An Equivalent-Satellite Method Exploiting Spatial Distribution to Reduce Fault Modes for ARAIM
Hangtian Qi, Xiaowei Cui, Xiang Wang, Gang Liu, Mingquan Lu, Tsinghua University, China
Location: Beacon B
To ensure the GNSS application in the safety of life, integrity technology receives more attention. Receiver autonomous integrity monitoring (RAIM), as a technique of integrity, can perform fault monitoring by utilizing the observation information inside the receiver or some simple auxiliary information methods with a short alarm time. Compared with traditional RAIM, advanced RAIM (ARAIM) employs dual-frequency and multi-constellation satellite signals to provide vertical services for navigation users. However, the multiple hypothesis solution separation (MHSS) algorithm, the basis of ARAIM, monitors more numerous fault modes to meet the probability of hazardously misleading information (PHMI) requirements, and users need to solve the position solution for every fault mode by weighted least squares to obtain the separated statistical information. In the case of four constellations and high fault probabilities, the quantities of fault modes even reach thousands which poses a severe challenge to the computing power of the receiver. Therefore, it arises to relieve the pressure of ARAIM calculation without loss of protection levels (PLs). Some fast calculation methods like iteration and approximation in the ARAIM process play a limited role, and the direct and most effective way is to reduce the number of fault modes. Obviously, it is simple to discard some visible satellites without changing the module of fault mode determination but this approach compromises measurement redundancy and damages the performance of ARAIM.
Currently, most research on reducing ARAIM fault modes focuses on subset consolidation although there are a few other methods that are either not effective enough or not suitable for single point positioning. The subset is another notation for fault mode as the fault modes are subsets of all-in-view satellites. Subset consolidation is a powerful measure that merges multiple fault modes into one wide fault mode. Naturally, the wide fault and contained multiple faults make a decisive effect on the complexity, capability of reducing the number, and PL for a specific subset consolidation method. There are mainly four ways with unacceptable shortcomings in the current literature. The first one utilizes one constellation wide fault to replace any faults in the constellation. However, the defect that the weak satellite geometry severely after separating satellites makes the PL increased drastically limits it to apply in practice. The second improved the first method and only two and more satellites faults instead of any faults are included in a constellation fault to compensate for the geometry. It not only becomes complicated but also impairs the ability to reduce the number of subsets, so it is inapplicable for the multi-constellation situation. The third is clustered ARAIM which divides visible satellites into some clusters where a cluster fault contains any fault in the cluster. Unfortunately, clustered ARAIM does not provide a foundation for dividing clusters so that it is hardly practical. In addition, the performance of clustered ARAIM is not stable enough and sometimes worse than baseline. The last one introduces the orbital plane as a basis for cluster division which reputes all satellites in the same plane as a cluster. The orbital method is a kind of clustered ARAIM on the whole and the orbital plane is difficult to be theoretically related to ARAIM performance.
For the existing subset consolidation methods, there is a common feature that the performance of ARAIM becomes wobbly. PLs are sometimes higher or lower than baseline at different times or in different regions. Available research does not explain this phenomenon and it is consequently difficult to control. Our attention is caught by periodical high performance which indicates that not only the number of subsets but PL is also reduced. If we can analyze the cause of this unsteadiness and design a new algorithm to suppress poor performance based on this reason, it mutually benefits the computational burden and PLs of ARAIM. This is the main objective and goal of our research.
2 An Equivalent-Satellite Method Exploiting Spatial Distribution to Improve the Fault Mode Determination
In this work, we propose a stable and easy-to-operate method that improves ARAIM performance while significantly reducing fault modes.
As mentioned above, we first explain the fluctuation of the subset consolidation method by mathematical analysis to provide design considerations for new methods. It is one of our innovations that clarifying two key parameters affecting the performance of ARAIM, i.e., the number of subsets and the geometric dilution of precision (GDOP) after separating satellites. From this, a criterion of subset consolidation is obtained, that is, the increment of the GDOP after separating multiple satellites is minimized as much as possible while the number of fault modes is drastically reduced. Then we propose an equivalent-satellite method based on spatial distribution, where equivalent satellite and spatial distribution guarantee the ease of operation and geometry respectively. One equivalent satellite is composed of multiple satellites of which any faults are consolidated into the equivalent satellite wide fault. In fact, the multiple satellites are regarded as one satellite in the whole ARAIM processing, which simplifies the procedure. Thus we note it as an equivalent satellite method while it is named clustered ARAIM in some articles. Afterward, we devise a meticulous generation of equivalent satellites. Every 4 satellites and the remaining satellites less than 4 are taken as an equivalent satellite which has the following characteristics:
(1): All satellites in the equivalent satellite belong to the same constellation.
(2): Satellites from one constellation are divided into four groups according to their elevation and azimuth angles. The allocation strategy refers to the 4-satellites selection theory to guarantee the geometry. The four satellites of an equivalent satellite are composed of each of the four groups.
The two characteristics can be considered from two aspects. On the one hand, each equivalent satellite after removing hypothetical satellites can be applied for a precise position due to superior geometry brought by satellite selection. This ensures that the GDOP of the remaining satellites maintains good. On the other hand, there are still multiple satellites in the same groups to compensate for the separated geometry of taking out malfunctioning satellites. This means that geometric damage cannot be much. It should be pointed out that the entire division refers to the tactic of sub-optimal satellite selection which just needs judgment and sorting, so the additional complexity is extremely low. After obtaining the divided equivalent satellites, the rest of the process is almost the same as the baseline. The mathematical analysis and flow of the entire algorithm shall be described in detail in my paper. It is our main innovation.
We employed the baseline, the second and fourth of the above four methods, and the equivalent satellite based on spatial distribution proposed for comparative analysis. Further, they are noted as Baseline, Cutdown, Orbit, and Geometric Equivalent-Satellite (GES) method respectively. The simulation conditions were set to four constellations and high failure probability. First, we compared the number of subsets which is taken as the maximum under the same number of visible satellites. Orbit and GES bring a sharp decrease in fault modes because they are both derived from the equivalent-satellite method. However, there are too many equivalent satellites in the Orbit, 17 at all, to result in a higher number of subsets than the GES. GES which adopts a more reasonable equivalent way makes the number fell most to 6% of the Baseline while the number is reduced to 1/14 in theory after calculation. Taking into account the fault feedback and the situation that an equivalent satellite contains 4 faults, the actual number shall be smaller than1/14. Therefore, the value of 6% is credible. Second, the Earth's 99.5% availability of each method experimented. The 99.5% value means the value can cover 99.5% of all values in 24 hours and 99.5% availability is a property of an area of which the 99.5% value meets the LPV-200 requirements. In the case of four constellations, all regions in the world are available. 99.5% values of the Baseline and Cutdown are larger and GES's are the smallest generally. Third, we provided statistics for each method. From the statistics, the performance of the Cutdown is almost equal to that of the Baseline while the Orbit method is better than the Cutdown and Baseline methods overall. However, the maximum value of the statistic of the Orbit is the largest, indicating that it is not stable enough. This is because the satellite geometry of each simulation point cannot be satisfactory steadily. The GES method is undoubtedly the best regardless of whether it is from the average or the maximum value. Last, to reflect the superiority of the method proposed, the results of 188624 simulation points from the GES were all compared with the Baseline. We find ratios of vertical PL (VPL), horizontal PL (HPL), and effective monitor threshold (EMT) are all less than 1 after dividing the results of the GES by the Baseline. Simulation results show that compared with the Baseline method this method can not only greatly reduce the fault mod but also decrease the PLs stably.
Aiming at the computational burden caused by a large number of fault modes with multi-constellation conditions in ARAIM, we propose an equivalent-satellite method based on spatial division from two key factors obtained by mathematical analysis. With the result of reducing the number of subsets substantially, this way firmly dwindles PLs by remaining the geometry after separation as much as possible. We choose three algorithms to compare with the GES method. Simulations show that the number of subsets is down to 6% compared with Baseline, as well, the performance of ARAIM is stably highest. The proposed method can be a helpful extension of multi-constellation ARAIM in the future.