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AIAA 82-4127
J. GUIDANCE VOL. 5, NO. 3
Doppler-Aided Low-Accuracy StrapdownInertial Navigation System
Itzhack Y. Bar-Itzhack, * Daniel Serfaty,t and Yehoshua VitekJTechnion—Israel Institute of Technology, Haifa, Israel
This paper presents a covariance analysis of the performance of a Doppler-aided low-accuracy coarselyaligned strapdown inertial navigation system (INS), whose fine alignment takes place automatically in flight. Itis shown that the fine alignment in azimuth, which requires turns, consists of in-flight gyro calibration and in-flight gyrocompassing. The spacing of the turns is investigated. The influence of several position fixes isexamined, and it is shown that they can replace INS turns. It is also shown that the use of magnetic headingreference reduces system errors but is not necessary for the satisfactory performance of the augmented system.Two suboptimal Kalman filters are designed and evaluated. Their small performance degradation with respect tothat exhibited by the optimal filter and their low sensitivity to parameter changes is demonstrated by truecovariance simulation runs.
I. Introduction
THE need for an inexpensive midcourse navigation instandoff weapon systems promoted the development of
low-cost/low-accuracy strapcjown INS (like LCIGS1*2) whosegyro constant drift rate is in the range of 1.0 deg/h, and itsstability is in the range of 0.1 deg/h in 1 h. While in thosesystems the strapdown INS is the.primary navigational aid,such INS can be installed on other vehicles and be augmentedwith their own navigation system. The synergistic effect of theaugmented systems, whose individual contributions areweighed optimally, is well known from the theory ofminimum variance linear estimation.3
Many flight vehicles use a Doppler radar navigation systemas their standard navigational aid. To upgrade this system onecan add to it a low-cost strapdown INS and use a properKalman filter to integrate the two systems. Figure 1 illustratesthe benefit of such augmentation. The position-error of theDoppler navigation system grows linearly with the distancetravelled by the vehicle. If the Doppler system is replaced by alow-accuracy strapdown INS, which is only coarsely aligned,its position-error diverges rapidly with time. When thecontribution of both systems is weighed optimally by aKalman filter along a baseline trajectory, the resultingposition-error behaves according to curve b of Fig. 1. (Thenature of the baseline trajectory and the time where the peakposition-error of the augmented system occurs, will bediscussed in the sequel.)
The idea of augmenting a gimballed INS and a Dopplernavigation system has been discussed in the past.4"8 Recently,San Giovanni9 presented an accuracy performance analysis ofa ring laser strapdown attitude and heading reference systemthat included magnetic heading and airspeed sensors toprovide required heading and velocity reference data. Henoted that turns of the vehicle on which such a system ismounted generally assist, and in some cases are vital tocalibration of many of the system errors.
Using true covariance analysis, the present work in-vestigates the performance of a low-accuracy coarsely alignedDoppler-aided strapdown INS with and without magneticheading. The main contributions of this work are: 1) in-vestigation and resolution of the effect vehicle turns have onsystem operation and accuracy, 2) examination of the in-
Received June 1, 1981; revision received Nov. 17, 1981. Copyright© 1982 I.Y. Bar-Itzhack. Published by the American Institute ofAeronautics and Astronautics Inc., with permission.
* Associate Professor, Department of Aeronautical Engineering.t Graduate Student, Department of Aeronautical Engineering.^Research Fellow, Department of Aeronautical Engineering.
fluence time interval lengths between turns have on systemperformance, 3) investigation of the weight position fixeshave on system operation, and 4) design and analysis of twosuboptimal Kalman filters, one for the system with magneticheading and the other for the system without magneticheading reference.'The efficiency of position fixes and thesensitivity of the suboptimal filters are examined too. Ac-cordingly, the next section presents the error model of theaugmented system. Section III presents performance resultsof the augmented system. An analytic investigation of theimportance vehicle turns (turn updates) have in improvingsystem performance is carried out in Sec. IV. Section V dealswith the influence of several turn updates and their possiblespacing and of position fixes and their spacing. Section VIpresents the design of two suboptimal filters and their per-formance. The main results of this work are summarized inSec. VII.
II. System Error ModelFigure 2 presents the configuration of the augmented
system. The INS is a baro-inertial strapdown system. Thereare two modes of operation. In the first mode the magneticheading is used to resolve the Doppler velocity ouput intonorth and east components from which, then, thecorresponding INS computed north and east velocity com-ponents are subtracted to form the observables Zj and z2respectively. The indicated magnetic heading is subtractedfrom the INS indicated heading to form z3. These observablesare processed by an appropriate Kalman filter to yieldestimates of the system error states. The latter are then used ineither a feedforward or a feedback mode10 to correct thesystem-indicated navigation states (Fig. 2 presents the
1000 2000TIME (sec)
Fig. 1 Horizontal position error growth of a Doppler radarnavigation system (curve a) and a Doppler-aided low-accuracystrapdown INS (curve b).
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MAY-JUNE 1982 DOPPLER-AIDED LOW-ACCURACY STRAPDOWN NAVIGATION SYSTEM 237
Fig. 2 System configuration (switch positions are according to thefirst mode of operation)..
feedback mode). In the second mode of operation, magneticheading (or some other heading reference) is used only forpreflight coarse azimuth alignment of the INS but it is theINS-indicated heading which is used to resolve the Dopplervelocity output. Using the foregoing system description, it canbe shown4'7'11'12 that for the second mode of operation
(la)
where VN and VE are, respectively, the errors in the INS-computed north and east velocity components and <£ is thevector of attitude difference between the imaginary platformaxes13 and the true local level axis.14 Similarly, 60 is thedifference between the computer and local level axes and $ isthe boresight error which is, actually, the attitudemisalignment error between the Doppler-radar and the air-craft axes. V is the true-velocity vector. The quantities AKNand AK£ are, respectively, the north and east components ofthe actual measurement error associated with the Dopplerequipment and nN and nE are the respective white noisecomponent of these quantities. The subscripts N and Edenote, respectively, north and east components in local levelcoordinates.
For the first mode of operation another measurement isadded, namely:
where d\I/ is the magnetic heading error and the expressions forZj and z2 are identical to those given in Eqs. (la) and (Ib),respectively, only that <t>D, the third component of 0, isreplaced by 6^. (The minus sign in front of <t>D in Eq. (2) is dueto the fact that the positive direction of <t>D is downward.)
The dynamics of the variables which appear in Eqs. (1) and(2) constitute the augmented dynamics equation of the system.The vector 60 is expressed in terms of the INS position error asfollows:
R(3)
-tanL
where rN, rE and rD are the components of the INS positionerror vector expressed in local level coordinates, RE and RNare the east and north radii of curvature, h is the altitude andL is the latitude of the vehicle. The dynamics of r, v and <£ is
expressed by the well-known INS error model.8'14 The drivingfunction vector of the latter consists of the acclerometer andgyro errors expressed in local level coordinates. These errorsare related to the errors of the body-mounted gyros and ac-celerometer s as follows:
DBL 0
0 DBL
(4)
where eL and eB are the vectors of gyro errors expressed inlocal-level and body coordinates respectively, and VL andVf l are the same for accelerometer errors. The matrix Z>£ isthe transformation matrix from body to local-level coor-dinates. The gyro errors which constitute eB are assumed to bea combination of constant drift, g-sensitive and white noisecomponents. The accelerometer errors are a combination ofbias and white noise components which constitute V B.
The Doppler errors AVN and A VE are obtained from A VLwhere
(5)
The components of A VB contain a constant part and a partwhich is proportional to the aircraft velocity. Similarly, nNand nE are extracted from nL where
(6)
and nB is a white noise vector. The vector ]8 when resolved inthe local-level axes satisfies
and ctB is a vector of constant boresight error angles in bodyaxes.
The magnetic heading reference error is modeled as acombination of random bias, first order Markov and whitenoise components.9
As a result of the augmentation of the error models of theINS, the Doppler radar and the magnetic heading reference, asystem dynamics model is obtained whose state vector con-tains three position states, three velocity states, three attitudestates, three gyro bias states, two gyro g-sensitive states, threeaccelerometer bias states, three constant Doppler velocitystates, three constant boresight error states, a constant, and aMarkovian magnetic reference state.
III. Nominal ResultsTypical LCIGS and Doppler radar values were chosen as
nominal data in this analysis. The values of the initial statesare tabulated in Table 1, the values of the system driving noisecomponents are shown in Table 2, and the measurementerrors are given in Table 3. The vehicle was flown at anomimal velocity of 50 m/s to simulate a helicopter flight.The baseline trajectory on which the vehicle was flownconsists of a straight and level flight at a 45 deg heading for900 s, a 90 deg right turn executed in 30 s, a second straightand level flight for another 900 s, a 90 deg left turn executed in30 s, and a final leg of straight and level flight. The total flightduration is three quarters of an hour.
The main performance characteristics of the augmentednavigation system are presented in Fig. 3 where the horizontalerror in terms of CEP and the azimuth la estimation error areplotted, both as a function of flight time. These results wereobtained for the first mode of operation.
The first turn, which starts at 900 s is, evidently, a keyfactor in the operation of the augmented system. Without thisturn, the performance of the system is as good as the per-formance of the Doppler navigation system whose horizontal
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238 BAR-ITZHACK, SERFATY, AND VITEK J. GUIDANCE
Table 1 Initial state vector
Notation^
rT
VT
*T
*smT
V r
*yB
"IWo
Description
Position error vectorVelocity error vectorAttitude error vectorGyro bias error vectorGyro ̂ -sensitive drift
error vectorAccelerometer biasDoppler constant
velocity error vector
Boresight error vectorMagnetic heading ref-
Numerical values
[10,10,10][10, 10, 10] a
[5,5,5][1.5,1.5,1.5][1,1]
[1,1,1]0.025 V+ 0.100.0025 V+ 0.100.0010F+0.05
K-nominal velocity[0.3,0.3,0.3]
1.5
Units
mm/sdeg
deg/hdeg/h/g
mgm/sm/sm/s
degdeg
erence constant errorMagnetic heading ref-
erence Markov error(correlation time of1800s)
1.0 deg
aThese values were chosen to be unrealistically high in order not to imply too small attitude errors. After thefirst update, they decrease to their proper values with the right correlations.
Table 2 System driving noise components
Notation Description Numerical values Units
Accelerometer whitenoise component
Gyro drift whitenoise component
Driving white noiseof Markov com-ponent of magneticheading
[0.27,0.27,0.27] (m/s)/h'/2
[0.013,0.013,0.013] deg/h1/2
2.0 deg/h1/2
CEP(km)
Fig. 3 CEP and azimutherrors of nominal augmentedsystem as function of time.
1000 2000TIME(sec)
Table 3 White noise measurement errorsNotation
4Description
Magnetic headingDoppler radarExternal position
information (whenavailable)
Numerical values
15[0.2,0.2,0.2]
[20, 20]
Units
arc-minm/sm
position error as a function of time is shown in Fig. 1 (curvea). It is the reduction of the azimuth error, beginning at theturn, which completes the INS alignment and thus turning itinto an acceptable navigation system whose augmentationwith the Doppler radar yields an improved navigation device.The question why hitherto nothing happened and azimuthalignment begins only after the turn, is important to theunderstanding of the operation of the system. This question isaddressed in the next section.
IV. Analytic Investigation of the Turn UpdateINS azimuth self-alignment (as opposed to optical align-
ment) can be achieved through either transfer alignment orgyrocompassing. The difference between the two can bebriefly explained as follows. In transfer alignment, somevector is measured by both the INS and a certain referencesystem. The vector is thus coordinatized in both systems. Thedifference between the corresponding components of thisvector in the two systems is a measure of the misalignment
error. In gyrocompassing, though, the INS azimuthmisalignment, in collaboration with the north component ofthe sum of Earth rate and the rate of change of the longitude,cause a growing tilt about the east axis, which in turn yields agrowing north velocity-error component. A measure of thiserror component is obtained from the comparison of the INSnorth velocity with that of some reference system. (Furtherdetails are given in Ref. 15.)
Although vehicle maneuvers enhance azimuth observabilityduring transfer alignment,15 the notion that transfer align-ment occurs during the turn is ruled out at the outset. Thereason for rejecting this explanation stems from the fact thatin transfer alignment the final alignment can be only as goodas that of the reference system16 which, in our case, isdetermined by the magnetic heading reference whose accuracyis around 1.8 deg. The turn itself adds no additional in-formation on the azimuth misalignment. This conclusion canbe arrived at analytically, too. Consider point P0 in Fig. 4,which is a point close to the beginning of the turn. At thispoint, where the flight time is 900 s, almost all of the tilt errorhas been already estimated and removed. For simplicity it isassumed that the position error is zero, that the Doppler radaris perfect and its boresight errors are negligible. For, if undersuch tight conditions,the azimuth error is unobservable, it issurely so for more lax conditions. Under these conditions themeasurement equations of the first mode of operation, that is,Eqs. (1) and (2) can be reduced to the following:
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MAY-JUNE 1982 DOPPLER-AIDED LOW-ACCURACY STRAPDOWN NAVIGATION SYSTEM 239
Fig. 4 Velocity and ac-celeration components atthe turn.
follows:
where the subscript 0 denotes variables at point P0. From thedeterministic point of view, there are four unknowns; namely,(/>D, d\l/> VNO and VEO with only three equations, thus the systemof equations cannot be resolved. From the stochastic point ofview, it is observed that repeated measurements and updateson this leg do not change the lack of stochastic observ-ability 3«15«17 of <t>D.
To show that the turn does not yield immediate informationto enhance the observability of <t>D, consider the followingdevelopment. During the turn the errors <t>D and d\l/ remainbasically constant. However, the INS velocity error com-ponents VN and VE do vary. It can be easily shown15 thatessentially they vary according to the following dynamicsequations
(9)
Since (t>D stays constant, Eqs. (9) can be integrated to yield
n aNdt'0
aEdt (10)
where t0 and tl denote the time at P0 and P7 respectively, andthe subscript 1 denotes variables at point P;. The integrals inEqs. (10) are equal to the change in the nominal velocitycomponents between points P0 and P7 hence
aEdt=YE1-VEO (11)
but from Fig. 4
thus Eqs .(11) become
f ' 7 _ f ' 7 -J'o N N° «W
Substituting Eqs. (13) into Eqs. (10) yields
N1 = VNO
(12)
(1.3)
(14)
Now, similarly to Eqs. (8), and using the previous assump-tions, the measurement equations at point P]y which is afterthe completion of the 90 deg turn, are
05)
With the aid of Eqs. (12) and Eqs. (14) these measurementscan be expressed in terms of the previous measurements as
EO
(16)
A comparison between Eqs. (16) and (8) yields
that is, the new measurements at Pj are a linear combinationof the measurements at P0. No additional information hasbeen added, and we are still left with only three independentequations and four unknowns. In conclusion, the turn addedno new information on the INS azimuth alignment. Note thatfor simplicity we chose the initial heading to be 45 deg. Thepreceding analysis can, of course, be extended to include anarbitrary initial heading.
The preceding discussion raises an important point for theanalyst. In a co variance simulation, the change in t;^ and VEdue to aE and aN, which is given here in its simplest form byEqs. (10), is modeled, though in a more elaborate fashion, inthe dynamics equation of the INS error propagation.8'14 Onthe other hand, the nominal velocity components, which arealso functions of VN and VE and which are the entries of themeasurement matrix [see Eqs. (9) and (16)] are known andentered into the measurement matrix directly. If due tocomputation inaccuracies the computed transition matrix,and consequently the propagated covariance matrix, areinaccurate, the relations expressed in Eqs. (11) are notfulfilled exactly. Therefore, there is a slight incompatibilitybetween the covariance matrix and the correspondingmeasurement matrix. Hence, the relationship between themeasurements at P0 and those at Pj is, erroneously, not quiteas given in Eqs. (17). Consequently, this slight mismatchrenders a certain degree of azimuth observability which doesnot exist in reality. For this reason a covariance analysis, ifnot executed carefully, yields overly optimistic results afterturns.
After rejecting the possibility that vehicle turns enhancesystem operation through transfer alignment, we considernow the possibility that it is gyrocompassing that improvessystem performance. According to this assumption, duringthe first leg of the flight the equivalent north gyro drift isobservable. Therefore, its value is estimated or, in otherwords, the equivalent north gyro is calibrated during the firststraight and level flight segment. During this time theequivalent east gyro drift is unobservable and cannot becalibrated. It is well known that the east gyro constant driftrate sets a lower limit on <j>D. Therefore, as long as the straightand level flight continues, <t>D remains on a constant highlevel. The 90 deg turn places the equivalent north gyro of thefirst leg in the east direction, and the equivalent east gyro inthe north-south direction replacing the task of the twoequivalent gyros. The new equivalent east gyro is now acalibrated gyro; that is, its constant drift has been calibratedout considerably. This reduces the lower limit that (t>D canreach through gyrocompassing, which enables the estimationof the azimuth misalignment. With the reduction of ct>D theobservability of other INS errors increases, their values arereduced and the INS becomes a higher quality system.
What is the evidence to support this calibration andgyrocompassing explanation of the influence vehicle turnshave on system performance? To examine whether calibrationindeed takes place, let the vehicle fly north or east and observethe estimation error of the constant drift rates of the north-south and east-west gyros. In this case, the north and east
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240 BAR-ITZHACK, SERF AT Y, AND VITEK J. GUIDANCE
1000 2000TIME (sec)
Fig. 5 Reduction of theestimation error of the constantdrift rate of the horizontalstrapdown gyros.
Fig. 7 Behavior of theestimation error of theconstant drift rate of theequivalent north and eastgyros.
1000 2000
TIME (sec)
( rod)
0.8
0.6
0.4
0.2
0.0
-0.21000 2000
TIME (sec)
Fig. 6 Behavior of thecross-correlation betweenthe estimation errors of theconstant drift rate of thehorizontal strapdowngyros.
0.08
0.06
0.04
0.02
Fig. 8 ff^D (curve a) and itslower bound (curve b).
1000 2000TIME (sec)
equivalent gyros coincide with the strapdown gyros whosedrift errors are included in the state vector. The reduction ofthe north gyro drift prior to the turn and its substitution witheast gyro drift after the turn is evident from the simulationresults. What happens, though, when the azimuth of the firstleg is neither in the north-south nor in the east- west direction.In this case the equivalent north and east gyro drifts are alinear combination of the strapdown gyros and are notmodeled directly in the state of the system. To examine thecalibration effect in this case note that
€0Neq
where eONeq and eOEeq are, respectively, the equivalent northgyro constant drift and equivalent east gyro constant drift.Similarly e0x and e0y are, respectively, the constant drifts ofthe roll axis and the pitch axis strapdown gyros. Using Eqs.(18) it can be easily shown that the standard deviations aregiven by
Consider now the baseline trajectory where for the first leg^ = 45 deg. In this case Eqs. (19) become
in Eqs. (19) yields
tONeq '(20)
On the first leg both aeQx and af(fy decrease in an identicalmanner as shown in Fig. 5 whereas 0e0xeo grows as shown inFig. 6. The net effect as expressed by Eqs. (20) is shown inFig. 7. Observe that the large estimation error of eONeq isindeed calibrated out. It reaches a steady state at about 500 sinto the flight. Even the initial estimation error eOEeq goesdown, but it reaches a steady state, which is much higher thanrequired. (The reason for this initial reduction of aeOEeq will bediscussed in the sequel.) During the turn at 900 s, the azimuthangle changes from 45 deg to 135 deg. Using the new value of
(21)
A comparison between Eqs. (20) and (21) reveals how thephysical interchange between the north and east equivalentgyros is expressed analytically. Since from Fig. 5 it is clearthat neither a€Qx nor atQ change appreciably during the turnand since, from Fig. 6, me same can be said about o€0x€0y, it isclear that the amount by which 02
eOEeq decreases during theturn is nearly 2aeQ Qy. It is concluded, therefore, that when thefirst leg is parallel neither to the N axis nor to the E axis, thecross correlation between e0x and e0y, which is built during thefirst leg, is essential to the reduction of aeQEe .
To show that this calibration is followed by gyrocom-passing consider the lower bound which <j>D can reach throughgyrocompassing. It is well known from the INS errorequations that
\(/>D\> \
therefore
(22)
(23)
Curve b of Fig. 8 is a plot of the right-hand side of Eq. (23);that is, of the lower bound of o^D as a function of time. Curvea is a plot of o^D itself as a function of time. It is seen thatafter the first update, a0D goes down to the value of theheading reference, which is a part of the "master" referencesystem. This is why a^D in this region is lower than the lowestbound attainable through gyrocompassing. In fact, here it isthe better quality of azimuth information, which is the sourcefor the calibration (the decrease) of the equivalent east gyroconstant drift rather than vice versa. When, at about 400 s,the equivalent east gyro constant drift reaches its steady-statevalue, which is compatible with a0£>, it stays constant whilea^D is practically unchanged. After the turn, though, thelower bound on o^D is drastically reduced with the reductionof ffQEeq- Then, a0z? starts moving down to its lower bound ona curve typical, in its time behavior, to a gryocompassing
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MAY-JUNE 1982 DOPPLER-AIDED LOW-ACCURACY STRAPDOWN NAVIGATION SYSTEM 241
Table 4 The effect of turn spacing
Azimuth error a^D, mrad Position error (CEP), m
Time ofCase no. first turn, s
1234 (Baseline)5678
100200400900
1400 .200200400
Time ofsecond turn, s
18001800180018001800400600900
Initial3
2323232323232323
Beforesecond turn
• 25.312.77.35.14.79.38.98.0
Final
4.04.03.74.03.58.06.75.9
Maximal
592379320710
1077237237320
Final
1099689
140202218194171
a After the first update where reference heading updates 4>D.
curve. From the fact the o^D hunts but never decreases belowits lower bound set in gyrocompassing and from the timebehavior of the decrease of a^D after the turn, it is concludedthat it is a gyrocompassing process that improves theestimation of </>D after the calibration of the north gyroconstant drift rate> and the exchange of the north and the eastequivalent gyros.
If this explanation is correct, then there is no need for thewhole vehicle to turn since it is enough to turn just the INScase about the azimuth axis. Indeed, when the covarianceprogram simulates a straight and level flight along which theINS is turned at the times at which the whole vehicle turnedwhen it was flown on the baseline trajectory, the behavior of(70D as well as of ae0£eq is almost identical to their behavior onthe baseline trajectory. The position error propagation is alsosimilar to its behavior on the baseline trajectory. The onlydifference is that after its fast decrease, which follows the firstturn, the position error starts diverging slowly. This is due tothe fact that the rotation of the velocity vector which takesplace on the baseline trajectory, enables the estimation of theDoppler velocity error components A VBx and A VBy while thisdoes not happen on the straight and level flight. At any rate,calibration and gyrocompassing occur here too, as it should.
V. Influence of Several Turn Updatesand Position Fixes
Turn updates do not necessarily require the turn of theentire vehicle. Following the preceding explanation, it is clearthat for a turn update it is sufficient to turn the sensor clusteronly. Even so, one is interested in knowing how often turnupdates have to be executed. It has been found that for allpractical purposes two such updates suffice. The question is,then, when to execute them. Obviously, if the first turn iscarried out before calibration is completed, the successivegyrocompassing is not very effective after the first turn. Insuch cases, the second turn becomes more effective than onthe baseline trajectory. To examine the effect turn spacing hason performance, several covariance simulations were run. Thebasic features of the baseline trajectory were kept in thoseruns with the only exception that the two turns were executedat different time points. The results of those runs are sum-marized in Table 4.
The following conclusions are derived from Table 4:1) Enough time has to elapse before the first turn becomes
effective. In this system the optimal time for the first turn is400s.
2) If the time of the first turn is sufficiently large, the effectof the second turn is minimal. Then, however, one has to livewith large position errors for a longer time.
3) Lowest final position error is achieved when the first turnis executed at 400s.
4) Lowest final azimuth error is achieved when the first turnis executed late in the mission.
5) The efficiency of the second turn update increases withtime bet ween turns.
Table 5 Error growth (in percent) with respect to the optimal filterwhen using a suboptimal filter
AU = 900s(before the turn)
Resultanterror in
rNrEVN<t>N*E*D
SOF1
3.13.12.00.50.23.6
SOF2
L41.41.30.10.01.6
AU = 2700s(at end of flight)
SOF1
6.56.75.89.04.67.1
SOF2
4.85.04.17.04.14.9
The influence of discrete position fixes was examined too.Two cases were simulated. In the first case, high frequencyDoppler updates were performed as usual and, in addition,low frequency position updates were performed too. Thesecond case included a turn update, in addition to the twokinds of updates of case one. The results of the first case aretypical to those of a position updated ordinary INS (see, forexample, Figs. 8 and 9 in Ref. 18). As for the second case, thesimulation was performed on a trajectory which, like thebaseline trajectory, started with a 45 deg straight and levelflight then a position update took place at 180 s, a 90 degright-hand turn was performed at 360 s, and a second positionupdate took place at 540 s. The simulation was terminated at1200 s. The accuracy of the fix was 20 m in the north directionand 20 m in the east direction. This updating schedule yieldedfinal CEP of 30 m, and a final azimuth error of 4 mrad (la).It is concluded, therefore, that if within 30 km of the takeoffpoint there are identifiable landmarks, or there is anothermeans for position fix whose RSS is around 30 m, one turnupdate suffices, and a good quality augmented navigationsystem is obtained after just 10 min of flight time.
VI. Suboptimal Filter DesignThe advantages of using a suboptimal rather than an op-
timal Kalman filter are well documented.3'8 Of the severalpossible categories of suboptimal filters (SOF), the approachof reducing the order of the SOF was chosen in this work. Tomeet this end, the observability of the elements of the statevector of the "truth model" was examined. It was found thatthe estimation error of eOD, m, and t±VBy was hardly reduced(i.e., the reduction was between 0 to 30%). The estimationerror of V.,, d\I/0, and 6^7 was reduced moderately (i.e., 30 to60%). These states were, therefore, candidates for eliminationfrom the state of the SOF. Because of the weak correlationbetween the vertical and the horizontal channels of the INS,and due to the barometric damping of the vertical channel, itsstates were eliminated from the SOF state altogether.Adopting these considerations, several reduced order SOF'swere evaluated. The evaluation tool was a covariance
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242 BAR-ITZHACK, SERFATY, AND VITEK J. GUIDANCE
500Fig. 9 Time histories ofarN when the optimal filteris used and when SOF1 isused.
1000 2000TIME (sec)
Table 6 Final errors when using the optimal filter,SOF1 and SOF2
Final error in
CEP, mTilt, mradAzimuth, mrad
Optimal
1400.0854.0
FilterSOF1
1500.0924.30
SOF2
1470.0904.21
simulation program that runs simultaneously the SOF filterand the truth model19'20 and yields the true covariance of theactual estimation error, which is obtained when using the SOFto estimate the state of the "truth model." As a result of theevaluation, two reduced order SOF's were chosen; one for thefirst mode of operation, and one for the second. The state ofthe SOF for the first mode, which is denoted by SOF1 , is
x*T= [rN,rE,vN,vE,<t>Nt<l>E,<t>D,e0x,e0y,e0z, V x, V y,
(24)
and that for the second mode of operation, which is denotedby SOF2, consists of the same plus d\l/j. Table 5 presents theperformance of the two filters with respect to the optimal one.Figure 9 presents the time histories of arN for both SOF1 andthe optimal filter. Finally, Table 6 presents a comparisonbetween the final errors of the most important navigationvariables when SOF1, SOF2, and the optimal filter are used.An error sensitivity for SOF1 was carried out. Accordingly,the filter used the nominal parameters, whereas theparameters in the "truth model" were changed. Thesuboptimal filter exhibited low sensitivity to a wide range ofchanges. This in contrast with the well-known3 high sen-sitivity of the optimal filter which was verified too in aseparate sensitivity analysis. '
VII. ConclusionsIt is concluded that the augmentation of a low-cost coarsely
aligned strapdown INS with a Doppler radar constitutes avery good navigation and attitude reference system which, inour case, yields a few rnrad azimuth error and a few hundredmeters of position error after 2700 s of flight time. This is trueonly at the later phases of the flight when error reductiontakes place. One condition for the satisfactory performanceof the augmented system is the ability to turn the INS inazimuth, at least at two points along the flight path. Duringthe initial part of the flight, the equivalent north gyro is beingautomatically calibrated, and then after the first 90 deg turn itbecomes the equivalent east gyro. At this point gyrocom-passing commences, which brings down the azimuth as well asthe other navigation errors. A second turn helps in completingthis process. INS turns can be replaced by several positionupdates. A suboptimal filter can be used to augment the twosystems. It yields very good results; the errors are very close tothose generated when the optimal filter is used and the filter is
quite insensitive to parameter changes. A reference headingimproves the performance of the system but is not necessary.A turn of the INS alone can replace the turning of the wholevehicle since gyro calibration and gyrocompassing occur thisway too. Better position results, though, occur when thewhole vehicle turns since the turning of the velocity vectorenables the estimation of the Doppler horizontal errorcomponents.
AcknowledgmentsThis research was supported by TAMAM—Precision
Instrument Industries, Division of Israel Aircraft Industries,Ltd.
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2Kraemer, J. W., Roessler, N. J., and Brandin, D. M., "In-FlightAlignment/Calibration Techniques for Unaided Tactical Guidance,"Proceedings of the National Aerospace and Electronics Conference,Dayton, Ohio, 1978, pp. 705-711.
3Gelb, A., (ed.), Applied Optimal Estimation, MIT Press, Cam-bridge, Mass., 1974.
4Nash, R. A., Jr., D'Appolito, J. A., and Roy, K, J., "ErrorAnalysis of Hybrid Aircraft Inertial Navigation Systems," AIAAPaper 72-848, AIAA Guidance and Control Conference, Aug. 1972.
5Lochrie, W. D., "The Evaluation of Autonomous NavigationSystems for Cruise Vehicles," AIAA Paper 73-874, Aug. 1973.
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11 Benson, D. O., Jr., "A Comparison of Two Approaches to Pure-Inertial and Doppler-Inertial Error Analysis," IEEE Transactions onAerospace and Electronic Systems, Vol. AES-11, July 1975, pp. 447-455.
12Bar-Itzhack, I. Y., "INS Position-Error and Velocity-ErrorModels," Technical Report No. TAE 396, Technion, Dept. ofAeronautical Engineering, Haifa, Israel, Feb. 1980.
13Weinreb, A. and Bar-Itzhack, I. Y., "The Psi-Angle ErrorEquation in Strapdown Inertial Navigation Systems, " IEEE Trans-actions on Aerospace, and Electronic Systems, Vol. AES-14, May1978, pp. 539-542.
14Leondes, C. T., (ed.), Guidance and Control of AerospaceVehicles, McGraw-Hill, New York, 1963, Chap. 2.
15Bar-Itzhack, I. Y. and Porat, B., "Azimuth ObservabilityEnhancement During Inertial Navigation System In-Flight Align-ment," Journal of Guidance and Control, Vol. 3, July-Aug. 1980, pp.337-344.
16Bar-Itzhack, I. Y. and Malove, E. F., "Accurate INS TransferAlignment Using a Monitor Gyro and External NavigationMeasurements," IEEE Transactions on Aerospace and ElectronicSystems, Vol. AES-16, Jan. 1980^ pp. 53-65.
17Bryson, A. E., Jr. and Ho, Y. C., Applied Optimal Control,John Wiley, New York, 1975, pp. 369, 370.
18Gelb, A. and Sutherland, A. A., Jr., "Software Advances inAided Inertial Navigation Systems," Navigation: Journal of theInstitute of Navigation, Vol. 17, No. 4, Winter 1970-71, pp. 358-369.
19D'Appolito, J. A., "The Evaluation of Kalman Filter Designs forMultisensor Integrated Navigation Systems," TASC TR-183-1, TheAnalytic Sciences Corp., Reading, Mass., Jan. 1971.
20Medan, Y. and Bar-Itzhack, I. Y., "Error and SensitivityAnalysis Scheme of a New Data Compression Technique inEstimation," Journal of Guidance and Control, Vol. 4, Sept.-Oct.1981, pp. 510-517.
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