An Area Computation Based Method for RAIM Holes Assessment

Transcription

1 Joural of Global Positioig Systems (2006) Vol. 5, No. 1-2:11-16 A Area Computatio Based Method for RAIM Holes Assessmet Shaoju Feg, Washigto Y. Ochieg ad Raier Mautz Cetre for Trasport Studies, Departmet of Civil ad Evirometal Egieerig, Imperial College Lodo, Lodo, Uited Kigdom, SW7 2AZ Abstract. Receiver Autoomous Itegrity Moitorig (RAIM) is a method implemeted withi the receiver to protect users agaist satellite avigatio system failures. Research has show that traditioal methods for the determiatio of RAIM holes (i.e. places where less tha five satellites are visible ad available) based o spatial ad temporal itervals (grids) compromise accuracy due to the costrait of computatio load. Research by the authors of this paper has addressed this ad developed a ew algorithm to determie RAIM holes usig bouded regios istead of approximatio based o grid poits. This paper uses the ew algorithm ad proposes a area based method for the computatio a RAIM satellite availability statistic based o the ratio of the total area of RAIM holes ad the coverage area (regioal or global area). Assessmet over time is based o the iterpolatio usig a model geerated from sapshot spatial statistics at a relatively log temporal iterval. Test results show that the area-based method for the calculatio of the RAIM satellite availability statistic is sigificatly more accurate with less computatioal load tha the traditioal grid poits based approach. Key words: Itegrity moitorig, RAIM hole, GNSS, performace assessmet 1 Backgroud Receiver Autoomous Itegrity Moitorig (RAIM) is a receiver-based scheme to provide timely ad valid warigs to users whe a Global Navigatio Satellite System (GNSS) is ot able to meet the required avigatio performace (RNP). RAIM algorithms are based o statistical cosistecy checks usig redudat measuremets. Cosistecy checks require five or more visible satellites, while i the case of Failure Detectio ad Exclusio (FDE), at least six visible satellites are required. A RAIM hole occurs whe there are isufficiet avigatio satellites i view to provide a itegrity check at a give locatio. It is defied as the area (or period) i which less tha five GNSS satellites are i view above a mask agle of 7.5 degrees (Air Force Space Commad, 1997). RAIM holes are the result of iformatio shortage causig a RAIM algorithm to be uable to perform its fuctio. Accordigly, a FDE hole ca be defied as the area (time) i which less tha six GNSS satellites are i view above a mask agle of 7.5 degrees. The method commoly used is to overlay a grid over the area of iterest ad to search at the grid poits (odes) over time. The spatial ad temporal itervals that have bee used i RAIM availability assessmet iclude: 5 degrees (spatial) ad 5 miutes (temporal) (Ochieg et al., 2002), 3 degrees ad 5 miutes (TSO-C129a,1996, RTCA/Do-229C, 2001) while Eurocotrol employs 0.25 degrees ad 2.5 miutes i the AUGUR software (AUGUR). These samplig itervals are relatively large ad are therefore susceptible to RAIM holes. Hece, if the spatial ad temporal itervals are too large, some RAIM holes could pass udetected if they lie either betwee the spatial or temporal samplig poits. However, smaller itervals result i a large umber of sample poits requirig icreased computatioal resources. The grid-based search method is therefore always depedet upo a trade-off betwee accuracy ad computatioal load. A ew method developed by Feg et al., (2006a) determies RAIM holes with a very high degree of accuracy. The key cosideratios i the ew approach to determiig RAIM holes are: The descriptio of what costitutes a RAIM hole The determiatio of precise satellite coverage boudaries. The determiatio of the itersectio poits of satellites boudaries.

2 12 Joural of Global Positioig Systems Topological aalysis of the regios formed by the itersectio of these coverage boudaries The coverage boudary of a satellite is ormally cosidered to be a curve o the Earth surface. Ay poit o the curve has the same value of elevatio agle as that of maskig agle. A RAIM hole is a polygoal area o the Earth surface formed by the satellite coverage boudaries of less tha five satellites i view. Sice the Earth surface ca be modelled by a ellipsoid, the area of a hole ca be completely described by its borders, which is eclosed by segmets of satellite coverage boudaries as demostrated i figure 1. For each segmet, there is a start poit ad a ed poit. These poits are the itersectios of coverage boudaries shared by the relevat two segmets, e.g. i Figure 1 the itersectio poit A is both the start poit of segmet (AB) ad the ed poit of segmet (CA). The segmets (AB, BC, CA) form a closed area. poits o the segmet betwee two crucial poits pair (start, ed) are referred to as critical poits i this paper. Aalysis shows that there exist at least three crucial poits if a hole exists (Feg et al., 2006b). The method here is to determie a polygoal area formed by the crucial ad critical poits of the segmets o the coverage boudaries as show i figure 2 (if 6). This is a more accurate descriptio of the area of a RAIM hole tha grid-poits based descriptio as demostrated i figure 3. The area with (-2) satellites i view is about 6% of the total area while the grid poit based method gives about 4.7%. The RAIM holes could be missed if the area is very small or the grid is ot dese eough Crucial poit satellites i view B C Segmet -2 Itersectio poit -1 Critical poit -1-1 A -1 Figure 2. Key poits of segmets based method Figure 1. The boudaries itersectio poits ad segmets The topology of the area formed by coverage boudaries is described briefly below: If the boudaries of (m) satellites itersect at oe poit, there are (2m) regios aroud the itersectio poit. The itersectio poit has the maximum umber () of visible satellites. The differece of the umber of visible satellites betwee ay two adjacet regios is 1. For m=2 or 3, there must exist at least oe area where oly (-2) satellites are visible. Based o the RAIM hole descriptio ad topology aalysis, the determiatio of a RAIM hole is trasformed ito the determiatio of itersectio poits with six or less satellites i view ad the adjacet segmets of coverage boudaries. The determiatio of itersectio poits is extremely importat because they defie the start ad ed poit of each segmet. These poits are referred to as crucial poits i this paper. To determie the aalytical solutio of the segmet betwee the start ad ed poit is quite difficult i a ellipsoidal model. However, the segmet ca be described discretely with a umber of poits o the coverage boudary. The Figure 3. Grid poits based area approximatio For a satellite of kow positio, there is a footpoit o the surface of the Earth, which is the itersectio poit of the Earth cetre to satellite vector ad the surface of the Earth as show i figure 4. The positio of a poit o the coverage boudary has a determiistic relatioship with the elevatio agle, the positio of the satellite, ad the relative azimuth to the footpoit either usig a spherical model or a ellipsoidal model of the Earth. However, it is ot straightforward to determie the poit usig a ellipsoidal model. Two steps are ormally used to

3 Feg et al.: A Area Computatio Based Method for RAIM Holes Assessmet 13 determie the positio of a boudary poit o a ellipsoidal model. I the first step, the Earth is cosidered to be a sphere, ad the poit is assumed to be o the surface of the Earth. It is the straightforward to determie a approximate locatio of the poit. I a secod step, a iterative process is carried out to obtai the precise positio of the poit o a ellipsoidal model usig the locatio i spherical model as the iitial value. If the user is ot o the surface of the Earth (e.g. a aircraft) ad 3D locatio is kow the the height iformatio ca be added i the secod step to exted the coverage boudary from the surface of the Earth to ay height of cocer. The process above is used to determie the itersectio poits. Iitially, the itersectio poits are assumed to be o the surface of a spherical model of the Earth. The itersectio poits (As ad Bs as show i figure 4) of two boudaries ca be calculated from the locatio of footpoits (F1, F2) ad parameters derived from the maskig agle usig spherical trigoometry. I a secod step, the itersectio poits (A E, B E ) are calculated iteratively i the ellipsoidal model where local height iformatio ca also be cosidered. The accuracy of the locatio of these poits depeds o the umber iteratios. 4 Oe iteratio ca reach a accuracy level of degrees ad six iteratios ca reach 1 10 degrees level (measured with respect to a fixed maskig agle). North F1 Footpoit Towards satellite Bs AE O Boudary o Sphere Boudary o Ellipsoid BE As Earth Cetre Towards satellite Footpoit Figure 4. Itersectio of two satellite coverage boudaries Amog these itersectio poits, oly those with six or less satellites i view are cosidered to be crucial poits ad eed to be idetified. The positio of the itersectio poit, the umber of visible satellites, ad related idetities (e.g. PRN pair) of satellites whose coverage boudaries itersect at this poit are recorded. The existece of crucial poits oly idicates the existece a hole. There is o direct iformatio about the size ad the umber of crucial poits that ecloses a hole. It is difficult to get a solutio from the related idetities of satellites that itersect at each crucial poit. Oe reaso is that the total umber of crucial poits could be very large ad the hole could be formed by ay umber F2 (more tha two) of crucial poits. The other reaso is the existece of two poits with the same idetity sice there are ormally two itersectio poits for the coverge boudaries of two satellites. The ext step is the determiatio of critical poits at a azimuth iterval alog oe coverage boudary startig from oe crucial poit ad edig at ext crucial poit. The latter crucial poit is take as the start poit for the other coverage boudary which itersects with the previous boudary. The process cotiues util a closed polygoal is foud. 2 Area calculatio ad assessmet of RAIM holes Accordig to the Radio Techical Commissio for Aviatio (RTCA), the grid poits for curret availability test are sampled every three degrees from zero to iety degrees orth (RTCA/Do-229C, 2001). The poits o each latitude circle are separated evely i logitude defied as: 360 log.iter val = (1) ROUND( 360 ) mi(3 degrees / cos( latitude), 360) The logitude iterval determied here eables uiformly distributed grid poits i terms of area. Cosequetly, the area ca be approximated by the umber of poits. Obviously, the direct calculatio of the area covered by RAIM holes is more accurate tha poit based approximatios, ad further eables a more accurate quatificatio of the RAIM satellite availability (Feg et al, 2005). 2.1 Area calculatio of a polygo The calculatio of the polygo area of a RAIM hole o the surface of a ellipsoid is ot straightforward. This is accomplished through a equal area projectio from the ellipsoidal model to the spherical model. The projectio ivolves two aspects, the determiatio of the authalic latitude, which trasforms the latitude from the ellipsoidal model to the spherical model, ad the determiatio of the radius of authalic sphere which has the same area as that i ellipsoidal model (Syder, 1987; Malig, 1992). The authalic latitude β, havig the same surface area o a sphere as the ellipsoid, provides a sphere of equal area (authalic) to the ellipsoid: β = arcsi( q / q ) (2) where, p

4 14 Joural of Global Positioig Systems 2 siφ 1 1 esiφ q = (1 e ) l (3) e si φ 2e 1+ esiφ 2 1 e 1+ e q p = 1+ l 2e 1 e o q p is q evaluated for a φ of 90, φ is the geodetic latitude. e deotes the first eccetricity of the ellipsoid. The radius R q of the sphere havig the same surface area as the ellipsoid is calculated as follows: q p R q = a (5) 2 where a is the semi-major axis of the ellipsoid. The area of a RAIM hole regio ca be calculated usig the umerical itegratio based o Gree s Theorem for a polygoal area o a surface of a sphere. 2.2 Satellite availability assessmet The availability statistic based o the grid poits method is to calculate the ratio of the umber of available spacetime poits Na versus the umber of total sample poits, which ca be expressed as: N total N (4) a A = (6) Ntotal where A is the availability. The accuracy of the method is poor due to the approximatio of a area by uiformly distributed poits with a certai desity. Therefore, it s always a trade-off betwee accuracy ad the umber of total grid poits (desity). I cotrast, the availability assessmet method proposed here determies the exact areas of RAIM holes (uavailable regio), ad further calculates the ratio of the area of availability ad the total area of cocered, which ca be expressed as: AHole A = 1 (7) A c where, A is the availability, A Hole is the area of RAIM holes, A c is the area of the regio cocered, or the surface of the Earth if a global assessmet is performed. Expressio (7) eables a spatial determiatio of satellite availability at a istat i time. To carry out the availability assessmet to cover a full geometry, a umber of temporal samples (time domai) must be take at a certai time iterval. A relatively short time iterval, e.g. oe secod ievitably results i a very large sample size. For example, for the grid-based method the product of the spatio-temporal assessmet would be too large requirig very sigificat computatioal resources. This problem is egated by the area-based method which is able to use a relatively large time iterval e.g. 50 secods. Because of the accuracy of the area-based method ad the predictability of RAIM holes, the performace ibetwee sample times ca be iterpolated by a model derived through a curve fittig process. 3 Results A global RAIM holes calculatio was carried out to verify the algorithm. The omial costellatio of 24 Global Positioig System (GPS) satellites (RTCA/Do- 229C, 2001) was used. The Earth s ellipsoidal model WGS-84 was used. The RAIM holes startig at to Secods of Week (SoW) at a time iterval of 200 secods are show i figures 5a to 5f. The solid lies are the boudaries of satellites footprit. The umber of visible satellites at each itersectio poit is ext to each poit i the figure. Patches with differet shapes ad sizes are formed by the boudaries. The asterisks (*) are the crucial poits ad the dots are the critical poits. The RAIM holes are the areas bouded by the asterisks ad the dots, where oly four or less satellites are i view. Figures 5a to 5f also show how the shapes ad sizes of the RAIM holes chage over time. boudary critical poit crucial poit Figure 5a. RAIM hole at secod Figure 5b. RAIM hole at secod

5 Feg et al.: A Area Computatio Based Method for RAIM Holes Assessmet 15 correspodig iterpolated availability. The very small errors i the iterpolatio cofirm that a relatively log temporal iterval ca be used with the positive effect of sigificatly reducig the computatioal load. Table1. RAIM hole area ad availability Figure 5c. RAIM hole at secod Figure 5d. RAIM hole at secod Time SoW (s) RAIM hole Area (km 2 ) Availability Figure a b c d e f Figure 5e. RAIM hole at secod Figure 6. Curve fit of availability Table2. Compariso of computed ad iterpolated availability Figure 5f. RAIM hole at secod The areas of the RAIM holes ad the global availability for a time iterval of 100 secods are listed i table 1. I order to create a model to eable spatio-temporal determiatio of RAIM holes usig a relatively large iterval, a curve has bee fitted to the data i table 1. This is show i Figure 6. The asterisks (*) are the satellite availability at each sample time. Table 2 gives example errors betwee computed satellite availability ad the Time SoW (s) Computed availability Iterpolated availability Error

6 16 Joural of Global Positioig Systems Aother example is the RAIM hole at SoW. The RAIM hole is small at oly 1.6 square kilometres. The latitude ad logitude of each of the three crucial poits are: ( , ) ( , ) ( , ). At SoW, the RAIM hole is smaller at oly 36m 2. The latitude ad logitude of the three crucial poits are: ( , ) ( , ) ( , ). The area computatio based method easily captures these RAIM holes, while for the grid based method a very dese grid would be required. I this example, a grid of 5 less tha 1 10 degree is required equivalet to more 15 tha sample poits i order to capture this RAIM hole. 4 Coclusios This paper has preseted a ew algorithm for the quatificatio of satellite availability with a particular focus of RAIM holes. Sice the area of a RAIM hole chages i a determiistic way with the motio of the satellites whose boudaries form a regio, a full geometry assessmet (spatial ad temporal) is possible via iterpolatio i the time domai usig a relatively log temporal iterval. This has the positive effect of deliverig high accuracy with miimal computatioal load. The ew approaches for determiig RAIM holes ad quatifyig the RAIM satellite availability statistic i space ad time, are sigificatly superior to the traditioal grid-based method, both i terms of accuracy ad computatioal load. Refereces Air Force Space Commad (1997) Air force space commad capstoe requiremets documet for global positio, velocity, ad time determiatio capability. July, AUGUR. Feg S., Ochieg W.Y., Mautz R., Brodi G. ad Ioaides R. (2005) RAIM Holes Assessmet via a Accurate ad Efficiet Computatio Method, Proceedigs of the Iteratioal Symposium o GPS/GNSS 2005, Hog Kog. Feg S., Ochieg W.Y., Walsh D. ad Ioaides R. (2006a) A Highly Accurate ad Computatioally Efficiet Method for Predictig RAIM Holes, The Joural of Navigatio, The Royal Istitute of Navigatio, 59 (1), pp Feg S., Ochieg W. (2006b) A Geeral Method for Accurate Assessmet of GNSS FDE Holes, Proceedigs of ION GNSS 2006, Fort Worth, TX, 26-29, September, pp Malig D. H. (1992) Coordiate systems ad map projectios (2d editio), Pergamo Press,1992. Ochieg W.Y., Sherida K.F., Sauer K., ad Ha X. (2002) A assessmet of the RAIM performace of a combied Galileo/GPS avigatio system usig the margially detectable errors (MDE) algorithm. GPS Solutio (2002). Vol. 5No. 3, pp RTCA/DO-229C (2001) Miimum operatioal performace stadards for global positioig system/ wide area augmetatio system airbore equipmet. November, Syder J.P. (1987) Map projectios: a workig maual, U.S. Geological Survey Professioal Paper 1395, TSO-C129a (1996) Airbore supplemetal avigatio equipmet usig the global positioig system (GPS), Departmet of Trasportatio Federal Aviatio Admiistratio Aircraft Certificatio Service, Washigto, DC, February, 1996.