Preliminary Simulation Results of a Deeply Coupled GPS/INS System for High Dynamics

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1 Prelimiary Simulatio Results of a Deeply Coupled GPS/INS System for High Dyamics Stefa Kiesel, Adreas Maier, Ulrich Seiller, Gert F. Trommer Karlsruhe Istitute of Techology, KIT BIOGRAPHY Stefa Kiesel is a Dr.-Ig. (Ph.D.) didate at the Istitute of Theory ad Systems Optimizatio i Electril Egieerig of the Karlsruhe Istitute of Techology (KIT), Germay. He received his Diploma (M.S.) i Electril Egieerig i 26 from the Uiversity of Karlsruhe. His areas of research iclude strapdow iertial avigatio systems, GPS/INS itegratio, ad software GPS receivers. Adreas Maier is a Dr.-Ig. (Ph.D.) didate at the Istitute of Theory ad Systems Optimizatio i Electril Egieerig of the Karlsruhe Istitute of Techology (KIT), Germay. He received his Diploma (M.S.) i Electril Egieerig i 25 from the Uiversity of Karlsruhe. His areas of research iclude SAR/INS itegratio ad federated Kalma filter desig for data fusio. Ulrich Seiller received his Diploma i Electril Egieerig of the Uiversity of Karlsruhe i 29. He was ivolved i GPS/INS research projects. Gert F. Trommer received the Dipl.-Ig. (M.S.E.E.) ad the Dr.-Ig. (Ph.D.) degrees from the TU Muich, Germay i 1978 ad 1982, respectively. He joied EADS/LFK, formerly MBB/Dasa, where he was project maager for the IMU developmet i the UK ASRAAM program. He was director of the sectio Flight Cotrol Systems with the resposiility for the Navigatio, Guidace ad Cotrol Susystem, the IR Seeer ad the Fi Actuators i the Germa/Swedish Taurus KEPD 35 program. Sice 1999 he is director of the Istitute of Systems Optimizatio at the Karlsruhe Istitute of Techology, with research focus o INS/GPS/TRN itegrated avigatio systems ad UAVs. Sice 24 he is dea of the Departmet of Electril Egieerig ad Iformatio Techology. ABSTRACT Deeply coupled GPS/INS itegratio comies GPS sigal tracig ad the avigatio prolem ito oe processig step. Dyamic iformatio from a iertial measuremet uit (IMU) is used to compesate trajectory dyamics i the sigal tracig process. This offers ew possiilities i desigig the GPS asead processor. No-coheret deep itegratio uses discrimiator iformatio as direct or smoothed measuremet of the residual tracig error. This measured tracig error used i the Kalma filter measuremet step is asilly a measuremet averaged over the predetectio itegratio time. Discrimiator values resultig from correlator outputs i geeral descrie a average tracig error. For icreasig the predetectio itegratio time a systematilly correct descriptio of the measuremet is ecessary. I existig deeply coupled GPS/INS systems these averaged measuremets are used as puctual values which produces systematic errors especially visile i high dyamics. The proposed deeply coupled GPS/INS itegratio system uses these averaged values i a adequate measuremet step. By imitatig the approach of time differeced rrier phase measuremets, a measuremet formulatio was foud that accouts for the systematic icosistecy. I this formulatio trajectory dyamics are modelled i the measuremet step. The accuracy of the avigatio process i highacceleratio eviromets up to 35g is sigifitly icreased ad the velocity error i the simulatio is lower tha 5cm/s. 1. INTRODUCTION Curret research is coducted o the field of Kalma filter ased sigal tracig that icludes iertial sesors. The avigatio process ad the sigal tracig process are performed simultaeously which is also lled deeply coupled GPS/INS itegratio. The sigal tracig process is supported y trajectory dyamics measured y a iertial avigatio system. This eales ew possiilities i desigig the ase ad processor icludig GPS/INS itegratio.

2 I a tightly coupled GPS/INS itegratio system measuremets of stadard GPS sigal tracig loops are used as measuremet i the avigatio filter. These measuremets are ot filtered y a Kalma filter iside the GPS receiver. The resultig positio ad velocity estimatio e used as aidig iformatio for stadard sigal tracig loops. The asic structure of the loop desig remais utouched ad a estimated relative LOS velocity is used to compesate for the dyamics. As a result the adwidth of the loop filter e reduced or adapted dyamilly depedig o the oise eviromet [5]. A low adwidth results i a log time iterval etwee ucorrelated measuremets. However, time-correlated measuremets e a prolem for a Kalma filter used for data fusio. Deeply Coupled GPS/INS Itegratio solves the avigatio ad tracig prolem y a cetral avigatio filter istead. Measuremet residuals ad iertial data are used for lculatig positio ad velocity updates that are used for cotrollig the sigal correlatio process. Sigle loop filters i stadard tracig loops are replaced y a cetral avigatio filter that cotrols the NCOs (umerilly cotrolled oscillators). Usig I ad Q correlator outputs directly as measuremets is commoly referred as coheret deep itegratio. The Is ad Qs deped o the code phase error ad the rrier phase error. For coheret deep itegratio the rrier phase has to e ow i the Kalma filter. Pre-processig Is ad Qs i discrimiator fuctios avoids the ecessity of the rrier phase iformatio i the Kalma filter ad uses oly code phase errors ad rrier frequecy errors as measuremets. This is ow as o-coheret deep itegratio ad is the optimal itegratio level for poor sigal-to-oise ad high dyamic eviromets. I a commo hardware software GPS receiver desig I E,P,L ad Q E,P,L (early, prompt ad late) correlator outputs are provided every 1ms y each chael. For etter oise performace these 1ms samples are itegrated coheretly over a predetectio itegratio time (PIT). This predetectio itegratio time e up to 2ms without cosiderig datait iformatio ad e loger (reasoaly up to 1ms) whe usig a datait estimatio process [6]. Discrimiator fuctios usig these itegrated I ad Q samples output a average tracig error over the predetectio itegratio time. Especially i high dyamic situatio the averagig effect i the frequecy error ecomes apparet. Usig a average Doppler frequecy error with traditioal rage-rate (delta-rage) measuremet step produces systematic errors durig high acceleratios. This paper studies the usage of a refied measuremet formulatio which was origially iveted to process time-differeced accumulated delta rage iformatio [2, 3]. It uses the iformatio of the trasitio matrix to correctly predict the differece i positio i a specific time iterval. This e see as a good estimatio of the average relative velocity durig the measuremet iterval. This measuremet formulatio e adapted for processig averaged frequecy error discrimiator outputs. I moderized GPS sigal desig ad i upcomig Galileo sigals, pilot sigals with o avigatio data it modulatio will e availale for sigal tracig. With these sigals the predetectio itegratio time e easily exteded over the 2ms datait time iterval of the GPS C/A-code. But eve with the existig GPS sigal there are strategies for extedig the predetectio itegratio time y a data it estimatio process explaied i [6]. This offers etter oise performace ut emphasizes the eed for a adapted measuremet step accoutig for averaged measuremet errors. Although GPS/INS deep itegratio is developed for high ati-jammig eviromets, this paility is ot show here. This paper cocetrates o the dyamic ehavior. Nevertheless with the possiility of exteded coheret pre-detectio itegratio time, which e used with avigatio it estimatio or with moder GNSS pilot sigals, this will also rig a eefit i ati-jammig ehavior. I [6] a deeply coupled GPS/INS system is preseted that uses a dyamics EKF with trasitio matrix factorizatio together with GPS loc processig. For loc processig digital itermediate frequecy samples (IF) must e availale which results i a dramatilly differet hardware software setup of the receiver. Whe usig traditioally availale correlator hardware, these IF-samples are ot availale. Correlator samples with a rate of 1 Hz or lower must e processed istead, which is doe i the preseted approach. The remaider of the paper is structured as follows. Chapter 2 gives a overview of the system architecture. The applied sigal tracig process is explaied i geeral. Chapter 3 descries the measuremet step extedig time-differeced accumulated delta rage measuremet y the measuremet of a frequecy error. Chapter 4 details the potetial of exteded predetectio itegratio time i a rrier frequecy discrimiator. Chapter 5 aalyzes differet update rates for the rrier frequecy. Chapter 6 illustrates the simulatio eviromet ad chapter 7 shows simulatio results. 2. SYSTEM ARCHITECTURE A o-coheret deeply coupled GPS/INS system primarily processes outputs of discrimiator fuctios covertig I E,P,L ad Q E,P,L to a code phase error ad a rrier frequecy error. The received correlator output e modeled as [4] ( /2) cos( δφ ) ( ) cos( δφ ) ( /2) cos( δφ ) I = Ad R x d + IP = Adi R x + IP I = Ad R x + d + E i IE L i IL (1)

3 ( /2) si( δφ ) ( ) si ( δφ ) ( /2) si( δφ ) ( ) τ ( πδ τ ) Q = Ad R x d + QP = Adi R x + QP Q = Ad R x+ d + E i QE L i QL A= 2 a c/ sic f a a Where A is the sigal amplitude depedig o the rrier-to-oise ratio c/, ad the average frequecy errorδ f i the itegratio iterval of the correlatorτ a. The I E,P,L ad Q E,P,L also deped o the code correlatio fuctio R, the datait d i ad the average phase error δφ. Sice rrier frequecy tracig is ow to e more roust comparale to rrier phase tracig i high dyamic ad i high jammig situatios the I E,P,L ad Q E,P,L are coverted to code ad frequecy errors y discrimiator fuctios. For code tracig a ocoheret code discrimiator is employed that results i a code tracig error D = I + Q I + Q. (2) Disr E E L L The discrimiator output has to e ormalized y a estimatio of the amplitude. I strog SNR situatios 2 2 I + Q is suitale while i poor SNR situatios a c/ has to e employed. c psr DDisr (3) fco Δ ρ = trasforms the discrimiator output to a tracig error i meters. Although this tracig error could uild the iovatio for the Kalma filterig process, it is comied with the measured pseudorage of the repli sigal ρ psr. Thus positio estimatio refers to the same samplig time for each chael of a fudametal time frame give y the receiver hardware. Usig the measured repli code phase for cotrollig the code NCOs is also doe i [7], ut i a differet way. As a avigatio filter a closed loop error state space formulatio of a Kalma filter is used with the states: δ x positio errors (3) δ v e velocity errors (3) δψ attitude error (3) δ a i errors of estimated accelerometer iases (3) δω i errors of estimated gyroscope iases (3) cδ t error of receiver cloc error (1) cδ t error of receiver cloc error drift (1) After the measuremet step the strapdow is corrected ad the state vector is set to zero. The estimated positio e trasformed to a estimated pseudorage ˆ ρ psr icludig a positio update iduced y a tracig error. This estimated pseudorage ow e compared with the former pseudorage correspodig to the code phase of the repli sigal ρ psr. The error Δ ˆ ρ psr e employed to cotrol the code NCO towards miimizig the offset. This is also doe euse code phase steps are ot possile i most GPS receivers. The NCO must e cotrolled y adjustig the code frequecy. By aligig the repli code with the estimatio of the avigatio filter the discrimiator oly cotais the tracig error. The oserver (Kalma filter) ad the cotroller (NCO cotroller) are realized separately. The same is doe with the frequecy error received from a rrier frequecy discrimiator. A cross product formulatio is employed that either e ormalized y the estimated rrier-to-oise ratio c/ or e used i a 4 quadrat arcta-fuctio resultig i the rrier frequecy tracig error Δf. cross = I Q I Q dot = IPS,1 IPS,2 + QPS,1 QPS,2 1 Δ f = arcta2, Δt 2π 2 PS,1 PS,2 PS,2 PS,1 ( cross dot ) (4) By applyig the speed of light c ad the omial rrier frequecy f L1 a LOS velocity error e lculated. c Δ v dop = Δf (5) f I low dyamic situatios the velocity error icludig the satellite velocity v S, lever arm l, attitude estimatio C, ad agular rate ωi might e the iput for a velocity estimatio process i the Kalma filter accordig to the measuremet model L1 ( ( ω )) v = e v v + C l + c δt +. (6) T S U i v The updated velocity estimatio will e employed to lculate a estimated Doppler frequecy that directly cotrols the rrier NCOs. ˆ f f = f vˆ (7) L1 IF dop c I high dyamic eviromets with high acceleratios ad jers this leads to a systematic error sice Δ cotais the average tracig error i the correlatio time iterval. This lead to a misiterpretatio ad fially to iaccurate velocity estimatios. To solve this prolem, a method is applied that origially was developed to process time differeced accumulated delta rages (ADR) ad is ow exteded y a average frequecy error. f

4 3. REFINING THE MEASUREMENT STEP Oe possiility of icreasig velocity accuracy especially i high acceleratio eviromets is to employ a measuremet formulatio that accouts for averagig effects durig the time etwee measuremets, meaig the correlator time iterval descried aove. The derivatio is similar to [2, 3]. Accumulated delta rages e descried as ( φ + N) λ = rsa + cδt+ δcm + δmp + rcvr. (8) Buildig differeces of two cosecutive measuremets leads to ( φ φ ) λ = + δ δ + (9) By usig rrier phase time differeces i a error state Kalma filter it has to e compared with the estimated chage i positio icludig the cloc driftδ t 1 rsa, rsa, 1 c t c t 1 Δ y ˆ ˆ ˆ = rsa, rsa, 1+Δt cδ t ( φ φ 1) λ (1) = yˆ y Oe way to fid the appropriate measuremet matrix H i the measuremet formula i the measuremet equatio Δ y = HΔ x + v (11) is to apply a trasitio matrix factorizatio as follows: with H = H ( ˆ,,e δ L,, T ˆ t e ˆ Δ L -1 C,,, # T T T T + t # ( T, e T, T, T, T,,1) t, t dt 1 t 1, t t-1 ˆ L -1 H = ) (12) Φ Φ (13) Where is the lever arm at the previous time step ˆ ad δ L is the chage i the lever arm i two cosecutive time steps. The itegral of the trasitio matrix factorizatio truly accouts for the dyamics t, t. durig the time iterval [ ] 1 This formulatio is ow exteded y the measured rrier frequecy error Δf multiplied y Δ t which e iterpreted as a chage of the LOS distace Δ y = yˆ y + Δf λδt (14), The averagig effect of the rrier frequecy discrimiator is comied with a accurate formulatio of the relative movemet durig time iterval of the correlator. To distiguish etwee the origial ADR algorithm ad the derived versio, the authors lled it virtual-adr. High acceleratios ad jers are truly accouted for y umerilly itegratig the trasitio matrix cotaiig the iertial measuremets. I the followig chapter the correlatio time will e exteded, which emphasizes the ecessity of this formulatio. 4. REDUCING NOISE Extedig the predetectio itegratio time results i a decreased oise stadard deviatio of the measured frequecy error. The ifluece o the oise reductio of the velocity estimatio will e explaied y theoretil estimatios of the rrier frequecy discrimiator ad will e verified y simulated data. Followig a expasio of the itegratio time of the discrimiators from 2ms (2x1ms) up to 1ms (2x5ms) is rried out. Therefore ot oly the PIT is varied, also the estimatio step of the Kalma filter has to e sychroized, due to the usage of the discrimiator output values i the data fusio algorithm i a deeply coupled system. The aim of the exteded sigal itegratio time is a reductio of the oise level of the GPS-measuremets. I the simulatio there is o GPS avigatio data modulated upo the sigal. That meas that the ifluece of the algeraic sig of avigatio data o the I/Q-samples, used i the discrimiators, do ot have to e tae ito accout explicitly. A estimatio of the avigatio its allows a exteded itegratio time eve i se of it trasitios. Thus a expasio of the PIT e realized ad tested. The rrier frequecy discrimiator uses a fourquadrat arcta-fuctio. The processig is doe y meas of the dot- ad cross-product, whereas the utilized sums of the iphase- ad quadraturecompoets are divided ito two time-sectios. I PS,1 ad Q PS,1 represet the accumulated values of the first time-sectio of the o shifted sigal (prompt). O the other had I PS,2 ad Q PS,2 use the accumulated values of the secod half of the PIT. The worig area of this rrier frequecy discrimiator depeds o the reciprol value of the PIT [8]. A higher PIT leads to a smaller pull i rage. That meas that the worig area is restricted, which use prolems especially i durig high frequecy errors used y a low rrier frequecy update rate. But a higher itegratio time also leads to reductio of the oise level of the discrimiator output. The variace of the ormalized error sigal of the rrier frequecy discrimiator is descried accordig to the derivatio i [1] 2 1 N 1 σ { Δ f } = 1+ (15) 2 2 c 3 2 ctco ( 2π ) ( N 1) T CO A closer loc at equatio (15) shows that i se of high SNR the last factor withi the racets is smaller tha oe ad does ot have a sigifit ifluece. O the other had, at low SNR values, this term has a strog effect.

5 For high SNR the equatio e re-writte with a costat factor C : 2 1 σ { Δ f} = C (16) 3 TCO Usig two differet coheret itegratio timest CO oe otais: σ { Δ f1} = C, σ 3 { Δ f5} = C (17) 3 ( 1ms) ( 5ms) The relevat factor etwee a predetectio itegratio time of 1ms ad 5ms of the stadard deviatio σ Δ f for high SNR is: { } (18) 5 For the ifluece o the accuracy of the velocity estimatio the multiplitio with T CO has to e cosidered. The resultig theoretil oise reductio is (19) 5 Comparig these theoretil values with the simulatio results show i figure 1 leads to the followig stadard deviatio of the frequecy errors: { 1} { } TCO = 1ms σ Δ f = Hz T = 5ms σ Δ f =.1247Hz CO 5 (2) For the ifluece o the accuracy of the velocity estimatio the multiplitio with T CO has to e cosidered. The resultig experimetal oise reductio is (21) This is a sufficiet validatio of the theoretilly predicted oise performace improvemet. 5. CARRIER REPLICA UPDATE A deeply coupled system uses the estimated velocity ˆv ad the estimated cloc error drift δt ˆU of the Kalma filter solutio to lculate the Doppler-afflicted rrier frequecies f ˆi y usig the satellite geometry ad velocity. Thus the NCOs e adjusted directly y usig the determied satellite-idividual estimatiovalues. Beuse of this direct feedac there is o cotroller ecessary at this place. Beuse of the fact that ew sesor-measuremet-values of the IMU are availale every millisecod ad are used i order to lculate a avigatio solutio i the SDA, each millisecod a updated state space estimatio is received. It is immediately possile to use the high data rate of 1ms of a IMU istead of a rrier repli update rate of 2ms i the GPS-receiver. Figure 2 shows the structure of a deeply coupled system with a update rate of the rrier replis of 1ms. It e see that the potetial of a update rate of 1ms is used. To aalyze the system ehavior ad the roustess of the system a output of the fuctio h is realized every millisecod ad processed i the software receiver. I real hardware systems there is of course a depedecy of the IMU data rate ad the used GPS-receiveriterface. Received Sigal IMU Discr. ωi 1ms a i Δf, i 2ms-1ms Carrier 1ms Replic ( 1) NCO φ φ λ fˆi Predictio Estimatio Kalma filter Navigatio 2ms-1ms Solutio v ˆ, δ tˆ U 1ms Fig. 2: Carrier repli update with 1ms 1ms 6. SIMULATION ENVIRONMENT Fig. 1: Measured Carrier frequecy error with differet predetectio itegratio time itervals (2x1ms ad 2x5ms) durig a uacc. flight For low SNR the latter term of equatio (15) will e relevat. The relative reductio of the velocity error stadard deviatio will e eve more: =.2 (22) 5 The aim of this research is the developmet of a deeply coupled system with the aility of a roust sigaltracig i extreme flight maoeuvres. Especially i corerig lateral acceleratios occur. This leads to strog chages of the relative movemet etwee a satellite ad the receiver. I order to aalyze the quality of the implemeted tracig ad avigatio algorithms of the deeply coupled system i Emedded Matla/Simuli, a flight sceario was created, that represets the trajectory of a fast movig ad strogly accelerated oject. Therefore the IMU sesor data over time ad the correspodig received GPS-sigals,

6 Fig. 3: Groud trac, velocity i avigatio frame ad acceleratio (up to 35g) i ody-frame of the trajectory which are Doppler-afflicted euse of the relative movemet of the satellite ad the receiver, time shifted, damped ad superposed y oise, are geerated. The overlaid GPS-sigals of differet satellites are used i the user segmet o the ases of itermediate frequecies, whereas the IMU data is processed i the strapdow algorithm (SDA) to lculate a asolute avigatio solutio. Followig the characteristics of a tactil grade IMU, the ideal sesor data is overlaid y white oise, ad sesor iases are added. The created trajectory comprises several situatios with extremely high acceleratio ad jer values. The flight starts straight o i a orthoud directio with a iitial velocity of 1m/s ad a iitial attitude of the roll-, pitch- ad yaw-agle of degrees. The further progressio of the flight shows six 9 degree spiraligs with varyig duratios of these idividual maeuvers. I etwee the oject is accelerated up to a velocity of 225m/s. All i all, the etire sceario lasts 46 secods. Durig that time the SNR of all received satellite sigals remais costat at a high level. Figure 3 ad figure 4 visualize the characteristics of the used trajectory. It e see that the values of the Fig. 4: 3D trac, velocity i ecef frame ad jer (up to 5 m/s 3 ) i ody-frame of the trajectory acceleratios give the ody-frame icrease i every spiralig over time, due to the risig velocity ad the alteratio of the duratio of the sigle flight maeuver. At secod 35 a maximum lateral/trasverse acceleratio of 35g ad a jer of up to 5 m/s 3 is reached. Previous simulatios showed that stadard tracig loops loose the trac of the satellite sigals i such extreme situatios. As a cosequece there is o aidig iformatio of the GPS availale, which results i a divergece of the avigatio solutio. The simulatio eviromet provides a full 6-degrees of freedom (DOF) simulatio of satellite orits ad receiver trajectory. I a first step, the receiver trajectory is geerated accordig to a give maeuver schedule. The startig poit, startig velocity ad attitude e chose aritrarily i WGS-84 coordiates. From the iitial setup the trajectory geeratio follows give chages i velocity, agle or height. I additio, attitude variatios e simulated ut this is ot used i the preseted sceario. To examie tracig loop characteristics i the presece of dyamic stress, a smooth chage i acceleratio is

7 essetial. Therefore the jer is ot modeled y a step fuctio ut y a siusoid durig trajectory dyamics. The resultig trajectory is used as a ideal referece ad is the asis i the 6-DOF simulatio of satellite receiver geometry. Based o the simulated satellite receiver geometry, the digital sampled IF sigal is geerated. This icludes the sigal time delay ad implicitly the Doppler effect. The sigal is filtered y a ad-pass filter with a adwidth of 4 MHz. The samplig frequecy is MHz ad the itermediate frequecy is set to MHz. After ad-pass filterig, a automatic gai cotrol is applied to optimally cover the rage of the followig A/D coverter. The resultig sigal is sampled i 2 its ad is fially processed i a software receiver cosistig of differet chaels ad a avigatio process as show i figure 5. Sigal Geerator AGC A/D 6 DOF Rover Trajectory Chael N Satellite Orits Software Receiver Navigatio Process Fig. 5: Simulatio Eviromet IMU-Data Geeratio (optioal) GPS/INS Itegratio (optioal) The simulatio platform for the used deeply coupled system is Emedded Matla/Simuli. I this eviromet several iteractig fuctio-locs are implemeted i order to allote differet tass. All i all the simulatio model is divided ito four locs, coected to roadst the appropriate iformatio. Besides a fuctio-loc Tracig there also exist the fuctios Satorits, Positio, ad mappig fuctio h. Tracig comprises the asic GPS-sigal-tracig with compoets lie the umerilly cotrolled oscillators (NCO), discrimiator fuctios ad the correspodig cotrollers. This allows evaluatig the Dopplerafflicted rrier frequecies ad the shift i phase for every sigle satellite, ased o a simulated GPS-sigal. I additio a estimatio of the quality - that meas the variace - of the lculated pseudorage ad deltarage is implemeted. Depedet o the structure of the itegrated system, differet fuctio-loc outputs are possile. I a deeply coupled system the output values of the discrimiators are also supplemetary used i the loc Positio, which comprises the data fusio algorithms ad the strapdow algorithm (SDA) i order to estimate a avigatio solutio. I se of the usage of virtual-adr measuremets the appropriate fuctio i the Positio loc is utilized i the filter step of the closed loop error state space Kalma filter (KF), istead of the delta-rage measuremets. The filter step of the KF is sychroized with the availaility of the GPS-measuremets of the Tracig loc. Beuse of the oliear mathematil models a liearizatio has to e doe i order to use the stadard Kalma filter model. As a result the error terms of the state space vector are estimated. Cosequetly the asolute avigatio solutio of the strapdow algorithm - ased o the sesor values of the iertial measure uit (IMU) - is corrected. I the deeply coupled system the fuctio-loc h provides the satellite-idividual error i distace ad the Doppler-afflicted rrier frequecies, ased o the estimated positio ad velocity solutio of the KF. The estimated cloc error ad drift are also cosidered. Thus a feedac to the Tracig loc is realized ad the cotrol circuit is closed. The fourth loc Satorits lculates the positio ad the velocity of sigle satellites over time y usig orit data. These values are required to lculate the distace etwee the receiver ad the satellites ad also the idividual relative velocity. I the simulatio a fixed step size of 1ms is used, equivalet to the data rate of the simulated IMU. 7. SIMULATION RESULTS I this sectio a comparative overview over the simulatio results of various implemeted system models with differet cofiguratios is show. The descriptio of the results is geerally divided ito deeply coupled systems with a update rate of the rrier repli of 2ms ad 1ms. To show the impact of various system cofiguratios o the avigatio result, two differet predetectio itegratio times (PIT) of the discrimiators ad two differet implemetatios of the Kalma filter estimatio steps are comied. O the oe had PITs of 2x1ms ad 2x5ms are used, i comiatio with the measuremet value processig of the GPS-delta-rage or virtual-adr. The listed figures elow show the asolute error. At each time step the Euclidea distace etwee estimated ad the true trajectory is lculated. I figure 6 the asolute velocity error i coordiates of the ecef-frame is illustrated. I deeply coupled systems usig discrimiators with a PIT of 2x5ms a icreasig error over time e recogized. These implemetatios are ot ale to trac extreme dyamics of the trajectory exactly eough i se of a update rate of the rrier replis of 2ms. This results i a divergece of the lculated avigatio solutio. Beuse of the smaller operatig rage of the rrier frequecy discrimiator due to the icreased PIT of 2x5ms istead of 2x1ms, a loss of loc occur. This meas that the correct operatig rage of the frequecy-accurate adaptio is left. I order to avoid this ehavior ad to stay withi the pull-i-rage of the discrimiator a exacter - that meas faster - update/feedac of the rrier replis is ecessary. I the followig sectio the results of this higher update rate will e show. The use of discrimiators with a icreased PIT, i order to lculate GPS measuremet values with a low oise level, is ot a reasoale step for the descried sceario ad the chose update rate of the rrier replis. However systems with a PIT of 2ms deliver - aalog to figure 6 - coverget estimatios of the system states. But especially i the

8 se of processig delta-rages - also lled Dopplermeasuremets - strog iflueces of the trajectory dyamics o the estimated solutio e recogized. These iflueces correspod with the appearace of high acceleratios durig the flight. A etter result is achieved y usig the virtual accumulated delta rage (virtual-adr). Fig. 7: Velocity error with 1ms rrier frequecy update time CONCLUSIONS Fig. 6: Velocity error with 2ms rrier frequecy update time The simulatio results i figure 7 with a rrier repli update of 1ms cosistetly show coverget avigatio solutios. That meas that withi the simulatio time the asolute velocity error stays elow a certai threshold. However there exist differeces cocerig the quality of the estimatios. Deeply coupled system implemetatios with a PIT of the discrimiators of 2x1ms deliver a oticeale higher oise level. O the other had the use of discrimiators with a higher PIT - i this se 2x5ms - leads to a reductio of the oise level of the discrimiator outputs. These discrimiator outputs are processed i the estimatio step of the Kalma filter, i order to support the avigatio solutio. As a cosequece the oise level of the appropriate state space solutio is also reduced. I figure 7 this characteristic is depicted i the form of smoother error patters for the two models Doppler 1ms ad Virtual-ADR 1ms i compariso to the models with a PIT of 2x1ms. Overall, the use of a higher update rate of the rrier repli sigals i the deeply coupled system leads to a higher accuracy i tracig. As a result the Dopplerafflicted rrier frequecy is adapted i a faster way ad shows a smaller frequecy error. Such a system allows tracig satellites eve i the occurrece of extreme trajectory dyamics. Hece a cotiuous support i the KF estimatio step is possile, ad a divergece of the short-time accurate iertial avigatio solutio e preveted. Especially the processig of virtual-adr measuremets geerates very low velocity errors over time. The avigatio solutio is ot strogly iflueced y high dyamics durig the simulated maeuvers. I the ed a roust, reliale ad log term accurate result is otaied. The results show a error value uder the threshold of 5cm/s ad also a smooth sigal characteristic over time. I additio the asolute positio error of the estimatio also is ot strogly iflueced y high dyamics. The positio estimatio shows a smooth error as well, ut the positio solutio will e offset y ucorrected ioosphere ad troposphere errors. By reformulatig the velocity measuremet step of a o-coheret deeply coupled GPS/INS system it was adapted to correctly accout for the averaged measuremet values of the rrier frequecy discrimiator. This results i a decreased velocity error especially i high dyamic eviromets. With this method the predetectio itegratio time i the asead processor may e exteded without sufferig from dyamic stress i the measuremet step. REFERENCES [1] Misra, P., ad Ege, P., Gloal Positioig System - Sigals, Measuremets, ad Performace. 2. Editio. Licol, Massachusetts: Gaga-Jamua Press, 26, CD Ege, P.: White Noise Aalysis of the Frequecy Loc Loop, 24 [2] Farrel J.L. GPS/INS-Streamlied. I NAVIGATION Joural of the Istitute of Navigatio, Vol. 49 No. 4, 22 [3] Wedel J. et. Al. Time-Differeced Carrier Phase Measuremets for Tightly Coupled GPS/INS Itegratio. I Proceedigs IEEE PLANS, April 26 [4] Groves, P.D. Mather, C.J. ad Maulay, A.A. Demostratio of No-coheret Deep INS/GPS Itegratio for Optimised Sigal-to-oise Performace. I Proceedigs ION GNSS, Septemer 27 [5] Groves, P.D. ad Log, D.C. Adaptive tightlycoupled, a low cost alterative ati-jam INS/GPS itegratio techique. I Proceedigs ION NTM, Jauary 23 [6] Soloviev A., Guawardea, S., ad, va Graas, F. Deeply Itegrated GPS/Low-Cost IMU for Low CNR Sigal Processig: Cocept Descriptio ad I-Flight Demostratio. I NAVIGATION Joural of the Istitute of Navigatio, Vol. 55 No. 1, 28 [7] Ladis D. et. Al. A Deep Itegratio Estimator for Ura Groud Navigatio. I Proceedigs IEEE PLANS April 26 [8] Kapla, E.: Uderstadig GPS Priciples ad Applitios. Artech House, 1996.

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