Reducing Computational Load in Solution Separation for Kalman Filters and an Application to PPP Integrity

Transcription

1 Reducing Compuaional Load in Soluion Separaion for Kalman Filers and an Applicaion o PPP Inegriy Juan Blanch, Kaz Gunning, Todd Waler. Sanford Universiy Lance De Groo, Laura Norman. Hexagon Posiioning Inelligence ABSTRACT This paper invesigaes wo echniques o reduce he compuaional load of running muliple faul oleran Kalman filers in order o provide inegriy. These approaches are hen exploied in he implemenaion of a soluion separaion inegriy monioring algorihm in a PPP Kalman filer soluion. We evaluae he echniques using GNSS daa colleced in saic and driving condiions. In our scenarios, hese echniques lead o compuaional load reducions of a leas 70% a he expense of proecion level degradaions of abou 50%. INTRODUCTION Unil recenly, Precise Poin Posiioning (PPP) echniques [1] have mosly been used o provide high accuracy. There is a growing ineres in ranslaing he benefis of PPP o inegriy and enabling is applicaion o safey criical applicaions in rail, auomoive, mariime, and even air navigaion [2], [3], [4], [5]. In [5], we demonsraed how echniques developed for aviaion applied o PPP can produce meer-level proecion levels in auomoive and aviaion scenarios. This was achieved by implemening an inegriy monioring algorihm based on soluion separaion, akin o he one used o analyze Advanced RAIM performance [6], o he PPP Kalman filer soluion. The principle of soluion separaion is o run a bank of filers, where each filer is faul oleran o a faul or se of fauls. The faul deecion saisic is he difference beween each of hese soluions and he all-in-view soluion. In addiion o heir opimaliy properies [7], soluion separaion algorihms offer a sraighforward proof of inegriy, and good performance [5]. However, hey can also be expensive in erms of memory and processing ime, because hey require he receiver o compue a bank of filers (or a process compuaionally equivalen o a bank of filers, as in [11]). In he wors case, he compuaional load will be proporional o he number of filers. In [5] we showed ha i was possible o dramaically reduce he cos of running he bank of filers: depending on he filer complexiy (ha is, he number of esimaed saes), we could run 20 o 50 addiional filers for he cos of one. This was obained by exploiing he fac ha, in PPP, many of he elemens in he compuaion (error models, correcions, ec) are common o he all filers, so ha i is sufficien o compue hem once for he all-in-view filer. Also, all he measuremens are linearized wih respec o he all-in-view posiion soluion, which furher simplifies he subse soluion filers. The goal of his paper is o inroduce and invesigae echniques o reduce even furher he cos of he soluion separaion for Kalman filer soluions. When he number of saes is large (larger han 50), which is he expecaion in a PPP mulifrequency user algorihm, here are a leas wo seps ha are compuaionally expensive: he deerminaion of he Kalman gain, and he deerminaion of he new error covariance. The firs echnique under invesigaion consiss in using a se of subopimal filers for he subse soluions (insead of he opimal filer), where he Kalman gain for each subfiler is derived from he all-in-view and does no require a full marix

2 inversion. The second echnique ha we will evaluae is he consolidaion of fauls ino a few subses. This is anoher ype of subopimal subse soluion ha has already been evaluaed for snapsho soluions in Advanced RAIM [8], [9]. For he experimenal evaluaion, we will use our sequenial PPP filer implemenaion [5], which is based on a simple exended Kalman filer wih esimaed parameers comprising he receiver posiion, clock biases for each consellaion in use, a ropospheric delay, and floa ambiguiies for each racked carrier phase. Dual-frequency measuremens are incorporaed from GPS, GLONASS, Galileo, and BeiDou. The precise orbi and clock esimaes are drawn from he IGS MGEX analysis ceners. SUBOPTIMAL SUBSET SOLUTIONS: FIRST APPROACH For he approach oulined here, we only consider he measuremen updae sep of he Kalman filer. For he all-in-view filer (indicaed by he index 0), we have he following Kalman filer equaions: Where 1 1 is he a poseriori sae esimae, xˆ T xˆ xˆ C G W y Gxˆ (1) is he a priori esimae, G is he observaion marix, W is he inverse xˆ 1 of he measuremen noise marix, y is he vecor of measuremens, and sae esimae. We have: Where 0 C 1 1 is he error covariance of he a priori esimae. 1 1 is he error covariance of he a poseriori C 1 T C 1 1 C 1 G WG 1 (2) Subse filer soluions In his paper we will consider probabiliy fauls of 10-5 and 10-4 per hour. For an inegriy of 10-7 per hour, his means ha we need o compue he soluion separaion saisic of all one ou subses in he case of 10-5 and wo ou subses for 10-4 [8]. The subse filer Kalman filer equaions (indexed by k) are similar o he all-in-view ones: k k k kt k k k xˆ xˆ C G W y G xˆ (3) 1 T C C G W G (4) k k k k k The only difference wih respec o he all-in-view is ha we only use a subse of he available measuremens o updae he sae esimae. One of he mos onerous seps in his process is he compuaion of he covariance as wrien in Equaion (4). As can be seen, we need a leas wo marix inversions, where boh marices are n by n wih n being close o a 100. (We noe ha he geomery marix is ofen as large, so a marix updae formula will no subsanially reduce he compuaional load). 1

3 The firs approach consiss in using a subopimal filer for we define i as follows: Where he marix k 1 1 insead of he opimal one defined above. More precisely, xˆ k1 1 k k k k T k k k k xˆ xˆ G W y G xˆ (5) is no longer given by Equaion (4). Insead, we aemp o find a marix ha will resul in a reasonable esimaor bu ha is cheaper o compue. One possible approach is o compue his marix as if he prior of he esimaed saes was given by he prior of he all-in-view filer: 1 T C G W G (6) k k k k The advanage is ha his marix can be obained wihou a full marix inversion. We have: 1 C G WG G W G G WG (7) 1 k T k T k k T k k k In mos cases, he rank of he marix T T G W G G WG is considerably smaller han he rank of. For example, in he one ou case of our PPP filer, he rank of his marix is 4. We can wrie: Using he Woodbury marix ideniy, we ge: The use of his formula can speed up he calculaion of k k k k k k T T T G WG G W G G W G 1 C 1 1 T G WG (8) 1 T T C G WG G W G (9) k k k k T C C G W G C G G C (10) k 0 0 k T k 1 k 0 k k k 1 1, because he marix o inver is usually much smaller han he whole covariance marix. Sandard marix inversion algorihms require around 2/3n 3 basic operaions, so he compuaional load is significanly reduced (from almos a million o less han a hundred). The new Kalman gain is given by: 1 T k k kt k k k k K G W C G W C G W I W G C G G C G W (11) 0 k T k 0 k T k 0 k T k Where we have highlighed wha has already been compued in he all-in-view filer. New subse covariance As opposed o he opimal filer, he covariance afer he updae is no given by k 1 1. Insead, i is given by:

4 1 k k T k T C I K C I K KW K (12) SUBOPTIMAL SUBSET SOLUTIONS: SECOND APPROACH The second approach can be described much more succincly. I consiss in grouping he fauls so ha we do no need o run as many filers. For example, insead of running a filer for a faul in saellie i and anoher one for a faul in saellie j, we run a filer ha is faul oleran o boh i and j. This will resul in a weaker soluion posiion, and herefore larger proecion levels. This second approach can be considered a subopimal subse soluion approach because each faul is accouned by a subopimal filer. For his paper, he groups were formed based on he PRN number, which is mosly equivalen o a random grouping wih regard o geomery. DATA AND PROTECTION LEVEL CALCULATION We used wo ypes of GNSS daa: one colleced by a saic receiver and one colleced by a receiver insalled in a car. The GNSS daa colleced in road condiions is described in [5] and briefly summarized here: Receiver: NovAel OEM Hour Driving Daa on March 1, 2018 GPS (L1 C/A -L2P semi-codeless), GLONASS (L1 C/A-L2P) a 1 Hz Truh posiions provided by NovAel OEM729 wih acical-grade IMU wih forward and reverse processing Specifically, we choose he open sky condiions. The saic GNSS corresponded o he following condiions: Receiver: Trimble NeR9 6 hours of saic daa on November 7, 2018 a Sanford GPS (L1C-L2W), GLONASS (L1C-L2P) a 1 Hz Truh posiion from IGS saion soluions The error models are and he proecion level calculaion is also described in [5]. I is a sraighforward adapaion of he ARAIM algorihm described in [6] o a Kalman filer soluion. The algorihms were run in MATLAB in a PC (Windows 10, 64 bi OS, wih Inel Core i7-8700k CPU@3.70GHz 3.70GHz wih 16.0 GB RAM). EXPERIMENTAL RESULTS Baseline vs subopimal subse Kalman filer The baseline resuls use he opimal subses as described in Equaion (4). The subopimal Kalman filer subse soluion was implemened as described in Equaion (11). Figures 1 a) and b) show he subse soluions corresponding o he baseline approach and he subopimal Kalman filer approach. We can observe a degradaion in he accuracy of he subse posiion soluions, and also in he covariances (no ploed here). Figures 2 hrough 4 show he proecion levels for boh he baseline and he subopimal approach, as well as he raio beween he wo (righ side plo).

5 Figure 1 a) and b). Subse posiion soluions for he baseline resuls (a), and he subopimal subse soluion (b) Figure 2. PL for he saic scenario wih one ou (lef side plo), and raio beween subopimal and baseline (righ side plo). Run ime was 6045 s and 5011 s (for baseline and subopimal respecively).

6 Figure 3. PL for he driving scenario wih one ou (lef side plo), and raio beween subopimal and baseline (righ side plo). Run ime was 319 s /291 s (baseline/subopimal). Figure 4. PL for he saic scenario wih wo ou (lef side plo), and raio beween subopimal and baseline (righ side plo). Run ime was s /6100 s (baseline/subopimal). The improvemen in execuion speed was modes, as we only saw a 10% reducion in execuion ime for he one ou case and a 50% reducion for he wo ou case. I is possible however ha a more significan benefi may be observed in real ime code (he prooype we use is implemened in MATLAB, which is very well suied for marix operaions). Subopimal subses: faul grouping In his secion, we evaluae he proecion levels resuling from he faul grouping echnique. The grouping of he fauls was no opimized. Figures 5 hrough 7 show he resuling proecion levels for group sizes of 2, 3, 5 and 10 fauls. Up o group sizes of 3, he degradaion appears o be accepable.

7 Figure 5. PL for he saic scenario wih one ou (lef side plo) for each of he grouping opions (1 is he baseline), and raio beween subopimal and baseline (righ side plo). Run ime was 6045 s/4324 s/ 3673 s/3352 s/ 3240 s (baseline/2/3/4/5 groupings). Figure 6. PL for he driving scenario wih one ou (lef side plo) for each of he grouping opions (1 is he baseline), and raio beween subopimal and baseline (righ side plo). Run ime was 519 s/337 s/ 297 s/273 s/ 265 s (baseline/2/3/4/5 groupings).

8 Figure 7. PL for he saic scenario wih wo ou (lef side plo) for each of he grouping opions (1 is he baseline), and raio beween subopimal and baseline (righ side plo). Run ime was s/3619 s/ 2241 s/1787 s/ 1458 s (baseline/2/3/4/5 groupings). Figure 7. PL for he driving scenario wih wo ou (lef side plo) for each of he grouping opions (1 is he baseline), and raio beween subopimal and baseline (righ side plo). Run ime was 3800 s/1280 s/ 820 s/ 560 s/490 s (baseline/2/3/4/5 groupings). As in he firs approach, he execuion speed improvemen in he one ou case was modes in our MATLAB implemenaion, bu i is significan in he wo ou case. For groupings of size wo, we observed an improvemen of 70% in run ime Forming faul groups wih sizes larger han 3 leads o very large proecion levels. The proecion level degradaion due o faul grouping is no negligible in any of he cases, bu i may be accepable for groupings of wo fauls, especially given he poenial reducion in compuaional load. Table 1 summarizes he resuls of our simulaions for he wo ou case. Table 1. Run ime improvemen for he differen echniques and corresponding performance degradaion

9 Algorihm Decrease in run ime compared o baseline Raio of proecion levels (max and median) Baseline soluion separaion 0 % 1 1 Subopimal filer 49% Faul grouping (2) 70% Faul grouping (3) 81% Faul grouping (4) 85% Faul grouping (5) 88% SUMMARY We have described and invesigaed wo echniques o reduce he compuaional load of a soluion separaion algorihm for Kalman filer posiion soluions. Boh echniques consis in using subopimal filers in he subse filers insead of he opimal filers. We apply hese echniques in a soluion separaion inegriy monioring algorihm in a PPP Kalman filer soluion wih daa colleced in road condiions. The resuls sugges ha, alhough he use of subopimal filers does increase he proecion levels and herefore degrade performance -, he degradaion may be accepable given he poenial compuional load savings. REFERENCES [1] Kouba, J. and Heroux, P. (2001). Precise Poin Posiioning Using IGS Orbi and Clock Producs, GPS Soluions, vol. 5, no. 2, pp [2] Madrid, P. F. Navarro, Fernández, L. Marínez, López, M. Alonso, Samper, M.D. Laínez, Merino, M.M. Romay, "PPP Inegriy for Advanced Applicaions, Including Field Trials wih Galileo, Geodeic and Low-Cos Receivers, and a Preliminary Safey Analysis," Proceedings of he 29h Inernaional Technical Meeing of The Saellie Division of he Insiue of Navigaion (ION GNSS+ 2016), Porland, Oregon, Sepember 2016, pp [3] D. Calle, E. Carbonell, P. Navarro, P. Roldán, I. Rodríguez, G. Tobías, Facing he Challenges of PPP: Convergence Time, Inegriy and Improved Robusness Proceedings of he 31h Inernaional Technical Meeing of The Saellie Division of he Insiue of Navigaion (ION GNSS+ 2018),Miami, Florida, Sepember [4] Barrios, J. e al Updae on Ausralia and New Zealand DFMC SBAS and PPP Sysem Resuls Proceedings of he 31h Inernaional Technical Meeing of The Saellie Division of he Insiue of Navigaion (ION GNSS+ 2018),Miami, Florida, Sepember [5] K. Gunning, J. Blanch, T. Waler, L. de Groo, L. Norman, Design and Evaluaion of Inegriy Algorihms for PPP in Kinemaic Applicaions Proceedings of he 31h Inernaional Technical Meeing of The Saellie Division of he Insiue of Navigaion (ION GNSS+ 2018),Miami, Florida, Sepember [6] Working Group C, ARAIM Technical Subgroup, Milesone 3 Repor, February 26, Available a:

10 hp:// hp://ec.europa.eu/growh/ools-daabases/newsroom/cf/iemdeail.cfm?iem_id=8690 [7] Blanch, Juan, Waler, Todd, Enge, Per, "Theoreical Resuls on he Opimal Deecion Saisics for Auonomous Inegriy Monioring", NAVIGATION, Journal of The Insiue of Navigaion, Vol. 64, No. 1, Spring 2017, pp [8] Blanch, Juan, Waler, Todd, Enge, Per, "Fixed Subse Selecion o Reduce Advanced RAIM Complexiy," Proceedings of he 2018 Inernaional Technical Meeing of The Insiue of Navigaion, Reson, Virginia, January 2018, pp [9] Orejas, Marin, Skalicky, Jakub, "Clusered ARAIM," Proceedings of he 2016 Inernaional Technical Meeing of The Insiue of Navigaion, Monerey, California, January 2016, pp [10] Orejas, Marin, Skalicky, Jakub, Ziegler, Ue, "Implemenaion and Tesing of Clusered ARAIM in a GPS/Galileo Receiver," Proceedings of he 29h Inernaional Technical Meeing of The Saellie Division of he Insiue of Navigaion (ION GNSS+ 2016), Porland, Oregon, Sepember 2016, pp [11] Tanil, Cagaay, Khanafseh, Samer, Joerger, Mahieu, Pervan, Boris, "Sequenial Inegriy Monioring for Kalman Filer Innovaions-based Deecors," Proceedings of he 31s Inernaional Technical Meeing of The Saellie Division of he Insiue of Navigaion (ION GNSS+ 2018), Miami, Florida, Sepember 2018, pp