Impact of one satellite outage on ARAIM depleted constellation conf igurations

Qin MENG , Jinye LIU , Qinghu ZENG ,*, Shojun FENG , Rui XU

a Navigation Research Center, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China

b Centre for Transport Studies, Imperial College London, London SW7 2AZ, UK

KEYWORDS Advanced receiver autonomous integrity monitoring;Depleted constellation;One satellite outage;Position estimator;Weighted solution

Abstract Advanced Receiver Autonomous Integrity Monitoring (ARAIM) is a new technology that will provide worldwide coverage of vertical guidance in aviation navigation.The ARAIM performance and improvement under depleted constellations is a practical problem that needs to be faced and researched further. It is a shortcut that improves the availability in position domain whose key idea is to replace the conventional least squares process with a non-least-squares estimator to lower the integrity risk in exchange for a slight increase in nominal position error. The contributions given in this paper include two parts: First, the impacts of one satellite outage on different constellations are analyzed and compared. The conclusion is that GPS is more sensitive and vulnerable to one satellite outage. Second, a constellation weighted ARAIM (CW-ARAIM)position estimator is proposed. The position solution is replaced by a constellation weighted average solution to eliminate the constellation difference.The new solution will move close to the constellation solutions with respect to the accuracy requirement.The simulation results under baseline GPS and Galileo dual-constellation show that the one GPS satellite outage will knock the availability from 91%to only 50%.The performance remains stable with one Galileo satellite outage.With the assistance of the CW-ARAIM method, the availability can increase from 50% to more than 80% under depleted GPS conf igurations. Even without any satellite outage, the proposed method can effectively improve the availability from 91.29% to 98.75%. The test results under optimistic constellations further verify that a balanced constellation is more important than more satellites on orbit and the superiority of CW-ARAIM method is still effective.

1. Introduction

With the development of Global Navigation Satellite System(GNSS), including American GPS, Russian GLONASS,Chinese BeiDou System (BDS) and European Galileo, the increased redundancy and accuracy will dramatically improve the satellite receiver performance.1,2Since the update of BDS in December 2017, the new open service signals B1C and B2a have been transmitted by the Inclined Geosynchronous Orbit(IGSO)satellites and Medium Earth Orbit(MEO)satellites of BDS-3.3These two signals are at the same frequencies with GPS L1/L5 and Galileo E1/E5a, which allows easier and cheaper implementation of multi-standard GNSS receivers.Advanced Receiver Autonomous Integrity Monitoring(ARAIM) aims at achieving worldwide aviation operation capability as a promising candidate of navigation for enroute flight and terminal operations.4,5It is based on dualfrequency multi-constellation GNSS. It also has potential to support lateral and vertical guidance during approach operations, especially LPV-200 approach (localizer approach with vertical guidance with a decision height as low as 200 ft)without the need for barometric vertical guidance.6The ARAIM Technical Subgroup (TSG) created by the leadership of the EU-US Cooperative Working Group (WG)-C has performed a feasibility study of ARAIM and concluded that ARAIM has the potential to achieve signif icant operational benef its worldwide.7,8In contrast to the classical RAIM which can provide horizontal guidance, the vertical guidance will likely require conservative threat models, which are combined with weak geometries,causing large position error bounds and loss of vertical guidance availability.9,10

To realize worldwide coverage of availability, ARAIM is strongly dependent on the number of satellites in view. But the update and satellite removal in constellation are practical problems that need to be faced.11,12Especially one fault satellite has been detected and removed, and how to compensate and maintain the ARAIM performance is a realistic problem.In TSG Milestone 2 and Milestone 3 reports, the above circumstance is defined as depleted constellation,which is considered as the nominal constellation with one satellite removed to account for outages.The above two reports also compared and analyzed the impact of depleted constellations on ARAIM performance under different availability criteria and ISM parameters.7-13But the depleted constellation defined above removes two satellites in total,one satellite from GPS constellation and the other from Galileo constellation. The constellation differences and one satellite outage to single constellation are not considered in the current literature.Compared to two satellites outage, one satellite outage is more likely to happen.

Furthermore, how to improve the ARAIM performance under depleted constellations is another problem that needs to be researched.The classical RAIM is evaluated by assuming that the position solution is obtained via least squares.14,15This approach guarantees the best accuracy under nominal conditions. However, the position that minimizes the error under nominal conditions is not in general the one that minimizes the integrity error bounds. It is therefore possible to reduce the protection level by choosing a different position solution, especially for ARAIM, which has definite accuracy condition for availability. It is possible to f ind a new position estimator in the limitation of accuracy tolerance. One of the techniques to lower the position error bounds in RAIM is to modify the position solution, which is also known as Non-Least-Squares (NLS) estimator. A weighting concept is introduced into NLS estimator by applying the additional weights to the range measurements or with weights analytically derived in the position domain.16,17Both of them can achieve higher availability than conventional RAIM methods, especially the Optimal Weighted Average Solution (OWAS) concept presented in Ref.17which can work with two independent constellations and provide integrity monitoring even in the presence of a common mode fault affecting multiple satellites in one of the two constellations. The same situation has been accepted by ARAIM. But ARAIM has much stricter vertical positioning performance criteria and accuracy constraints.18,19The NLS estimator is more realistic in depleted constellations as the worldwide constellation performance itself decreases.

NLS estimator is just employed heuristic approaches to reduce integrity risk in Ref.16,17Furthermore,the NLS estimator design is casted into a multi-dimensional optimization problem in Ref.20,21which greatly promotes the development of NLS estimator. The above two references use protection level and direct integrity risk evaluation respectively to solve this problem. Both of them can improve the availability dramatically under depleted constellation. But the algorithm is more complex and needs a large amount of computation as the process of f inding the optimal solution involves several iterative processes.A simpler method to determine a position estimator is described in Ref.22and the idea consists in restricting the search of the estimator to all aff ine combinations of the allin-view least squares estimator and a fault-tolerant estimator.However, the simpler method only considers the fault mode with the largest contribution to the integrity risk,and the availability improvement is limited by the fault mode with the largest standard deviation.

A practical position estimator is proposed in this paper.Considering the constellation conf iguration differences and the impact of one satellite outage on the constellation performance,the new position solution is a NLS estimator with constellation subset solution weighted.The proposed method also includes the ARAIM accuracy constraint. The proposed method can improve the ARAIM availability dramatically without any extra burden in calculations and memory resources. The rest of this paper is organized as follows: Section 2 discusses the GPS and Galileo constellation conf iguration and analyzes the impact of one satellite outage on the single constellation. In Section 3, based on the baseline ARAIM user algorithm and the LPV-200 requirements, the proposed position estimator with constellation weighted is presented. Worldwide simulation results and analysis under GPS and Galileo dual-constellation are given in Section 4. Finally,Section 5 is the conclusion.

2. Impact of one satellite outage on depleted constellations

ARAIM is based on multi-constellation. But four global navigation systems: GPS, GLONASS, Galileo and BeiDou have different constellation conf igurations.They have different performance around the world.For ARAIM application,the performance after removing one fault satellite is a practical problem with great concern. The constellation conf iguration is one of the important parameters affecting the ARAIM availability results. The constellation difference and the impact of one satellite outage are analyzed in this section.

2.1. Depleted constellation conf igurations

Fig. 1 GPS and Galileo constellation conf igurations.

Considering the nominal constellation with one satellite removed to account for outages, the concept of‘depleted constellation conf iguration’ is accepted by the EU-US Cooperation on Satellite Navigation to evaluate the performance under imperfect constellation circumstance. As shown in Fig. 1, the baseline conf igurations have 24 GPS satellites and 24 Galileo satellites, where GPS is the 24-slot nominal constellation and Galileo is a Walker 24/3/1. In the depleted conf iguration, a‘24-1’GPS satellites constellation and ‘24-1’Galileo satellites constellation are assumed, and one arbitrarily chosen satellite has been removed from the baseline in each constellation,which is shown as the traditional classif ication in Table 1. As BDS and GLONASS system have a similar constellation conf iguration to Galileo, and the number of MEO satellites is reduced to 24 in BDS-3, and the above baseline constellation combination of 24 GPS satellites and 24 Galileo satellites is accepted to analyze the impact of one satellite outage on different constellations.23,24

Besides the satellite number, the orbital parameters are another important factor that impacts the constellation performance. Compared to GPS, Galileo has a higher orbit altitude and longer repeatability. The main difference between GPS and Galileo is that GPS has six orbits while Galileo has three.Research shows that the worldwide average improves with altitude and three orbital planes seem to be the best solution for almost all altitude considered since an increase in the number of planes would cause degradation in the performance.25The constellation design of Galileo seems to have more advantages than that of GPS.

To further research the impact of depleted satellite from different constellations on the ARAIM performance,a refined classif ication is presented in this paper. As shown in Table 1,the depleted conf iguration will be refined to three situations:depleted GPS conf iguration, depleted Galileo conf iguration and depleted whole conf iguration, corresponding to one GPS satellite outage,one Galileo outage and one GPS satellite plus one Galileo satellite outage.

2.2.Number of satellites in view and vertical dilution of precision

In this subsection, number of satellites in view and Vertical Dilution of Precision(VDOP)are taken to evaluate the performance of two independent constellations. The satellite DOP depends solely on geometry and is independent of receiver position algorithm according to its def inition. The availability of ARAIM focuses on the vertical position coordinate, for which the aircraft approach navigation requirements are often the most difficult to fulf il. Number of satellites in view and VDOP are chosen further to evaluate the impact of one satellite outage on the whole constellation.The VDOP is defined as

where G is the geometry matrix.

The statistic results of four constellation conf igurations are shown in Fig.2 and Fig.3.The mask angle is 5°and the results are the average values of a day and the time interval is 600 s.The range oflatitude is 70°S-70°N and the interval is 5°. As the performance is symmetric in the northern hemisphere and the southern hemisphere, the f igures only show results at four classical latitudes: 0°, 20°N, 40°N and 60°N, which can represent the performance in the equator,low latitudes,middle latitudes and high latitudes respectively. The western hemisphere is shown in negative longitudes in f igures.

About the satellites in view,GPS and Galileo have the similar trend.In low and high latitudes,the number of satellites in view is more than that in middle latitudes.Both two constellations are affected by the one satellite outage. But the trend remains unchanged. Compared to GPS, Galileo has more visible satellites at the same latitude, no matter the system has a baseline conf iguration or a depleted conf iguration.Galileo has a wider amplitude of variation around the equator.

Table 1 Constellations of depleted and baseline conf igurations.

Fig. 2 Number of satellites in view under four constellation conf igurations.

Fig. 3 VDOP under four constellation conf igurations.

Focusing on the VDOP, in terms of the baseline constellations, the performance of GPS and Galileo at different latitudes is different. Compared to the vertical performance of GPS which is between 1.5 and 1.6 at every latitude and has average performance worldwide,Galileo has the lowest VDOP at high latitudes with respect to any latitude of GPS, which is less than 1.5.But the performance at low and middle latitudes is lower than that of GPS. We will pay more attention to the impact of one satellite outage on two constellations. With one satellite outage,the performance of both GPS and Galileo declined. But the most obvious phenomenon is that GPS has two clear peaks at 40°N corresponding to two ‘black holes’in performance.The VDOP of most areas at 40°N is less than 1.7,but the VDOP in two‘black holes’rises to near 2.0,which means that GPS is no longer a balanced constellation and the vertical performance in middle latitudes has a dramatic decline. The two peaks are symmetric and in the same longitude circles (the specific longitudes are less important as the removed satellite is random). On the other hand, though VDOPs at different latitudes rise in varying degrees,the global performance of Galileo is still balanced in depleted conf iguration and there is no obvious area that has severe performance loss.

Overall,there are differences between the worldwide performance of baseline GPS and Galileo constellations.The impact of one satellite outage will expand this difference where GPS is more sensitive and vulnerable. The depleted Galileo has more balanced performance compared to the depleted GPS. As integrity examines the consistency of every measurement, the difference between constellations is negative to ARAIM performance. In the next section, the baseline ARAIM algorithm is analyzed and a new NLS position estimator considering the constellation performance differences is proposed.

3. NLS position estimator with constellation weighted

3.1. Baseline ARAIM user algorithm

The baseline ARAIM algorithm is based on multiple hypothesis solution separation (MHSS).26In ARAIM multiple hypothesis, the fault modes that need to be monitored will be calculated with the help of integrity support message(ISM). The parameters in ISM include the constellation prior fault probability and satellite prior fault probability.Different from the traditional RAIM which can only monitor single satellite fault, in ARAIM, the constellation faults and satellite faults are independent events in fault mode determination. A fault mode is a combination of events. Not all fault modes need to be monitored.ARAIM introduces PTHRESas threshold for the integrity risk coming from unmonitored faults. The ARAIM can work when the probability of unmonitored fault modes is less than PTHRES.

In ARAIM, one fault mode determined in the multiple hypothesis corresponds to a fault-tolerant subset calculated in solution separation. The fault-tolerant position solution is defined as

where S（k）= (GTW（k）G)-1GTW（k）is the vector of coefficients that projects the pseudorange residuals y onto the positionG and W（k）are the geometry matrix and the weighting matrix respectively, and k represents the k th subset.

In ARAIM solution separation, the associated outputs of the subsets corresponding to the fault modes will be computed.The subset contains all satellites except the fault event corresponding to the fault mode. If the fault mode is GPS constellation fault, all the GPS satellites will be removed in the faulttolerant position. The outputs of the k th subset are shown in Table 2.

In Table 2, the latter three values should consider the east,north and up components respectively; Pevent,iand idx are the prior fault probability of the i th fault event and the event index corresponding to the fault mode; bnom,iis the maximum nominal bias for satellite i used for integrity.

Among the outputs of all subsets,is used for threshold tests and the determination of EMT. The other three parameters are used to calculate the protection levels. The solution separation tests are defined as

The thresholds are defined by

where Kfais not the same for different components. Let the index 1, 2 and 3 designate the east, north and up components respectively, and then

where PFA HORand PFA VERTrepresent the continuity budget allocated to the horizontal and vertical mode respectively,Q is the right hand side cumulative distribution function of a zero mean unit Gaussian,and Q-1is the inverse of Q function.If any of the tests fails, exclusion must be attempted. Only if all the tests pass, the PL is obtained by taking the maximum across the PLk, where PLkprotects the user against fault mode k.

Table 2 Subset outputs of ARAIM solution separations.

The step-by-step specification about above solution separation calculation can be found in Ref.26Only when all the four outputs meet the requirements of LPV-200, the ARAIM is available.

3.2. Requirements of LPV-200

The target operational level for ARAIM is LPV-200. The ARAIM studies to date have focused on meeting the requirements for LPV-200 procedures.27These requirements include:the Vertical Alert Limit (VAL) of 35 m; the Horizontal Alert Limit (HAL) of 40 m; the Effective Monitor Threshold(EMT) of 15 m and the vertical accuracy of 1.87 m. ARAIM is considered to be available in the corresponding environment and LPV-200 operation, while above index criteria calculated

3.3. Constellation weighted ARAIM position estimator

As shown above, the output position solution in baseline ARAIM algorithm is the all-in-view position solution based on the traditional Least-Squares (LS) method. To illustrate the impact of position solution on the ARAIM availability,the calculations of outputs are combined and evaluated together.For the protection levels, we will pay more attention to the VPL as the calculations of VPL and HPL are similar and the global vertical guidance for aviation is the most ambitious goal in LPV-200. Actually, the ARAIM availability is more affected by VPL according to the existing literature. As S（0）/S（k）are vectors mapping the pseudorange residuals to position solutions, for the convenience of expression,S（0）/S（k）are regarded as position solutions in this section.

by the ARAIM algorithm are satisfied. So the outputs of ARAIM should include VPL/HPL, EMT and σacc.

(1) VPL/HPL: denotes the Vertical/Horizontal Protection Level and the limit on the position error is 99.99999%.

where PHMI represents the probability of hazardously misleading information, and Nfaultsrepresents the total number of fault modes.

(2) EMT: the Effective Monitoring Threshold (EMT) test prevents faults that are not suff iciently large to be detected from creating vertical position errors greater than 15 m more than 0.0001% of the time. The EMT can be defined as the maximum of the detection thresholds of faults that have a prior probability equal to or above PEMT.

(3) σacc: the standard deviation of the position solution under the 10-7fault-free positional error bound.

where e3denotes a vector whose 3rd entry is 1 and all others are 0.

As shown above,on one hand,this LS position solution S（0）is used for the calculation ofand ΔS（k）.Then ΔS（k）will be used to calculate EMT and VPL further.Particularly,the maximum ΔS（k）determines the final EMT and affects the final VPL as VPL output is the maximum among all VPLs obtained from subsets. The deviation of S（0）will affect EMT, σaccand VPL simultaneously, and then affect the ARAIM availability.It is feasible to adjust S（0）to reduce VPL and EMT, and improve the ARAIM availability further.But the new solution should meet the accuracy requirement σacc,reqat the same time.

On the other hand, ΔS（k）is determined by the geometry matrix G and the diagonal weighted matrix W（k）.If the satellite is assumed to be fault in one subset, the corresponding diagonal element in W（k）will be set to zero. The result is that the above satellite is removed and will not be involved in positioning. ΔS（k）ref lects the difference of geometry conf iguration after the removal of fault satellites. Among the fault modes determined in ARAIM,it is easy to understand that the impact of constellation faults on the all-in-view position solution is much more serious than other satellite faults as all the satellites which belong to the corresponding fault constellation will be removed in the subset positioning (G must be redefined by moving the corresponding column). Reducing the ΔS（k）calculated in constellation fault subsets will effectively contribute to EMT and VPL, and then improve the ARAIM availability.

As S（0）is a matrix of m×n（ ）,where m and n are unknowns in position calculations and number of satellites,the NLS position estimator in ARAIM is a multi-dimensional optimization problem.The resolution is a complex iteration process and has a heavy computational load.As the MHSS itself has heavy calculation burdens, it is unwise and unbearable to spend more processing time and memory resources to solve a new position estimator.

A NLS position estimator, called constellation weighted ARAIM (CW-ARAIM), is proposed in this section. This method accepts most of the MHSS algorithm in baseline ARAIM to ensure that it can improve the ARAIM performance signif icantly without any extra complexity. The proposed position estimator with constellation weighted method is an improved OWAS method. The concept of OWAS was first proposed in Ref.17Two methods, named OWAS-1 and OWAS-2,are presented.OWAS-1 works with a single constellation and assumes a single fault at a time, which is not applicable to ARAIM. OWAS-2 works with two or more independent constellations. The navigation solution would be obtained not from the all-in-view solutions but from a weighted average of the subset solutions where the weights would be optimized to minimize the protection levels. However, the OWAS method does not consider the performance differences of constellations.The weight ratio given to constellation is derived to minimize the VPL.But in ARAIM,the user algorithm is a new structure. The final VPL is determined by probability of fault mode and outputs of all subsets, which is shown in Section 3.2. Compared to OWAS-2, a new weight,which is chosen from the subset outputs considering the constellation performance differences, is adopted in CWARAIM. Furthermore, CW-ARAIM is based on ARAIM concept and inherits most of the baseline ARAIM user algorithm. It is to meet LPV-200 requirements.

As analyzed in Section 2, the performance of GPS and Galileo and the impact of one satellite outage on the above two constellations are different. So, it is practical and necessary to put weights of constellation into the NLS position estimator to reduce this difference. A vivid concept map of the proposed method is shown in Fig.4.Note that this is a concept map, not a result diagram as it is actually a m×n（ ） dimensional space.

Firstly, a weighted average solution S（w）is calculated with the Galileo constellation fault subset and GPS constellation fault subset, and their corresponding position solutions are S（GPS）and S（GAL）respectively.The weight is σ2ss,corresponding to the difference between the all-in-view position solution ^x0（）and the fault tolerant solution^xGPS（）and^xGAL（）.S（w）is given as

Fig. 4 Concept map of constellation weighted position estimator.

Secondly, with respect to the accuracy constraint, a new position estimator S（N）between the all-in-view position solution S（0）and weighted average solution S（w）is searched with the explicit ARAIM accuracy requirement. The solutions are given by

It is straightforward that it is a quadratic in equation with one unknown and the maximum t will be chosen as we want that S（N）is close to S（w）as much as possible.

Finally, the new position estimator S（N）will replace S（0）to calculate the protection levels and EMT for ARAIM availability again.The calculation and judgement of σ2acccan be ignored as it has been executed in the second step.

In CW-ARAIM, the new S（N）will reduce the ΔS（k）corresponding to the constellation fault solution. The purpose is to reduce the geometry difference caused by different GNSS constellations. To guarantee the best accuracy and reduce the computational burden as much as possible, the weighted position estimator will not be triggered until the ARAIM performance based on all-in-view position solution fails to meet the LPV-200.

4. Worldwide simulation and verif ication

The simulation conditions include three parts: the constellation conf igurations, the availability criteria and the receiver parameters for ARAIM. The refined classif ication in Table 1 is chosen as the constellation conf igurations in this section.The availability of ARAIM is the core evaluation index. The availability criteria for LPV-200 are given in Table 3. At the same time, the choice of ARAIM receiver parameters is made from TSG Milestone 3 Report7and they are given in Table 3.

Users are simulated on a 5 by 5 degree grid,for a period of 10 sidereal days,which is the repetition rate of the Galileo constellation with a time step of 600 s. So 2592×144×10 = 3732480 points are available for analysis.In turn,the main outputs about ARAIM include 99.5%availability coverage, VPL, HPL, EMT andσacc. Then, for each user, the 99.5% availability means that ARAIM is available during 99.5%of the time at the user grid point.For coverage,the user grid points are weighed by the cosine of the latitude to account for the relative area that they represent.

Table 3 Simulation conditions.

4.1. Results under baseline constellation conf igurations

The coverage results under baseline constellation conf igurations (GPS 24+Galileo 24) are shown in Table 4. With the baseline LS method, the availability under baseline constellation is 91.29%. But the impacts of one GPS satellite outage and one Galileo satellite outage are greatly different. Instead of affecting the availability, one Galileo satellite outage will increase the availability slightly. The trends are similar from baseline constellation to Galileo depleted constellation and from GPS depleted constellation to depleted whole constellations. The reason is that integrity evaluates the consistency of every subset solution. One Galileo outage can just reduce this difference partly and improve the consistency with solution of GPS to a certain degree, which results in a slight increase in ARAIM availability. On the other hand, it can be also found that the loss of availability is mainly caused by GPS satellite outage. The loss exceeds 37% and it is a heavy hit to the ARAIM performance. The Galileo constellation with 3 orbital planes shows obvious superiority compared to the GPS constellation with 6 orbital planes.

Focusing on CW-ARAIM, both the baseline constellation and Galileo depleted constellation have a nearly 100% availability. Compared to the LS method, the growth is more than8 percentage points. Even though with a GPS satellite outage,the availability can reach 80.74% from only about 50%. The increase is up to nearly 27%. Furthermore, the CW-ARAIM can eliminate the difference caused by constellation conf iguration as a trade-off is reached between two constellations.With the limitation of space, only the ARAIM coverage under depleted whole constellations (GPS 23+Galileo 23) is given in Fig. 5. It can be seen that except the ‘black holes’ caused by GPS satellite outage, the CW-ARAIM method can compensate the availability loss in other areas effectively.

Table 4 LPV-200 coverage of two ARAIM methods under different constellation circumstances.

The same conclusion can also be obtained from the VPLs calculated by two methods under four conf igurations.=gives the maximum VPLs in 99.5%of the time at the user grid point over a 10-day period. These results are clear in baseline constellation and depleted Galileo constellation. Note that 35 m is the VPL threshold in LPV-200, which is marked by black dotted line in Fig. 6. A lot of user points at 20°N and 40°N exceed this threshold and cause availability loss. The CWARAIM method can reduce the VPL obviously and this improvement is global and position-independent. The improvements have an important signif icance to the civil aviation guidance as most of the airports and population are in mid-latitude regions. On the other hand, the GPS unbalance caused by one satellite outage is also ref lected in ARAIM availability. The VPLs near (40°N, 120°W) and (40°N, 60°E)exceed 150 m. This ‘trajectory’ just corresponds to the VDOP peaks in depleted GPS constellation.This loss cannot be made up by constellation weighted solutions as LPV-200 has definite requirement in accuracy. The availability loss caused by depleted GPS constellation needs other means to compensate,maybe from the space segment or control segment. But in other areas, the CW-ARAIM method can still reduce the difference between constellation solutions and improve the availability positively.

4.2. Results under optimistic constellation conf igurations

Particularly, considering the fact that 24 satellites are only baseline conf iguration,current GPS has more than 30 satellites on orbit and Galileo has more than 27 operational satellites in the future. We evaluate the impact of one satellite outage on the optimistic conf igurations proposed in TSG Milestone 3 Report, where both GPS and Galileo have 27 satellites. The coverage results are shown in Table 5.

Fig. 5 Global ARAIM availability under depleted whole constellations.

Fig. 6 VPLs calculated by two methods under four constellation conf igurations.

Table 5 LPV-200 coverage of two ARAIM methods under optimistic conf igurations.

Fig. 7 Global ARAIM availability under depleted ‘GPS 26+Galileo 26’ conf iguration.

The performance of the baseline method is analyzed first.When both the GPS and Galileo are in optimistic status, the coverage of baseline method is 95.88%. One Galileo satellite outage has no negative impact on the worldwide performance but a slight improvement.The impact of one GPS satellite outage is obvious and the ARAIM availability coverage decreases by nearly 7%.The trend of the impact of one satellite outage is similar to the baseline constellations. On the other hand, the CW-ARAIM method can improve the coverage under optimistic conf igurations to 99.51%, almost 100%. With one GPS satellite outage, the coverage can improve to 95.08%from below 90%. The improved performance approaches to the performance of baseline method under optimistic conf igurations.The CW-ARAIM method still shows great superiority in dealing with depleted conf igurations.The ARAIM coverage under ‘GPS 26+Galileo 26’ conf iguration is given in Fig. 7.The experiments of baseline and optimistic constellation conf igurations show that GPS is often more vulnerable to the one satellite outage. The constellation weighted method can reduce the differences between two constellation solutions and improve the availability coverage effectively.

One result that should be noted is shown in the first row of Table 4 and the last row of Table 5. With CW-ARAIM method, conf iguration ‘GPS 26+Galileo 26’ has four more satellites than conf iguration ‘GPS 24+Galileo 24’, but the coverage of the former is lower than that of the latter(95.08% versus 98.75%). The reason is that the former is depleted constellations whose nominal conf iguration is 27 satellites.One satellite outage destroys the balance of satellites distribution globally.On the other hand,the latter is a nominal constellation where the 24 satellites are under a most reasonable distribution. The result further verif ies that the impact of one satellite outage on ARAIM coverage is that the balance of constellation is destroyed. Compared to more satellites on orbit, a balanced constellation is more important to ARAIM performance. Another supporting fact is that the number of MEO satellites in BDS-3 has been simplif ied to 24 from 27,which means the ‘optimistic’ BDS can provide global services with only 24 MEO satellites.

5. Conclusions

(1) As the constellation conf igurations of GPS and Galileo are different, the analysis about vertical performance between baseline constellations and depleted constellations show that the global performance of GPS is easy to lose balance in middle latitudes with one satellite outage. The impact of one satellite outage on GPS constellation is more serious than that on Galileo.

(2) A practical NLS position estimator, called constellation weighted ARAIM(CW-ARAIM), aiming at f inding the optimal weights in the trade-off between integrity and other constraints, is proposed in this paper. As the constellation fault will cause large geometry difference, the purpose of the proposed method is to give more weights to the constellation subset to decrease this difference.The weighted solution considers the GPS constellation solution, Galileo constellation solution and LPV-200 accuracy constraint synthetically.

(3) Under baseline constellation circumstance,on one hand,the GPS constellation is vulnerable. The availability is knocked by the GPS satellite outage seriously and the Galileo satellite outage does not affect the availability.On the other hand, the proposed CW-ARAIM method can improve the availability effectively under both baseline constellation and depleted constellation. The availability can reach 98.75% under baseline constellation.Even under depleted GPS constellation, the CWARAIM method can increase the availability by 27 percentage points, which can compensate the performance loss caused by the satellite outage to the maximum.The experiments under optimistic constellation conf igurations further verify the above conclusions and the effectiveness of the proposed CW-ARAIM method is conf irmed ulteriorly.

(4) Compared to more on-orbit satellites, a balanced constellation is more important to ARAIM availability.Compared to GPS of six orbits, the Galileo of three orbits has more balanced performance with respect to one satellite outage.

(5) As the proposed CW-ARAIM method has no increase in complexity and calculation, engineering applications can be expected simultaneously.

Acknowledgements

This research has been funded by the National Natural Science Foundation of China (Nos. 61533008, 61374115, 61328301 and 61603181, the Funding of Jiangsu Innovation Program for Graduate Education of China (No. KYLX16_0379), and the Open Fund of State Laboratory of Information Engineering in Surveying, Mapping and Remote Sensing, Wuhan University of China (No.17P02).