ICAS 2002 CONGRESS

ROBUST INTEGRATED INS/ ACCOUNTING FAULTS AT THE MEASUREMENT CHANNELS

Ch. Hajiyev, R.Saltoglu

(Istanbul Technical University)

Keywords: Integrated , INS/, Robust Kalman Fillter, Error Model,

Abstract was a milestone, and we were witnesses to these improvements in the near past. A great amount of In this study, the integrated navigation system, study has already been made about this issue. consisting of and INS , is Many more seem to be observed in the future. As presented. INS and the radio altimeter have many of these studies were examined, and some different benefits and drawbacks. The reason for useful information was reached. integrating these two navigators is mainly to Integrated navigation systems combine the combine the best features, and eliminate the best features of both autonomous and stand-alone shortcomings, briefly described above. systems and are not only capable of good short- The integration is achieved by using an term performance in the autonomous or stand- indirect Kalman filter. Hereby, the error models alone mode of operation, but also provide of the navigators are used by the Kalman filter to exceptional performance over extended periods estimate vertical channel parameters of the of time when in the aided mode. Integration thus navigation system. In the open loop system, INS brings increased performance, improved is the main source of information, and radio reliability and system integrity, and of course altimeter provides discrete aiding data to support increased complexity and cost [1,2]. Moreover, the estimations. outputs of an integrated navigation system are At the next step of the study, in case of digital, thus they are capable of being used by abnormal measurements, the performance of the other resources of being transmitted without loss integrated system is examined. The optimal or distortion. Kalman filter reacts with abnormal estimates to In the paper [3] integrated navigation this situation as expected. To recover such a system issue has been discussed. In the paper it is possible malfunctioning, the Robust Kalman stated that, with the demand from the aviation Filter (RKF) algorithm is suggested. industry, Instrumental Landing Systems (ILS) tend to be improved. This could not only be 1 Introduction through replacement with the Microwave Landing System (MLS), but also by integrating The new century’s improvements in computer Global Positioning System to the ILS that is technology, and increased data processing rates practically in service. brought the ability to improve the navigation Due to the paper, the majority of current systems of air vehicles in precision, correctness precision landing research has exploited stand- and reliability.The integrated navigation systems alone GPS receiver techniques. This paper, as an concept with the application of the Kalman filter improvement, exploits the possibilities of using

684.1 Ch.Hajiyev & R.Saltoglu

and Extended Kalman Filter (EKF) that group named ‘High Precision Navigation’ integrates an Inertial Navigation System (INS), showed some experimental effort to prove the GPS, Barometric Altimeter, and Radar Altimeter improvement of navigation parameters in an for precision approaches. As a result, it is integrated navigation system. The main tool was seen that Federal Aviation Authority (FAA) the Kalman filter as usual, but this time the requirements for Category I and II approaches attitude of the vehicle was monitored as shown in could be met through this new approach. their paper dated 1994 [5]. The work in the paper is conducted through a NASA was also interested in the application computer simulation. The simulation program is of Kalman filter in the navigation systems. The basically developed on Kalman filter algorithms. need for the precision altitude determination at The plotted outputs are generated by the low altitude flight phases was the main issue. On commercial software package MATLAB. In this a multi-sensor navigation suite, again using manner, regarding the subject and the tools, this Kalman filter as the main tool for integrating paper is very close to the issue discussed in this navigation data from different origins, radar study(3) . altimeter and INS data was used for selecting the In the paper [4] development of a Kalman most similar digital map profile in obtaining filter for optimal combination of GPS, INS and horizontal position. Radar Altimeter data is presented. Due to the A close subject was discussed by a working paper, being two independent navigation systems, group of NASA and U.S. Army in 1993. The GPS and INS have their own shortcomings when improvement of a terrain-referenced guidance used in a stand-alone mode. The ever-growing system with the implementation of radar drift in position accuracy of the INS, and the altimeter into the traditional navigation system by possible unavailability of the GPS signals are the help of the Kalman filter was exploited. discussed. The author suggests that, these Starting from mathematical models, this group shortcomings would be eliminated, and each was able to accomplish some flight tests and system’s best performances combined through experiments on a Blackhawk Helicopter, as a the Kalman filter. navigation test bed [6]. The benefits of integrating GPS with a The integrated systems designed in the past strapdown INS are significant. However, altitude studies do not have robust character towards accuracy can further be improved by integrating abnormal measurements. Generally, more than the GPS, baro-inertial loop aided strapdown INS, 10% of the radio measurements, and 5% of the and radar altimeter data. An error model of the whole navigation measurements are expected to strapdown INS plays an important role in the be abnormal in the navigation systems [7]. In development of a Kalman filter for optimal some measurement periods, the abnormal rate of combination of navigation data provided by GPS, the radio measurements might exceed %10. This strapdown INS and radar altimeter. Integrating fact certainly decreases the efficiency of most of the error models of each system with the use of the standard statistical processing methods.Thus, the Kalman filter simulates this. The simulation these abnormal measurements resulting from results show an undeniable improvement in the some possible failures at the measurement demanded properties. The approach, tools and the channels significantly decrease the effectiveness data are identical to the ones in this study and the of the integrated navigation systems. In this results are somehow in the same manner [4]. study, to overcome such a problem, design of an Another work on the integration of INS and integrated system, which is robust to the GPS was done in [5] by a group of scientists at abnormal measurements, is intended as an the Technical University of Darmstadt. The improvement.

684 .2 ROBUST INTEGRATED INS/RADAR ALTIMETER ACCOUNTING FAULTS AT THE MEASUREMENT CHANNELS

2.1 INS Error Model, The Altimeter ()kg −∆ 1 : Measurement error of the gravitational acceleration Having numerous output parameters out of ∆ ()kU −1 : White Gauss noise with different channels, the INS has a complex error az model. This model is nevertheless useless in this zero mean ()− study, because the issue covers the INS altimeter. ∆g kU 1 : White Gauss noise with For this reason, only the vertical channel of the zero mean INS is considered. α β , g : The terms for the Although the subject is about the INS altimeter, correlation period not only the vertical position (altitude), but also ∆t : Discrete time the other vertical channel parameters will be += HRR (H - the flight altitude, R - the discussed, and included in the calculations. The I 0 0 INS vertical channel parameters are, radius of earth)

1. Vertical position (altitude). H I

2. Vertical speed. Wz 3.2 Error Model, The Radio Altimeter 3. Vertical acceleration. a z The error model of the radio altimeter, which 4. Gravitational acceleration. g will be used in our calculations, is as follows(14). The system design in the following sections will be in discrete form, so the Kalman filter. The ()()β ()kHtkHkH 11 +−∆∆−−∆=∆ linear and discrete error model of the INS RRR (5)

∆ ()ktU −∆+ 1 altimeter is given as follows [8] H R () ( ) (−∆∆+−∆=∆ ) zII kWtkHkH 11 (1) In the error model expressions, the following are the explanations for the terms. 2 ∆tg ∆ () () ( )+−∆=∆ ()+−∆ R kH : Measurement error of the zz kWkW 1 I kH 1 RI (2) radio altitude () ()−∆∆+−∆∆+ ∆ ()kU −1 : White Gauss noise with z kgtkat 11 H R zero mean ()()α ()+−∆∆−−∆=∆ β zzz katkaka 11 : The term for the ()−∆+ (3) correlation period ∆a ktU 1 z ∆t : Discrete time () ( )β ( )+−∆∆−−∆=∆ g kgtkgkg 11 (4) ()−∆+ 4 Radio-INS Integration ∆g ktU 1 The core task of this study is to combine two In the error model expressions, the following are different navigation sources with the use of the explanations for the terms. Kalman filter. In fact, the main task for any kind ()−∆ z kW 1 : Measurement error of the of navigation study is to fight with the vertical speed disadvantages of a navigation equipment, thus to ∆ () increase correctness and reliability. By I kH : Measurement error of the altitude integrating Radio and INS altimeters, the ()−∆ objective is to benefit from the advantages of z ka 1 : Measurement error of the vertical acceleration

684 .3 Ch.Hajiyev & R.Saltoglu

both of the systems, while eliminating the 1. INS altimeter shortcomings. 2. Radio altimeter There are two possible Kalman filter designs for 3. Data (central) processing unit integrating such systems. 4. Control block (Kalman filter) 1. Total state space Kalman filter (direct 5. Display ( users) method). 2. Error state space Kalman filter (indirect method). In the direct method, all of the states of the system are used by the Kalman filter, where the INS accelerometer and the aiding navigator signals are the sources. But in the indirect method, which is basically chosen in this study, Kalman parameters are the errors. While the INS is the main source for determining vertical navigation information, with the discrete aiding Figure 1. Integrated Radio-INS altimeter signals from the Radio altimeter, Kalman filter estimates the instantaneous errors.Through the block diagram design, the outcoming dynamical characteristic of the INS is proposed to be carried to the complete Here, the INS altimeter is not a dedicated stand- system, and the possibility of a total system alone instrument, but the altitude or the vertical breakdown in case of a Kalman filter failure is position channel of the Inertial Reference Unit eliminated. (IRU) of a certain aircraft. The switch sign on the By integrating INS and Radio altimeter block diagram refers to the discrete character of through an indirect feed forward Kalman filter, the data provided by the radio altimeter for the 1. The good characteristics of the INS with high integrated navigation system. However, Kalman bandwidth and frequencies are maintained. filter could be applied in both continuous and 2. In certain operation periods, acceptable discrete systems. The main source of altitude data accuracy is attained. is INS, and the radio altimeter acts as a correcting 3. A dynamic system, which is easily adapted to element supporting the Kalman filter for the computer environment, is developed. estimations. By the use of the data processing The two navigators of the integrated Radio-INS unit, radio altimeter output is transformed to the altimeter provide different types of altitude data. main coordinate system of the aircraft navigation The radio altimeter measures the real altitude and control. from the ground level, but the INS measures the The radio-INS integration is feasible for the cases relative altitude from an initial alignment when the aircraft route is over landscapes such as location. In a system where these two navigators seas, oceans, lakes and deserts where the altitude are combined, this difference could easily be above ground level does not have fluctuating eliminated. This elimination is done in the data character. (Not over mountains, valleys, etc.). processing unit of the integrator. When this condition is set, the real altitude The block diagram of an integrated Radio-INS value at the Kalman filter input will be altimeter is shown in the Figure 1.The integrated compensated, and the input signal will be Radio-INS altimeter contains the following designated as the difference between the altitude components: measurements of the two sources.

684 .4 ROBUST INTEGRATED INS/RADAR ALTIMETER ACCOUNTING FAULTS AT THE MEASUREMENT CHANNELS

−= RI tHtHtr )()()( (6) One of the most important improvements of the ∆+−∆+= tHtHtHtH ))()(())()(( (7) integrated radio altimeter is that, a dynamic error RgIg of measured altitude is not generated. As this is ∆−∆= RI tHtH )()( (8) an open loop system for the measured altitude, the filter will not limit the operation rate of the

In the expressions (6), (7), and (8), I (tH ) and navigators. For this reason, the radio-INS altimeter will have outstanding dynamic R tH )( are the altitude measurements of INS and radio altimeter, ∆ (tH ) and ∆ (tH ) are the character like a stand-alone INS. Another I R important feature is that, an in-flight failure at the measurement errors of the two sources Kalman filter will not result in a whole system respectively. tH )( is the real altitude value. g breakdown. In such a case, when the Kalman In this block diagram of the integrated system, block at the integrated system fails, the INS INS is the main error source. In the previous measurements could be used without corrections sections, dead-reckoning character of the INS for a certain period of operation. was mentioned. This causes a severe error of measurement, increasing by time. 5 Optimal Kalman Filter for the Integrated The mathematical models of the INS (1)-(4) System and the radio altimeter (5) given in this study are first or second order stochastic Markov processes The state space model of the integrated radio-INS [8]. The model parameters are estimated by the altimeter system is expressed with the following use of the Kalman filter. The filter will give the equations. optimum (estimated) value of the INS error ∆ ˆ ()()()()()φ −−+−−= I tH )( , in the condition where the standard error kUkkGkXkkkX 11,11, (11) is minimized. Finally, the estimated value of ∆ ˆ () () ()+= ()kVkXkHkZ (12) error I tH )( is subtracted from the altitude measurement of the INS. Equation 11 is the combination of error models of INS and the radio altimeter, (1)-(4) and (5), ˆ ∆−= ˆ )( IIInt tHtHH )( (9) which were given in the previous sections. This expression contains all of the error parameters The calculated altitude value is the output of the that act on the integrated system. The definitions integrated system, and it would be used for of the terms at the state space model of the navigation and control purposes. system are, As the INS error increases with time in a cumulative manner, the integration diagram is φ()kk −1, : System transfer matrix valid and effective for short periods of operation. ()kX : State vector (For example, 1 hour period). ()kkG −1, : Noise transition matrix In such periods when the linear operation regime of INS is provided, the output signal will be as kV )( : Noise of the system follows. The elements of the vector kX )( , (11) are the measurement error components, where the open ∆+= Re IlI tHtHtH )()()( (10) form of the vector is depicted in the following expression. where, Rel (tH ) is the relative altitude.

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T ()[] () () () ()∆∆∆∆∆= () RzzI kHkgkakWkHkX R (k) : Measurement noise (13) covariance Q (k-1) : Process noise covariance These terms in (13) are the main four error parameters of the INS altimeter, which are the Equation 12 is the measurement equation of the ∆ () error of finding the altitude I kH , vertical system. The measurement matrix ()kH in this ∆ () ∆ () speed z kW , acceleration z ka , and expression is highly important, as it gives a gravitational acceleration ∆ ()kg ; and the altitude general idea of the whole design of the integrated ∆ () system.In this design, the measurement matrix is, finding error of the radio altimeter R kH . The Kalman filter designed for the integrated ()kH −= 10001 navigation system of this study is optimal,linear and discrete. The following are the final As easily seen by combining this matrix with expressions of the Kalman equations to be used equation (13), the system integrates two in the calculations. independent navigation sources in means of subtracting the radio altimeter error from the INS ˆ ()ˆ ( )()()∆+−= / 1/ kkKkkXkkX (14) altimeter error. However, this operation is carried () () ()ˆ (kkXkHkZk −−=∆ 1/ ) (15) out through the Kalman filter, and the divergent − TT 1 error character of the INS is compensated. This is () ( ) ()[] ()(1/1/ ) ()+−−= ()kRkHkkPkHkHkkPkK one of the ideas that should be underlined in this (16) study. ˆ ()()()φ 1,1/ ˆ kkXkkkkX −−−=− 1/1 (17) 6 Robust Kalman Filter Design ( )[] () () ()kkPkHkKIkkP −−= 1// (18) In case of a failure at the measurement channel of a system, a robust Kalman filter is developed to ( )()()()φ φ T +−−−−=− kkkkPkkkkP 1,1/11,1/ consider, and take action against these ()()()T kkGkQkkG −−−+ 1,11, measurement errors.Let’s assume the observation (19) (measurement)model to be like the following [9].

The definitions of the terms at the Kalman += γ kvkkxkHkz )()()()()( (20) equations are presented below. Here, the parametric variable γ(k), is an ˆ () / kkX : The estimation of state independent and random variable at each step. vector In case of normal operation, parametric ˆ ()− kkX 1/ : Extrapolation value vactor variable is equal to 1 and in case of abnormal ()kK : The gain matrix of the operation, parametric variable is equal to σ. Kalman filter ∆() The variance of the measurement error is highly k : Innovation sequence effective on the error formation, rather than the ()kkP −1/ : The covariance matrices of normal operation. Thus, when γ(k)=σ>1 the extrapolation errors following suboptimal filter algorithm is valid : ()/kkP : The covariance matrices of ˆ )/( ˆ ∆+−= kkKkpkkxkkx )()()/1()1/( (21) estimation errors

684 .6 ROBUST INTEGRATED INS/RADAR ALTIMETER ACCOUNTING FAULTS AT THE MEASUREMENT CHANNELS

~ The other filter parameters are the same as the [])....1/()()(1 kkkPkHkK )( Ω∈∆−− expressions (14)-(19). kP )( =  ~  ...... )...... 1/( kkkP )( Ω∉∆− γ Here, p(1/k), is the posterior probability of (k) (24) variable to be equal to 1, so the normal operation of the measurement channel at the k time step. It 7 Simulation could be seen that, only the matrix with the p(k) The simulation section of this study is mainly parameter in front of the filter gain coefficient is based on the error models of each of the separate missing. When p(1/k)=1, this filter will be navigation sources of the integrated radio-INS exactly same with the optimal Kalman filter, but altimeter. Error models of the INS and the radio when p(1/k)=0 it disregards the new altimeter are combined using Kalman filter, so measurements, acting as an extrapolator. the estimated values of error is calculated. In the At normal operation of the measurement channel, algorithm of the filter design at this study, INS is ~ ∆ k)( normalized innovation sequence a primary; radio altimeter is a secondary source − 2/1 of navigation. The main tool for the simulation is ~ []T ∆+−=∆ kkRHkkPkHk )()()1/()()( the MATLAB software. will provide normal distribution N(0,1). The whole algorithm is exactly the same with the one for the optimal conditions, but errors are For the measurements z(k), the robust Kalman implemented into 101th, 201th, 301th, and 401th filter is designed which basically depends on the ~ iterations. These abrupt errors, which represent ∆ k)( to fit in the tolerance region (confidence the failure condition at the measurement channel, interval) Ω where the following relations are are formed by increasing the standard deviation accepted [9]: of the normal error by 100 times. In this study, integrated navigation system for ∆~ ∈Ω when k)( then p(1/k)=1 altitude determination of an aircraft is simulated. ~ The following are the data, and the initial when ∆ k)( ∉Ω then p(1/k)=0. (22) conditions for the system. In the expression (22), it is seen that, when the ~ −6422 −8422 ∆ σ ∆ [ sm ]=10 ;σ ∆ [ sm ]=10 ;τ []s = 10 normalized innovation sequence k)( , is settled az g s in the confidence interval, p(1/k), will be equal to − 1 β[]s 1 == 1.0 ;τ []s =1000 ; ()= 1000mkR 2 1. In that case, the filter will be exactly the same τ az s as the optimal Kalman filter (14)-(19), designed − 1 γ α[]s 1 == 001.0 ;τ []s = 200 ; [ smg 2 ]= 89.9 for the normal operation, when ( (k)=1). τ g a At the abnormal measurement conditions, z − 1 when there becomes a failure at the measurement β []s 1 == 005.0 ; = 1kmH ; g τ channel, p(1/k) will be equal to 0. This time, the g Kalman filter will disregard the new =+=+= []=∆ 0 637216371 kmHRR ; st 1.0 . measurements and consider extrapolation values The initial conditions for the measurement for the output vector. The resulting term of the system include the following errors. filter is as follows. ()=∆ ()=∆ I 230 mH ; z /002.00 smW ; ˆ )/( ˆ kkxkkx −= )1/( (23) () =∆ 2 ()=∆ 2 z /001.00 sma ; /0001.00 smg ; ()=∆ Then, the correlation matrix of the error is R 20 mH . calculated by the help of the following equality.

684 .7 Ch.Hajiyev & R.Saltoglu

7.1 Optimal Kalman Filter Simulation

In the simulation of the integrated system equipped with the optimal Kalman filter (OKF), the results were quite satisfactory. In all altimeter parameters, Kalman estimations were able to catch the model values, and this tendency was kept until the last iteration. As the estimations are correct, the integrated system error is kept around zero through the simulation. That is one of the goals of this study, as the system error has been stabilized, thus the drifting character of the INS error has been eliminated. The resulting data are given in table 1 and figure 2-3. In figure 2, INS altimeter’s drifting Figure 3. Altitude error (INS) error (solid line) character could easily be seen. (OKF was used) Meanwhile, the integrated systems error of altitude (dotted line) keeps values around zero, Table 1. INS error versus integrated system error which is a success of the filter design. Figure 3 is (Altitude) the combined graph of model error (solid line), Iterations INS Integrated estimated error (dotted line), their differences, abs.error Altimeter and error variance of estimation for altitude. (m) abs.error (m) 50 23.8841 11.1194 100 26.7612 -1.6229 150 31.9571 4.3391 200 39.9773 7.3407 250 51.5466 -5.0445 300 67.6660 1.5723 350 89.6901 -10.7381 400 119.4324 9.7124 450 159.3070 0.1315 500 212.5156 -2.1919

7.2 Failure Conditions Simulation

To simulate a possible failure at the measurement channel of the system, abrupt errors with random magnitude were implemented into the algorithm. At the steps of failures, the estimations diverted from the model enormously. In all simulation Figure 2. INS error versus integrated runs, these peaks were observed higher or lower, system error (OKF was used) but they were anyway unacceptable. The high peaks of estimated error (dotted line) at the steps, where abrupt error was introduced to the measurement channel, are seen in figure 4.

684 .8 ROBUST INTEGRATED INS/RADAR ALTIMETER ACCOUNTING FAULTS AT THE MEASUREMENT CHANNELS

acceptable estimations.Dislike the previous simulation in the section 7.2, estimations did not divert from the model value, as observed in Figure 6. In this figure, again the solid and dotted lines stand for INS and integrated system errors respectively.

Figure 4. INS error versus integrated system error (Failure conditions)

Figure 6. INS error versus integrated system error (RKF was used)

Figure 5. Altitude error (INS) (Failure condition)

In figure 5,the combined graph of model(solid line) and estimated (dotted line) errors, their differences, and error variance; the effect of Figure 7. Altitude error (INS) (RKF was measurement failures at the system is clearly used) seen. Figure 7 is the combined graph of altitude error, 7.3 Robust Kalman Filter Simulation model (solid line) and estimated (dotted line), in The results of the RKF simulation were almost RKF simulation. The success of this robust perfect, as the system was successful to get rid of algorithm could again be concluded on this the measurement failures, and it produced graph.

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Table 2. Integrated altimeter errors in case of a 2. The open loop mechanization of the failure at the measurement channel (Altitude) system tolerates failures at the Kalman filter computations, without causing a Iterations Altitude abs. Altitude abs. total system breakdown. err.(m),(OKF) err.(m),(RKF) 3. Ground or space based navigation aids are not used, and this brings autonomy 100 31.5445 31.5445 to the entire system. 101 789.7662 31.9989 4. The Kalman filter compensates the 200 29.4787 33.3060 instant lack of radio altimeter 201 -280.5042 33.4452 information, as this data is discretely 300 48.4811 52.8860 supplied to the system. 301 -255.2256 53.0908 5. Abnormal measurements, in cases of 400 111.9313 115.8765 measurement channel failures, are 401 625.2949 116.5206 detected and corrected by the system, 500 205.7108 202.9625 without affecting the good estimation behavior. The simulation results, in case of a failure at the measurement channel, are given in table 2. 9 References [1] Kayton, M. and Fried, W. R., Avionics Navigation 8 Conclusion Systems, 2nd Edition, John Willey & Sons, Inc., New York, 1997. The integrated navigation system, consisting of [2] Biezad, D. J., Integrated Navigation and Guidance Systems, Ed. Przemieniecki J. S., American Institute of radio and INS altimeters, is presented. The Aeronautics & Astronautics,Inc.,Reston,Virginia, 1999 integration is achieved by using an indirect [3] Gray, R. A. and Maybeck, P. S., Integrated Kalman filter. Hereby, the error models of the GPS/INS/BARO and radar altimeter system for aircraft navigators are used by the Kalman filter to precision approach landings, Proceedings of the IEEE estimate vertical channel parameters of the 1995 National Aerospace and Electronics Conference, Part 1, Dayton, KY, USA, May 22-26,1995. navigation system. [4] Deergha, R. K., Integration of GPS and baro-inertial To recover such a possible malfunctioning, loop aided strapdown INS and radar altimeter, IETE the RKF algorithm is suggested. In the robust Journal of Research, v 43 n 5 (1997) 383-390. algorithm, when the abnormal measurement is [5] Sohne, W., Heinze, O. and Groten, E., Integrated detected, it is disregarded, and instead of INS/GPS system for high precision navigation applications, Proceedings of the 1994 IEEE Position estimated value, extrapolation value is used. Location and Navigation Symposium, Las Vegas, NV, As a conclusion, this study suggests the USA, April 11-15, 1994. application of an integrated Radio-INS [6] Zelenka, R. E., Design and analysis of a Kalman filter altimeter, where for terrain-referenced positioning and guidance, 1. The integrated system combines the best Journal of Aircraft, v 31, n 2 (1994) 339-344. [7] Brandin V. N., et al., Experimental Ballistics of the features of two different navigation Space Apparatus, Mashikostroyeniye, Moscow (in sources, INS and Radio altimeter.The Russian),1984. estimated altitude is more correct than [8] Zhukovskiy, A. P., and Rastorguev, V. V., Complex the radio altimeter outputs. While the Radio navigation and control systems of aircraft, MAI, divergent error characteristic of the Moscow (in Russian), 1998. [9] Gadzhiev (Hacıyev), Ch. M., Simplified filtering INS is restrained, the system has algorithm in the presence of faults in the measuring dynamical characteristics as good as channel, Measurement Techniques, v 32 n 6, 1989, pp. the INS. 505-507.

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