector is

There are many methods to construct the equivalent weight matrix, but the robust estimates are much the same, and a “normalized” residual index

The following fault detection and identification methods are similar to the least squares RAIM method, and construct test statistics:

No fault assumption

A fault assumption

This paper designs the following five programs, using the self-compiled software for simulation analysis.

The Beidou regional constellation BD2, GPS constellation, and GLONASS constellation all adopt the broadcast ephemeris of 2015-01-12. The Beidou system, BDS, for the future global navigation satellite system will be simulated with 35 satellites (5GEO, 3IGSO, and 27MEO); the Galileo system is simulated with 27 satellites; the specific parameters are shown in Table

Constellation parameters for BDS and Galileo system.

Navigation system | BDS | Galileo system | ||
---|---|---|---|---|

Orbit type | GEO | IGSO | MEO | MEO |

| ||||

Orbital plane | 1 | 3 | 3 | 3 |

Satellite number | 5 | 3 | 27 | 27 |

Semimajor axis/km | 42164 | 42164 | 27906 | 29978 |

Orbit eccentricity | 0 | 0 | 0 | 0 |

Orbit inclination | 0° | 55° | 55° | 56° |

Ascending node | Fixed in the longitude of 58.75°, 80°, 110.5°, 140°, 160° | The intersection longitude is east longitude 118° | 70°, 190°, 310° | 0°, 120°, 240° |

Mean anomaly | The initial time near point angle of first satellite in each orbit is, respectively, 0°, 15°, 30°, followed by an increase of 40° | |||

Argument of perigee | 0° | 0° | 0° | 0° |

First, select Program 2 (GPS + BD2) double constellations, and two kinds of RAIM methods are simulated and compared; the basic simulation conditions are shown in Table

Basic simulation conditions.

Program | Condition |
---|---|

Ephemeris reference time | 2015.1.12 00:00:00 |

Simulation constellation | GPS + BD2 |

Simulation area | (40°N, 116°E) |

Standard deviation of pseudorange noise | 5 m |

Obstacle angle | 10° |

Simulation step | 5 s |

Simulation cycle | 24 h |

False alarm probability | 1/3000000 |

During the specified simulation time interval, 60 m deviation is injected in a satellite. Test statistics and detection threshold statistics of all sampling points of two methods are shown in Figure

Test statistic and threshold comparison of the two methods separately.

It can be seen from Figure

In addition, the M-estimation robustness performance is better; when the fault is not eliminated, an iteratively weighted method based on the deviation is selected for M-estimation, and the impact of the deviation on positioning performance can be reduced eventually, as shown in Figure

Positioning errors of the two methods on the condition that the fault is not excluded.

Least squares

M-estimation

As shown in Figure

Therefore, this paper selects the multiconstellation RAIM method based on M-estimation to simulate and analyze five programs separately. Add the deviation from 5 m to 120 m in the faulty satellite, and the step is 5 m. Simulation with the Monte Carlo method, fault detection probability, and fault identification probability corresponding to five programs are shown in Figures

Fault detection probability of M-estimation.

Fault identification probability of M-estimation.

From Figures

This paper firstly introduces the RAIM method of multiconstellation based on traditional least squares and deduces the test statistic and threshold calculation process of fault detection and fault identification. Then, a multiconstellation RAIM method based on M-estimation is proposed in this paper; at the same time, the fault detection and identification process are deduced. Finally, for the five programs, including BD2, GPS/BD2, and GPS/BD2/GLONASS at the current stage and Beidou Global Navigation Satellite System and GPS/BDS/GLONASS/Galileo System in the future, the RAIM method based on M-estimation is compared with the traditional RAIM method based on least squares by simulation. The simulation results show that the availability of RAIM method increases with the number of constellations, and the fault detection probability and fault identification probability also increase. Under the same condition, the fault detection probability and fault identification probability based on M-estimation method are higher than those based on least squares method, and the M-estimation is more sensitive to the minor deviation than the least squares estimation.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation, China (no. 61502257).