Analysis of the Effect of Time Delay on the Integrated GNSS/INS Navigation Systems

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1 the Interntionl Journl on Mrine Nvigtion n Sfety of Se Trnsporttion Volume 7 Number June 13 DOI: / Anlysis of the Effect of Time Dely on the Integrte / Nvigtion Systems C.K. Yng & D.S. Shim Chung Ang University, Seoul, Kore ABSTRACT: The performnce of tightly couple / integrtion is nown to be better thn tht of loosely couple / integrtion. However, if the time synchroniztion error occurs between the receiver n (Inertil Nvigtion System), the sitution reverses. The performnce of loosely couple / integrtion n tightly couple / integrtion is nlyze n compre ue to time synchroniztion error by computer simultion. 1 INTRODUCTION (Globl Nvigtion Stellite System) n (Inertil Nvigtion System) re wiely use s stnlone nvigtion systems, respectively. The integrtion of n not only les to high ccurcy of vehicleʹs position, velocity, n ttitue, but lso provies position error boun. To mximize the performnce of / integrte systems, time synchroniztion between n is n importnt issue. There hs been some reserch for time synchroniztion of / mesurements for loosely couple / systems or tightly couple / systems. However, there re few reserch to consier both. This pper compres the time ely effect of loosely couple / systems n tightly couple / systems. For loosely couple / systems, we nlyze the effect of time ely for the cse when only position informtion is use s Klmn filter mesurement. For tightly couple / systems, we nlyze the effect of time ely for the cse when only pseuornge informtion is use s Klmn filter mesurement. To compre the performnce of the two integrtion systems, it is nlyticlly stuie how the time ely between n hs effect on the Klmn filter innovtion for ech integrtion metho. Then bse on the nlysis results, computer simultions re performe to chec how ech integrtion metho hs effect on the nvigtion performnce such s position, velocity, n ttitue of the vehicle. The two / integrtion methos re reviewe briefly in section, n the effects of time ely re nlyze in section 3. Computer simultions re performe in section 4 n conclusions re given in section 5. INTEGRATED / NAVIGATION SYSTEMS Severl pproches re possible for the integrte / systems epening on which informtion is shre between n. Loosely couple n tightly couple integrte systems re consiere 199

2 in this pper. The position n velocity clculte from re use to upte the filter in the loosely couple integrtion, while rw pseuornge is use in the tightly couple integrtion for Klmn filter..1 Loosely couple / nvigtion system The bloc igrm of the loosely couple / integrtion is shown in Figure 1. The position n velocity is clculte in the receiver n then use s mesurement in the Klmn filter. As seen in Figure 1 the structure is simple n moulriztion is possible. The computtion time is reuce since the lgorithm in the receiver cn be use. The Klmn filter in the loosely couple / integrtion hs the error moel with imension of 15. P, V z x P, V Figure 1. Loosely couple / integrtion (1) x F x w, w ~ N(, Q ) LC LC LC n n x [ Llhv B B ] () ccel gyro The stte vrible contins the position error with ltitue, longitue, n height ( L, l, h), n n n velocity error ( v ), ttitue error( ) expresse in the nvigtion frme, n bis errors ( Bccel, B gyro ) of ccelerometer n gyroscopes(refer to Knight 1997 for etil error moel). The mesurement eqution of the loosely couple / integrtion cn be enote s follows:, ~ (, ) z x v v N R, (3) LC LC LC LC where HLC I In the cse of loosely couple integrtion there re some rwbcs. When there re short of visible stellites, nvigtion informtion cnnot be clculte in the receiver. Uner the lrge ynmic environment nvigtion solution becomes inccurte, thus the integrtion performnce becomes worse.. Tightly couple / nvigtion system Antenn Aie Trcing Loop Receiver Inertil Reference x Klmn Filter - z + PR, PR rte P, V Correction Nvigtion Informtion Rnge, Rnge rte Preiction Figure. Tightly couple / integrtion The bloc igrm of the tightly couple / integrtion is shown in Figure. The pseuornge n pseuornge rte re use irectly in the mesurement of Klmn filter n the cloc error of the receiver shoul be inclue in the stte vrible to be estimte. Thus the Klmn filter error moel contins position error, velocity error, ttitue error, ccelerometer bis, gyroscope bis, cloc bis n cloc rift of receiver, resulting in 17 stte vribles s in (4). x F x w, w ~ N(, Q ) 15 x cloc 15 F cloc x cloc TC TC TC xcloc cbis c rift (5) where cbis, crift enotes for cloc bis n cloc rift of receiver n F, F re cloc escribe in Knight The mesurement is given s in (6)., ~ (, ) z x x v v N R GPS TC cloc TC TC TC In the cse of positioning, the former position is use s the reference of lineriztion n thus the nvigtion error increses for lrge ccelertion n ngulr rte. However, for the tightly couple / integrtion, informtion with goo ynmic performnce is use s the reference of lineriztion of mesurement, thus the sme problem oes not hppen. Even though the visible stellites re not enough, the nvigtion clcultion is still possible. So it provies more efficient structure in the usge of informtion thn the loosely couple integrtion. The rwbc of tightly couple integrtion is the complexity of the integrtion structure n thus computing effort increses s the number of stellites increses. (4) (6)

3 3 EFFECTS OF THE TIME DELAY IN THE INTEGRATED / SYSTEM IMU Acceleromete rs Gyros Antenn t=ts Receiver Inertil Nvigtion System M t= Ts T t=ts T1 Integrte / Klmn Filter Estimte Errors (Nvigtion Errors, Sensor Errors) Figure 3. Structure of time synchroniztion error T(=T T1) in the integrte / nvigtion system For the / integrtion there exist time ely between receiver n since not only smpling time but lso the signl processing time is ifferent s in Figure 3. For simplifie nlysis, the receiver smpling perio is ssume to be multiple M of the IMU smpling perio Ts. T1 n T re processing time elys of the inertil nvigtion system() n the receiver, respectively n T =T T1, i.e., the ifference of the n processing time. 3.1 EKF lgorithm Consier the error moel of n EKF eqution to nlyze the effects of time ely in mesurement. The error moel cn be escribe s follows: x x G e (7) 1 z x w (8) Here, is the error stte trnsition mtrix, e is the process noise, G is the process noise gin, H is the mesurement mtrix, n w is the mesurement noise. The subscript implies th smpling sequence. The process noise e n the mesurement noise w re consiere s white noise, uncorrelte with ech other n the covrince mtrices re R { ww } n Q { ee }, where, enotes the expecttion opertor. The EKF lgorithm for the / integrtion is given in Tble 1. Tble 1. EKF Algorithm for Integrtion between the n the 1 Klmn gin upte ( ) Difference of the two mesurements Estimtion of the stte errors R z pˆ pˆ xˆ,. rˆ z ˆ Error correction rˆ rˆ xˆ ˆ ˆ Covrince upte ( ) Sensor error compenstion Nvigtion eqution upte Sensor error upte uˆ g( u, ˆ ) rˆ ( ˆ, ˆ ) 1 f r u ˆ 1 ˆ Covrince preiction 1 GQG 3. Effects of time synchroniztion error Vectors r ˆ, ˆ n u ˆ in Tble 1 enote the estimte nvigtion stte, the estimte sensor errors, n the IMU mesurements, respectively. Effects of time synchroniztion error between receiver n cn be nlyze s follows. Let the time ifference of smpling mesurement between receiver n be T n suppose tht smpling hs time synchroniztion error less thn 1sec. Then in the cse tht vehicle position is use in loosely couple integrtion, the estimte position in is given s in (9), where p() t mens true trjectory. pˆ p ( ) p v.5 w (9), S where T 1, v is the velocity of the vehicle n is the ccelertion. Suppose tht the pseuornge, which is the mesurement of tightly couple integrtion, hs time synchroniztion error less thn 1sec, then pseuornge estimte in cn be escribe s (1), where () t is true pseuornge. ˆ ( ).5 w (1), S where the pseuornge rte is ( v, v) l, pseuornge ccelertion is, (, ) l, l is LOS(line of sight) vector between stellite n vehicle, n v, n, re velocity n ccelertion of stellite, respectively. Thus the true observtion z of two integrtion systems re obtine s in (11). 1

4 For loosely couple integrtion z pˆ pˆ (11),, p v w p ˆ, For tightly couple integrtion z ˆ ˆ (11b),, ˆ w, Let ˆ x p p or, ˆ x,, then (11) becomes (1). z x w (1) In (1) the resiul error vector of position or the resiul error vector of pseuornge is introuce n efine s v or (13) In Sog & Hänel (11), the errors of the closeloop system re foun to be ˆ 1 ( ) x x x G e (14) x z G e Let p,, then (14) becomes (15). x ( ) x w G e (15) 1 p, p, Here, by substituting z with z in (14) n inserting (1) into this, the following ifference eqution for the error stte cn be obtine: x ( ) x w G e (16) 1 p, p, p, If we introuce p,, then the MSE(men squre error) of the nvigtion solution cn be expresse s in (17). { x ( x ) } R G QG p, p, x x p, p, p, p, (17) cn be neglecte in the loosely couple integrtion n MSE vlue in (17) becomes R G QG. 1 p, p, However, for the tightly couple integrtion, the resiul error vector oes not become zero even though the velocity of vehicle is lmost zero becuse of high spee of GPS stellite lie 3.9m/sec, i.e., ( l ) ( l ),, Thus we cn notice tht time ely in the tightly couple integrtion hs more effect on the nvigtion performnce thn in the loosely couple integrtion when velocity n ccelertion of vehicle re not lrge. 4 SIMULATIONS In this section computer simultion is performe to verify the nlysis result for the effects of time ely. Tble shows the specifiction of n GPS use in the simultion. The vehicle trjectory is ssume s circle n run time is sec. Vehicle spee is 1m/sec, m/sec, n 4m/sec n the time ely is ssume to be 1sec in mesurement. Tble. Specifiction of error sources of the n the GPS Error Sources 1 σ vlue initil position error 1m initil velocity error 1m/sec initil horizontl ttitue error.3 eg initil verticl ttitue error 5 eg ccelerometer bis 5 μg gyro bis 3 eg/hr GPS cloc bis 1m cloc rift 1m/sec Figure 4 shows the position errors in north irection n pitch errors in the loosely couple GPS/ ccoring to the vrious vehicle spees. Generlly the position error n ttitue error become lrge s the vehicle spee increse. This hppens becuse the resiul error vector in (13) becomes lrge ccoring to the vehicle spee n thus the optimlity of Klmn filter gets worse. The oscilltion of the position error comes from the circle trjectory. where the men error x { x} cn be clculte s in (18). x x (18) 1 p, Notice the resiul error vector in (13). For the loosely couple integrtion, time ely T hs little effect on in the cse of smll velocity n ccelertion, i.e.,. Thus for smll velocity

5 15 position error[m] 3 euler error[eg] 1 5 no time-ely 1sec time-ely(1m/hr) 1sec time-ely(m/hr) 1sec time-ely(4m/hr) 1 no time-ely 1sec time-ely(1m/hr) 1sec time-ely(m/hr) 1sec time-ely(4m/hr) North Pitch () the north position errors Pitch euler error[eg] no time-ely 1sec time-ely(1m/hr) 1sec time-ely(m/hr) 1sec time-ely(4m/hr) (b) the pitch ngle errors Figure 4. The position errors n ttitue errors in the loosely couple GPS/ ccoring to the vrious vehicle spees (1m/hr, m/hr, 4m/hr) (b) the pitch ngle errors Figure 5. the position errors n ttitue errors in the tightly couple GPS/ ccoring to the vrious vehicle spees(1m/hr, m/hr, 4m/hr) Figure 5 shows the position errors in north irection n pitch errors in the tightly couple GPS/ ccoring to the vrious vehicle spees. Notice the position error t first. The position error is lrge lie 3m even though the spee of vehicle is smll lie 1m/hr. The resiul error vector in (13) is smll, i.e., in loosely couple integrtion when vehicle spee is smll, while in tightly couple integrtion the resiul error vector, (, GPSl).5(, GPSl), is lrge since the stellite spee is lrge lie 3.9m/hr. Figure 5(b) shows tht the ttitue error becomes lrge ccoring to vehicle spee n the ttitue error is much lrger thn tht in loosely couple integrtion since the stellite spee is the ominnt term in. Figure 4 n Figure 5 show tht time ely error between GPS n cuses more effect on tightly couple integrtion thn on loosely couple integrtion North position error[m] no time-ely 1sec time-ely(1m/hr) 1sec time-ely(m/hr) 1sec time-ely(4m/hr) () the north position errors 5 CONCLUSIONS The estimtion performnce of loosely couple / integrtion n tightly couple / integrtion is compre ue to time synchroniztion error between receiver n. If the time synchroniztion error occurs between two sensor mesurements, resiul error exists. For loosely couple integrtion the resiul error vector increses ccoring to vehicle spee while for tightly couple integrtion the resiul error vector increses ccoring to stellite spee s well s vehicle spee. The GPS stellite spee is bout 3.9m/sec, which is much lrger thn the vehicle spee. Thus resiul error vector in tightly couple integrtion epens minly on the stellite spee. Simultions re performe to verify the nlysis result for the two GPS/ integrtion methos. The simultion shows tht time synchroniztion error hs effect more on the tightly couple integrtion thn the loosely couple integrtion 3

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