A Clock and Ephemeris Algorithm for Dual Frequency SBAS

Transcription

1 A Cloc and Ephemeris Algorithm for Dual Frequency SBAS Juan Blanch, odd Walter, Per Enge. Stanford University. ABSRAC In the next years, the new GPS and Galileo signals (L1, L5) will allow civil users to remove the ionospheric delay in the pseudoranges. his will have a large impact on the Satellite Based Augmentation Systems (SBAS), as the ionospheric delay is currently the largest error. Once this source of error is removed, the Vertical Protection Levels will decrease substantially, and other error sources will dominate. he remaining terms in the error bound were much less critical than the ionospheric delay error bound, so they have received less attention. It is therefore liely that they can still be optimized. his is true in particular for the User Differential Range Error (UDRE) algorithm which computes the cloc and ephemeris error bounds. In addition, new SBAS messages will be broadcast in the L5 channel, and their content is still not fixed. herefore, it is a good opportunity to determine whether changes can be made both in the UDRE and Message ype 8 (M8) computation and transmission to increase overall SBAS performance. In this wor, we propose an algorithm to compute the error bounds on the cloc and ephemeris in SBAS. As opposed to the current Wide Area Augmentation System (WAAS) UDRE algorithm, this algorithm computes the UDRE and M8 simultaneously and taes into account receiver failures explicitly. We will evaluate the performance of the algorithm and compare it to the current UDRE and M8 algorithm to determine whether its implementation for dual frequency SBAS would be worthwhile. L1-L5 WAAS is being developed [1] to tae advantage of the second civil signal in the L5 frequency band. his second signal will allow receivers to estimate and cancel the effect of the pseudorange delay induced by the ionosphere. Since this delay is the most important source of uncertainty in single frequency SBAS [], it is the largest contributor to the user position error bound. Once the ionospheric error bound is removed, the largest contributor will be the term bounding three different sources of error: the cloc and ephemeris error, the codecarrier coherence (CCC), and the signal deformation (SQM) [1]. his term is designated in the Minimum Operational Performance Standards [3] as σ t. In the case of WAAS, this term is the product of the UDRE and a shaping matrix contained in Message ype 8 [4]. he broadcast index UDRE is the maximum of the output of the UDRE algorithm and the oor imposed by both the CCC and SQM monitors. WAAS today provides vertical guidance in the conterminous U.S and Alasa with very high availability. However, if we want to either achieve better levels of service or be more rust against depleted constellations, it will be necessary to reduce the Protection Levels. his could be done by modifying the Vertical Protection Level [5], [6] or by reducing the term σ t. With the development of WAAS dual frequency, there is an opportunity to upgrade the algorithms. In this paper, we outline the broad lines of a cloc and ephemeris algorithm that has the potential to reduce the WAAS error bounds significantly. his algorithm uses ideas similar to the ones described in [11], but differs in some ey points. In the first section we will outline the threats and the message constraints that the current WAAS cloc and ephemeris algorithm accounts for, and must be accounted in a new algorithm. he second part will show how each of these constraints can be accounted for. In the third part we will summarize the algorithm. he fourth part will show the potential benefits of the new algorithm as compared to the current one. Finally, we will add a few remars on the implementation of this algorithm. INRODUCION HREA MODEL AND MESSAGE CONSRAINS he current cloc and ephemeris algorithm has evolved to account for: - Nominal error from the networ receivers - Nominal biases (antenna biases) - he use of corrections that are generated outside the safety processor - Possibly undetected errors in the networ receivers (one station is assumed to return erroneous measurements)

2 PROBLEM SAEMEN he pseudorange error due to the cloc and ephemeris for a user with a line of sight u LOS is given by: LOS ( ) u x x Broadcast he 4 by 1 vector x represents the true satellite cloc and ephemeris, and x Broadcast represents the cloc and ephemeris computed by the receiver after applying the SBAS corrections [7]. he prlem consists on finding an upper bound on this expression for each satellite-user pair. he error bound needs to be of the form: = σ = σ Cov 8 Lu K K u u LOS t UDRE LOS M LOS he matrix Cov M 8 is a 4 by 4 matrix that is sent every 10 s per satellite. Every 6 s, it is possible to modify it by multiplying it by σ UDRE. K is the factor assumed by the receiver and is Measurements and prior distribution Every second, the ground networ receivers collect pseudorange measurements to all GPS satellites in view in L1 CA and L semi-codeless. hese measurements are processed to tain an ionospheric delay free carriersmoothed estimate of each pseudorange [8]. Let y be the vector of smoothed measurements from the ground receivers to one satellite corresponding to one epoch. After linearization, the relationship between the ground pseudorange measurements and the satellite s true cloc and ephemeris can be represented by: y = Gx+ n G is a matrix where each row represents the line of sight to one of the WAAS stations. he vector n is the noise affecting each measurement. his noise is characterized by a Gaussian whose standard deviation is give by the Code Noise and Multipath (CNMP) curve [8], as well as an antenna bias in the order of tens of centimeters which is deterministic, but very difficult to calibrate [9]. he noise is modeled as a gaussian random vector with covariance W -1 and bias b. 1 (, ) n N b W he bias b is unnown but its magnitude is bounded by b max : b b max In addition to the measurements, WAAS assumes a conservative prior distribution of the position of the satellite, which we note x prior. he inverse of the covariance of the prior is given by P, and its magnitude can be found in [4]. ERROR BOUND DERIVAION Error bound on the estimation error in nominal conditions he approach taen in this wor is to estimate the cloc and ephemeris of the satellite using the above equations. If we neglect for the moment the nominal biases b, the optimal estimate is given by a minimum mean square estimator: x = x + P+ G WG G W y Gx Estimated prior prior he covariance of the estimation error is given by: So we have: 1 Cov = P + G WG ( ( ) ) = ( ) LOS Estimated HMI LOS LOS HMI P u x x K u Cov u Q K where Q is the cdf of a normal unit Gaussian. K is related to the integrity allocation PHMI alloc through the equation: PHMI = Q K alloc he error bound is then given by: 1 HMI L u = K u Cov u LOS HMI LOS LOS his error bound fits within the message format. In the next sections we will modify this error bound to account for the constraints cited above. aing into account the nominal biases he previous equation does not tae into account the nominal biases. he error bound must be increased to account for them. For a user s line of sight, the contribution of the biases is given by: Where: u LOS Hb 1 H = P+ G WG G W An upper bound on the error is then given by:

3 max max u LOS b b Hb ( ) = ( ) + ( ) u x x u x x u x x LOS Broadcast LOS Broadcast Estimated LOS Estimated However, this bias term does not fit within the message. An upper bound of this bias is given by the Cauchy- Schwartz inequality: 1 1 = LOS LOS LOS LOS u Hb u HW W b u HW Hu b Wb Since we have: 1 1 HW H= P+ GWG GWG P+ GWG P+ GWG = Cov we end up with: u Hb u Cov u b Wb LOS LOS LOS he next step is to compute an upper bound of the scalar bwb. For this we compute: Again the first term does not fit within the message. We proceed again using the Cauchy-Schwartz inequality: 1 1 LOS Broadcast Estimated LOS Broadcast Estimated ( ) = ( ) u x x u Cov Cov x x x x Cov x x u Cov u Broadcast Estimated Broadcast Estimated LOS LOS Because the error bound must be valid for 10 s, an upper bound of the first term is necessary. hat is we find K fa such that with a prability consistent with the false alarm requirement we have: ( x x ) Cov 1 ( x x ) K Broadcast Estimated Broadcast Estimated pfa After this additional term, the error bound is given by: 3 = ( + + ) L u K K K u Cov u LOS HMI bias pfa LOS LOS K bias = max b bmax bwb W is diagonal, so the upper bound is given by: K = max bwb= b Wb bias b bmax he error bound is now given by: = ( + ) max max L u K K u Cov u LOS HMI bias LOS LOS aing into account undetected measurement errors he error bound computed in the previous sections would be valid if all measurements were trusted. However, there exists the possibility that measurements used to assess the integrity might be corrupted. Although this happens very rarely, the WAAS threat model assumes that at all times one of the measurements might be erroneous. his can be taen into account by computing the pair: ( ) ( ) ( xestimated, Cov ) aing into account the broadcast cloc and ephemeris he error bound computed in the previous section does not yet account for the fact that the user uses x Broadcast instead of x Estimated. x Broadcast is computed by the Corrections Processor, whereas x Estimated is computed in the Safety Processor. For a more detailed description of the system architecture, please refer to [10]. For the purpose of this wor it suffices to say that x Broadcast is a more accurate estimate than x Estimated under nominal conditions. he role of the Safety Processor is to mae sure that the error bound associated to x Broadcast is valid under all circumstances. his is done by accounting for the difference between x Broadcast and x Estimated. for each subset where measurement has been excluded. he prlem consists now in finding a matrix Cov such that for all lines of sight over the footprint: ( ) LOS LOS LOS LOS u Cov u u Cov u such that uloscov ulos is as small as possible. he exact optimization prlem could be then written: minimize such that u Cov u LOS LOS u over footprint LOS ( ) u Cov u u Cov u LOS LOS LOS LOS for all u LOS and

4 he jective function is a linear function of Cov, so it can be written as the trace multiplied by a matrix A. he constraint can be relaxed by extending the constraint to any vector u (not only a line of sight). he resulting prlem is written: minimize trace such that ( Cov A ) ( ) Cov Cov for all Under this form, this prlem is a Second Order Cone Program (SOCP) [11]. It is a convex prlem and can be solved efficiently. Exploiting the structure of the set of matrices Cov () Although the prlem above can be solved for any set of definite positive matrices, it is worthwhile exploiting their structure, in particular the fact that they differ from the allin-view covariance by a ran two matrix in the general case and by a ran one matrix if the weighting matrix is diagonal. In the diagonal case, which is assumed throughout the paper, we have: ( ) ( ) ( ) ( ) ( ) Cov = P + G W G = Cov g w g Cov g w g Cov = Cov + 1 g w Cov g g he prlem above can therefore be simplified to: minimize trace such that hh where: ( Δ Cov A) ΔCov h = Heuristics to find Cov for all w 1 g w Cov g g Cov g In this section, we describe the method that was used to compute Cov at each step and for each satellite. Using the notations above, the following steps are performed: 1. For each compute: r = h Cov h By construction we have: hh rcov hh τ Cov Sort the r in I large in decreasing order. We renumber them to be r 1 to r p. 4. For from 1 to p we perform the following operations: 1 α = 1 hc h ( α ) C = C + max 0, h h he resulting matrix C p is an upper bound of the matrices hh. he final matrix is then: Cov = Cov + C he error bound computed by the user must be such that: = σ = ( + + ) L u K K K K u Cov u LOS t HMI bias pfa LOS LOS We found that the sub-optimal approach produced error bounds less than 5% larger than the optimal one. UDRE Floor implementation As indicated above, σ t has a oor imposed by the CCC and SQM monitors. Let us assume that the oor is given by σ oor. We must then find Cov + to account for the oor. Cov + must be such that: HMI bias pfa σ u Cov u oor LOS + K u Cov u u Cov u LOS LOS LOS + o meet this inequality, it is sufficient to have: σ oor HMI bias pfa I Cov K Cov Cov + p + o solve the above prlem, we form the singular value decomposition of Cov : Cov = U D U Changing basis, the above constraints are equivalent to:. Find the set I large of r that exceeds a threshold τ = Define C 0 as τcov. For outside of I large we have:

5 σ K oor HMI bias pfa D UCov U + We define the diagonal matrix D + as: D σ = max D, +, ii, ii I UCov U K oor HMI bias pfa Finally, the matrix Cov + defined as: +. Chec that ( ) ( ) ( ) ( x x Broadcast Estimated ) Cov ( x x Broadcast Estimated ) Kpfa 3. Compute K = b Wb bias max max 4. Compute the matrix Cov + as indicated above 5. Choose σ UDRE and compute: HMI bias pfa Cov = Cov M 8, pd + Kσ UDRE 6. Discretize Cov M8,od to tain Cov M8 Cov = U D U + + AVAILABILIY EVALUAION meets the conditions above. Composing the message and discretization he covariance and UDRE broadcast by WAAS must be such that we have: σ = + + t HMI bias pfa LOS + K K K K u Cov u As mentioned above, the user forms σ t by combining the UDRE and M8. he broadcast σ UDRE and Cov M8 must be such that: σ Cov + + u UDRE LOS M 8 LOS HMI bias pfa LOS + K u u K K K u Cov A sufficient condition is: Cov 8 K HMI bias pfa σ Cov UDRE M + Cov M8 is computed by choosing a value for σ UDRE and discretizing the matrix: HMI bias pfa Kσ UDRE Cov + he discretization of M8 is described in [4]. his discretization introduces a small penalty, so σ UDRE should be chosen to minimize it. In the implementation simulated below, a value of 0.91 m (which corresponds to the UDRE index of 5) was chosen. Summary of the algorithm Here are the main steps of the algorithm: ( ) ( ) 1. Compute ( x, Cov for each subset Estimated ) In this section, we compare the performance of the proposed algorithm with the current UDRE algorithm adapted to L1 L5. A description of the basic elements of the current algorithm can be found in [11]. he Service Volume Analysis tool MAAS was used to simulate the performance of the algorithm for 4 hours every 300 s over North America. he 4 satellite GPS constellation specified in [3] was assumed. Parameter settings he magnitude of W is determined by the CNMP curve and the cloc calibration error. K bias is computed in real time and is a function of W and the maximum biases [8], [9]. he resulting factor is between 3 and 4. K HMI is determined by the integrity allocation. In this wor, the value 5.5 was assumed and is an upper bound of what would need to be assumed. he oor for σ t was taen to be 0.68 m (UDRE index 4). Results he histogram shown in Figure 1 shows the ratio between σ t computed using the proposed algorithm and the current algorithm. he new values are up to 50% smaller. Figure and 3 show the 99% VPL quantile for the current and new algorithm respectively. here is a significant improvement, which suggests that such an approach could help WAAS achieve lower VPLs and, as a consequence, new levels of service.

6 ADDIIONAL REMARKS In this paper we have only presented the outline of the algorithm, which is sufficient to evaluate its potential. For its implementation, many decisions remain to be taen. For example, it will be necessary to chec the consistency of the measurements before computing x Estimated. If they are not consistent, it will have to be decided whether Fault Detection and Exclusion should be performed, or the satellite declared unfit for WAAS. Another point that will need to be specified is the external UDRE monitor. As mentioned above, the covariance can only be sent every 10s, but the multiplying factor σ UDRE can be sent every 6 s. Future wor should address the optimal way of updating σ UDRE and the fast correction. Figure 1. Histogram of New Error Bound/ Current Error Bound CONCLUSION In dual frequency WAAS, σ t which includes the cloc and ephemeris error, is the largest contributor to the error bound. In this wor, we propose a cloc and ephemeris algorithm that could reduce VPLs by 0%. he algorithm computes a covariance that bounds both the user estimation error in the presence of biases and receiver faults. he improvements presented here do not depend on a change in the message standards because the information produced by this algorithm fits in the current ones. ACKNOWLEDGEMENS his wor was sponsored by the FAA GPS Satellite Product eam (AND-730). REFERENCES Figure. 99% VPL quantile for the current algorithm [1] Walter,.,Blanch, J., Enge, P. Evolving WAAS to Serve L1/L5 Users. Proceedings of the Institute of Navigation GNSS-11, Portland, September 011. [] Lawrence, D., "Wide Area Augmentation System (WAAS) Status," Proceedings of the 3rd International echnical Meeting of he Satellite Division of the Institute of Navigation (ION GNSS 010), Portland, OR, September 010, pp [3] WAAS Minimum Operational Performance Specification (MOPS), RCA document DO-9D [4] Walter, odd, Hansen, Andrew, Enge, Per, "Message ype 8," Proceedings of the 001 National echnical Meeting of he Institute of Navigation, Long Beach, CA, January 001, pp Figure 3. 99% VPL quantile for the proposed algorithm [5] Walter,., Blanch, J., Enge, P., "Vertical Protection Level Equations for Dual Frequency SBAS," Proceedings

7 of the 3rd International echnical Meeting of he Satellite Division of the Institute of Navigation (ION GNSS 010), Portland, OR, September 010, pp [6] Blanch, J., Walter,., Enge, P., "Optimization of a Vertical Protection Level Equation for Dual Frequency SBAS," Proceedings of the 011 International echnical Meeting of he Institute of Navigation, San Diego, CA, January 011, pp [7] Walter,., "WAAS MOPS: Practical Examples," Proceedings of the 1999 National echnical Meeting of he Institute of Navigation, San Diego, CA, January 1999, pp [8] Shallberg, K., Sheng, F., "WAAS Measurement Processing; Current Design and Potential Improvements," Proceedings of IEEE/ION PLANS 008, Monterey, CA, May 008, pp [9] Shallberg, K., Grabowsi, J., "Considerations for Characterizing Antenna Induced Range Errors," Proceedings of the 15th International echnical Meeting of the Satellite Division of he Institute of Navigation (ION GPS 00), Portland, OR, September 00, pp [10] Schempp, imothy R., Pec, Stephen R., Fries, Rert M., "WAAS Algorithm Contribution to Hazardously Misleading Information (HMI)," Proceedings of the 14th International echnical Meeting of the Satellite Division of he Institute of Navigation (ION GPS 001), Salt Lae City, U, September 001, pp [11] Wu,., Pec, S., "An Analysis of Satellite Integrity Monitoring Improvement for WAAS," Proceedings of the 15th International echnical Meeting of the Satellite Division of he Institute of Navigation (ION GPS 00), Portland, OR, September 00, pp [1] S. Boyd, L. Vandenberghe. Convex Optimization. Cambridge University Press, 004. p 43.