Fusion of Onboard Sensors for Better Navigation

This paper presents simulation results of navigation sensors such as integrated navigation system (INS), global navigation satellite system (GNSS) and TACAN sensors onboard an aircraft to find the navigation solutions. Mathematical models for INS, GNSS (GPS) satellite trajectories, GPS receiver and TACAN characteristics are simulated in Matlab. The INS simulation generates the output for position, velocity and attitude based on aerosond dynamic model. The GPS constellation is generated based on the YUMA almanac data. The GPS dilution of precession (DOP) parameters are calculated and the best combination of four satellites (minimum PDOP) is used for calculating the user position and velocity. The INS, GNSS, and TACAN solutions are integrated through loosely coupled extended Kalman filter for calculating the optimum navigation solution. The work is starting stone for providing aircraft based augmentation system for required navigation performance in terms of availability, accuracy, continuity and integrity.


Keywords:  Extended Kalman filtermmultisensor data fusionintegrated navigation system global navigation satellite systemGPS ConstellationGPS Receiver 

Sensors onboard an aircraft are integrated navigation system (INS), global navigation satellite system (GNSS), TACAN, radio altimeter, surface radar tracker (ground proximity warning system), link 16, air data computer and forward looking sensors (LiDAR, RADAR, Visible/IR) which assists pilot for navigation. However, it is extremely difficult for the pilot to rely on any one instrument and the decision making depends on pilot’s proficiency. This study and simulation amalgamates outputs from these sensors and processes it through a navigation filter for providing best solutions for RNP in terms of availability, accuracy, integrity and continuity. The RNP parameters1,2 and area navigation3 are defined in literature. Many papers and books had discussed the integration of navigation sensors4, Kalman filter (KF) for fusion of INS and GPS5,6, fusion of different sensors5,7, loosely coupled integration of GPS and INS using EKF8, loose integration of INS, GPS and camera9, INS and GPS on ultra tightly system10-12. However, in this paper various navigation sensors are simulated and the navigation data fusion problem is deeply examined with the aim of developing an EKF suitable for exploiting existing navigation sensors in various manned and unmanned aircraft for both civil and military applications.


INS provides standalone solution. Long duration solution from INS is achieved by using very high quality accelerometers and gyros which increase the cost of navigation system. GNSS navigation solution depends on the availability of GNSS signals which in turn is dependent upon numerous factors such as location, antenna orientation and dilution of precision (DOP). TACAN is used by military for range and bearing with respect to TACAN ground station.


The scope of multi sensor data fusion is shown in Fig 1. The EKF is selected as a prime data fusion method to provide navigation solution.


Figure 1. Multi sensor data fusion.

The sensor models such as INS, GNSS constellation, GNSS receiver and TACAN are required. The GNSS receiver works in the earth centre earth fix (ECEF) coordinate system. The other sensors provide solutions in a local coordinate frame. The INS is simulated by simulating the accelerometers, gyros, bias, scale factor and white noise random drift. The generalised mathematical model which are used in simulation for gyros, accelerometer and derive velocity, acceleration, position (errors) are described in Appendix I. The GNSS receiver simulation requires input from GNSS constellation as GNSS receiver characteristic is dependent on satellites position, inclination and distance from the receiver. GPS almanac data is used to calculate approximate position of satellites in the orbit. The almanac data is downloaded from the US coast guard navigation centre13. Ephemeries error model is used from reference14. Satellite’s position and velocity are computed from equations defined in Appendix II. GNSS Receiver’s position and velocity calculations are also described in Appendix II.


The mathematical model for TACAN and VOR/DME is same. Only, the error characteristics for TACAN and VOR/DME are different. TACAN range and bearing models are defined in Appendix III. Sigma error calculation is defind in Appendix IV.


Figure 2. GNSS, INS and TACAN EKF fusion block diagram.

The integration of navigation sensors is achieved through navigation filters such as KF, EKF and UKF. The EKF model is based on measurement compensation. The measurement matrix z is a difference between INS and GNSS measurements. Matrix z is used to correct the INS solutions.

The EKF navigation solution requires system model and measurement model15. EKF is used to describe the system which has non linear states. In loose integration there is no interaction between the INS, GNSS and TACAN states therefore the system, transition and system noise covariance matrices may be partitioned as described by Eqn (1). to Eqn (17)16



x system [ Position Volocity Attitude BiasAcc BiasGyro TACANRANGE TACANAzimuth ] 17×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGZbGaamyEaiaadohacaWG0bGaamyzaiaad2gaaeqaaOWa amWaaqaabeqaaiaadcfacaWGVbGaam4CaiaadMgacaWG0bGaamyAai aad+gacaWGUbaabaGaamOvaiaad+gacaWGSbGaam4BaiaadogacaWG PbGaamiDaiaadMhaaeaacaWGbbGaamiDaiaadshacaWGPbGaamiDai aadwhacaWGKbGaamyzaaqaaiaadkeacaWGPbGaamyyaiaadohacaaM c8UaamyqaiaadogacaWGJbaabaGaamOqaiaadMgacaWGHbGaam4Cai aaygW7caaMc8UaaGPaVlaadEeacaWG5bGaamOCaiaad+gaaeaacaWG ubGaamyqaiaadoeacaWGbbGaamOtaiaaykW7caWGsbGaamyqaiaad6 eacaWGhbGaamyraaqaaiaadsfacaWGbbGaam4qaiaadgeacaWGobGa aGPaVlaadgeacaWG6bGaamyAaiaad2gacaWG1bGaamiDaiaadIgaaa Gaay5waiaaw2faamaaBaaaleaacaaIXaGaaG4naiabgEna0kaaigda aeqaaaaa@82EB@           (1)


where Position= [ x y z ] 3×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaiaad+ gacaWGZbGaamyAaiaadshacaWGPbGaam4Baiaad6gacqGH9aqpdaWa daabaeqabaGaamiEaaqaaiaadMhaaeaacaWG6baaaiaawUfacaGLDb aadaWgaaWcbaGaaG4maiabgEna0kaaigdaaeqaaaaa@4708@      (2)

Velocity= [ V N V E V D ] 3×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaiaadw gacaWGSbGaam4BaiaadogacaWGPbGaamiDaiaadMhacqGH9aqpdaWa daabaeqabaGaamOvamaaBaaaleaacaWGobaabeaaaOqaaiaadAfada WgaaWcbaGaamyraaqabaaakeaacaWGwbWaaSbaaSqaaiaadseaaeqa aaaakiaawUfacaGLDbaadaWgaaWcbaGaaG4maiabgEna0kaaigdaae qaaaaa@49A1@      (3)

Attitude= [ ϕ θ ψ ] 3×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaads hacaWG0bGaamyAaiaadshacaWG1bGaamizaiaadwgacqGH9aqpdaWa daabaeqabaGaeqy1dygabaGaeqiUdehabaGaeqiYdKhaaiaawUfaca GLDbaadaWgaaWcbaGaaG4maiabgEna0kaaigdaaeqaaaaa@4949@       (4)

BiasAcc= [ α x α y α z ] 3×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaadM gacaWGHbGaam4CaiaaykW7caaMc8UaamyqaiaadogacaWGJbGaeyyp a0ZaamWaaqaabeqaaiabeg7aHnaaBaaaleaacaWG4baabeaaaOqaai abeg7aHnaaBaaaleaacaWG5baabeaaaOqaaiabeg7aHnaaBaaaleaa caWG6baabeaaaaGccaGLBbGaayzxaaWaaSbaaSqaaiaaiodacqGHxd aTcaaIXaaabeaaaaa@4E49@      (5)

BiasGyro= [ g x g x g x ] 3×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaadM gacaWGHbGaam4CaiaaykW7caaMc8Uaam4raiaadMhacaWGYbGaam4B aiabg2da9maadmaaeaqabeaacaWGNbWaaSbaaSqaaiaadIhaaeqaaa GcbaGaam4zamaaBaaaleaacaWG4baabeaaaOqaaiaadEgadaWgaaWc baGaamiEaaqabaaaaOGaay5waiaaw2faamaaBaaaleaacaaIZaGaey 41aqRaaGymaaqabaaaaa@4D4C@      (6)

TACANRANGE= [ ρ ] 1×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiaadg eacaWGdbGaamyqaiaad6eacaaMc8UaamOuaiaadgeacaWGobGaam4r aiaadweacqGH9aqpdaWadaqaaiabeg8aYbGaay5waiaaw2faamaaBa aaleaacaaIXaGaey41aqRaaGymaaqabaaaaa@47D8@      (7)

TACANAzimuth= [ θ ] 1×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiaadg eacaWGdbGaamyqaiaad6eacaaMc8UaamyqaiaadQhacaWGPbGaamyB aiaadwhacaWG0bGaamiAaiabg2da9maadmaabaGaeqiUdehacaGLBb GaayzxaaWaaSbaaSqaaiaaigdacqGHxdaTcaaIXaaabeaaaaa@4A4D@      (8)

The linearised dynamic model of continuous state space model for dynamics of INS, GNSS and TACAN. is expressed by Eqn (9).

where

F system n   = F 11 I 3 Z 3 Z 3 Z 3 Z 32 F 11 F 22 F 23 Z 3 Z 3 Z 32 Z 11 Z 3 F 33 Z 3 F 35 Z 32 Z 11 Z 3 Z 3 F 44 Z 3 Z 32 Z 11 Z 3 Z 3 Z 3 F 55 Z 32 Z 23 Z 23 Z 23 Z 23 Z 23 F 66      (9)

Z 3 =[ 000 000 000 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwamaaBa aaleaacaaIZaaabeaakiabg2da9maadmaaeaqabeaacaaIWaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaIWaaabaGaaGimaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG imaaqaaiaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicda caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicdaaaGaay5waiaaw2 faaaaa@6F7C@      (10)

l 3 =[ 100 010 001 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiBamaaBa aaleaacaaIZaaabeaakiabg2da9maadmaaeaqabeaacaaIXaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaIWaaabaGaaGimaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGymaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG imaaqaaiaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicda caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaigdaaaGaay5waiaaw2 faaaaa@6F91@      (11)

Z 23 =[ 000 000 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwamaaBa aaleaacaaIYaGaaG4maaqabaGccqGH9aqpdaWadaabaeqabaGaaGim aiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGimaaqaaiaaicdacaaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaicdaaaGaay5waiaaw2faaaaa@5E9B@      (12)

Z 32 =[ 00 00 00 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwamaaBa aaleaacaaIZaGaaGOmaaqabaGccqGH9aqpdaWadaabaeqabaGaaGim aiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaaqaaiaaicdaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicdaaeaacaaIWaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaIWaaaaiaawUfacaGLDbaaaa a@56E5@      (13)

F 66 =[ 00 00 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaaI2aGaaGOnaaqabaGccqGH9aqpdaWadaabaeqabaGaaGim aiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7aeaaca aIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVdaa caGLBbGaayzxaaaaaa@50C2@      (14)

Details of F11, F12, F22, F33, F35, F44 and F55 are in. 17These are derived from equations described in Appendix I for position, velocity and attitude computation of INS. The system noise covariance matrix Q is defined by Eqn (15).



Q system = z 3 z 3 z 3 z 3 z 3 z 33 I 3 n ra 2 I 3 z 3 z 3 z 3 z 32 z 3 I 3 n rg 2 I 3 z 3 z 3 z 32 z 3 z 3 I 3 n bad 2 I 3 z 3 z 32 z 3 z 3 z 3 I 3 n bgd 2 I 3 z 32 z 23 z 23 z 23 z 23 z 23 Q 66      (15)

where n2ra, n2rg, n2bad and n2bgd and n2bgd are the power spectral densities (PSD) of the accelerometer random noise, gyros random noise, accelerometer bias variation and gyro bias variation respectively defined17. It is assumed that all the three accelerometer and gyros used for INS simulation have same PSD. It is assumed that TACAN range and bearing are linearly dependent. Therefore the derivative of TACAN states are zero. Q66 is defined by Eqn (16).



Q 66 =[ TACA N RANGE Errpr 20 0TACA N RANGE Errpr 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGrb WaaSbaaSqaaiaaiAdacaaI2aaabeaakiabg2da9maadmaaeaqabeaa caWGubGaamyqaiaadoeacaWGbbGaamOtamaaBaaaleaacaWGsbGaam yqaiaad6eacaWGhbGaamyraaqabaGcdaWgaaWcbaWaaSbaaWqaaiaa dweacaWGYbGaamOCaiaadchacaWGYbaabeaaaSqabaGccaaIYaGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaaqaaiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaamivaiaadgeacaWGdbGaamyqaiaad6 eadaWgaaWcbaGaamOuaiaadgeacaWGobGaam4raiaadweaaeqaaOWa aSbaaSqaamaaBaaameaacaWGfbGaamOCaiaadkhacaWGWbGaamOCaa qabaaaleqaaOGaaGOmaaqaaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7aaGaay5waiaaw2faaaqaaaaaaa@CB1F@      (16)

The measurement matrix is defined by Eqn. (17).



H= [ 1000000..0 0100000..0 0010000..0 0001000..0 0000100..0 0000010..0 ] 6×17 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2 da9maadmaaeaqabeaacaaIXaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caGGUaGaaiOlaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWaaabaGaaGimaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGymaiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaiOlaiaac6cacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGimaaqaaiaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaigdacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicdacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicdacaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaac6cacaGGUaGaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicdaaeaacaaIWaGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaIXaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caGGUaGaaiOlaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaIWaaabaGaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGymaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaiaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaiOlaiaac6cacaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGimaaqaaiaaicdacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicdacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicdacaaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaigdacaaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caGGUaGaaiOlaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaIWaaaaiaawUfacaGLDbaadaWgaaWcbaGaaGOnai abgEna0kaaigdacaaI3aaabeaaaaa@EFC2@      (17)

The statistical nature of INS, GNSS and TACAN are different. Therefore, errors are separated by EKF. The estimated INS error is subsequently subtracted from the INS output.

Performance of all sensors is based on input from aircraft dynamic model (ADM)18.

4.1 INS Simulation


Characteristics which are assumed for accelerometers and gyros for the simulation of INS are tabulated in Table 1.

Table 1. INS input characteristic for simulation20.



Sigma errors are tabulated in Table 2 for an update rate of 50 Hz for attitude, velocity and position using ADM as input.INS attitude, velocity and position are computated without vertical channel correction and do not meet the RNP.




Table 2. Velocity, position, attitude sigma error


4.2 GNSS Simulation

The YUMA almanac data14 is downloaded on 06 august 2011 and is used for GPS constellation simulations. The GPS satellites constellation simulation (visibility) is verified at longitude 0.6267° W, latitude 52.0703° N and geodetic height 139 m at 8 hrs, 27 min and 54 s on 7/08/2011 online19. The program is written to track the 32 satellites and can be easily modified to track more satellites in future. The program can be easily modified to simulate the GLONASS orbit.


The intermediate parameters require for satellite position calculation is performed for satellite number 3 (PRN 3) and data is taken from YUMA almanac file14.Table 4. lists the input parameters require for calculation of satellite coordinates. Similar calculation is performed for the remaining satellites.

Table 3. Values used for satellite coordinate calculation20




Table 4. YUMA Alamac data for PRN 03



The output generated from the Matlab program for calculating the satellite position in ECEF (WGS 84) coordinates is listed in Table 5. For details of terms used, ref20.
Figure 3. shows the orbits of all GPS saltellites (31 number). Axes measurements are in meters (ECEF).


Figure 3. GPS Satellites trace for one orbital period.




Figure 4. Attitude solution with filter.




Figure 5. TACAN range solution.




Table 5. GPS satellite position (PRN 03)




Figure 6. TACAN bearing solution.




4.3 GPS Receiver Simulation


Table 6.shows the position and velocity sigma error of the GPS receiver. DOP factor of less than 2.5 is considered for 4 best DOPs.

Table 6.. Position, velocity sigma error




4.4 TACAN Simulation


Typical TACAN range error and azimuth error are assumed as meters and respectively. In this analysis, TACAN ground station location is assumed as 519 m North, 153 m East and 5 m down in NED frame. Simulation result of range and bearing are shown in Figs 5. and Fig 6. as actual TACAN range and bearing index respectively.


4.5 EKF Simulation Results for INS, GNSS and EKF


GNSS position solution is converted into NED frame. Update rate of EKF filter is 1 Hz and INS update rate is 50 Hz. GNSS and TACAN update rate is 1 Hz. INS, GNSS and TACAN simulation parameters are same as used in section 4.1, 4.2, 4.3 and 4.4.


Figure 4. shows the attitude solution obtained from the EKF filter of INS, GNSS and TACAN. In this solution GNSS signal is temporarily unavailable from 4100 s to 4500 s. During this period the solution accumulates the similar errors as of standalone INS. Similar analysis is performed for velocity and position. attitude, velocity and position sigma errors are tabulated in Table 7..The sigma error is better than the standalone INS solutions.

TACAN range and bearing filter solutions are shown in Figs 5. and Fig 6., respectively.


Table 7. Attitude, velocity and position sigma error with filter



The loose integration of INS, GNSS and TACAN is performed using EKF. The data link may be added to the same navigation filter to provide the navigational fix in terms of position and velocity from cooperating platforms. The error budget of the platform can be computed based on sigma error of the cooperating platforms with different sensors accuracies and sigma errors. This will pave the way for development of decision support tool for flight operations.

1.     Gustavsson, P. Development of a Matlab bassed GPS constellation simulation for navigation algorithm developments. University of Technology, Lulea, 2005, Master Thesis.

2.     Kelly, R.J. & Davis, J.M. Requires navigation performance for precision approach and landing with GNSS application. J. Institute Navigation, 1994, 41(1), 1-30.

3.     Eurocontrol.http://www.eurocontrol.int/eatm/gallery/content/public/library/euro_stds/Eng/rnav/RNAV_Standard_Ed_22a_web.pdf [Accessed on 30/08/2012]

4.     Allerton, D. & Jia, H. A review of multisensor fusion. Journal Navigation, 2005, 58, 405-417.

5.     Seraji, H. & Serrano, N. A multisensor decision fusion system for terrain safety assessment. IEEE Trans.  Robatics, 2009, 25(1), 98-108.

6.     Babu, R. & Wang J. Ultra tightly GPS/INS/PL integeration: Kalman filter performance analysis.  GPS Solut.,  2009, 13(1), 75-82.

7.     Wang J. & Liang, K. Multi sensor data fusion based on fault detection and feedback for integerated navigation systems. In the International symposium on integrated information technology application workshops, Beijing, IEEE Computer Society, 21-22 Dec 2008. pp. 232-35.

8.     Lemay, L.; Chu, C.C & Egziabher, G.D. Precise input and output error characterization for loosely integerated INS/GPS/Camers navigation system.  In the proceedings of the national technical meeting; Institute of navigation, San Diego, California, 2011. 2, pp. 880-894.

9.     Lijun, W.; Huichang, Z. & Xiaoniu, Y. The modeling and simulation for GPS/INS integerated navigation system. In the ICMMT 2008 proceedings, IEEE, 2008. Nanjing, China, 21-24 April 2008, pp. 1991-994.

10.   Celestial Observer.http://www.calsky.com/ [Accessed on 06/08/2011].

11.   Edward, L.W.; Clark, J.B. & Bevly, D.M. Implementation details of a deeply integerated GPS/INS software receiver. In Position Location and Navigation Symposium (PLANS), 2010, IEEE/ION, 4-6 May 2010, pp. 1137-146.

12.   Han, L.J. & Wei, M. Adaptive EKF filter based on genetic algorithm for tightly coupled integrated inertial and GPS navigation. In the 2nd International conference on Intelligent computing technology and automation, 2009. Changsha, Hunan, 10-11 Oct 2009, 1, pp. 520-524.

13.   U.S Cost Guard Navigation Centre http://www.navcen.uscg.gov/?pageName=gpsAlmanacs[Accessed on 06/08/2011]

14.   Joerger, M.; Neale, J.; Pervan, B. & Datta, B.S. Measurement error models and fault detection algorithms for multi constelation navigation systems. In Position Location and Navigation Symposium (PLANS), 2010 IEEE/ION, Indian Wells, USA, 4-6 May 2010, pp. 927-946.

15.   Zarchan P. & Howard M. Fundamental of Kalman filtering : A practical approach. Ed 3rd, AIAA, 2005 765p.

16.   Shankar. R. Development of multi-sensor navigation system using on-board sensor resources. Cranfield University, U.K, April 2011. (MSc, Thesis).

17.   Paul, D.G. Principles of GNSS, inertial and multisensor integrated navigation systems. Artech house, London, 2008.

18.   Aerosond UAV.http://www.aerosonde.com/ [Accessed on 15/08/2011].

19.   Celestial Observer. http://www.calsky.com/ [Accessed on 06/08/2011].

20.   Mohinde, S.G.; Lawrence, R.W. & Angus, P.A. Global positioning systems, inertial navigation. Ed  2nd, 2007, A John Wiley & Sons Inc. 517 p.

21.   Titterton, D.H. & Weston, L.J. Strapdown inertial navigation technology. Institution of Electrical Engineers. Ed  2nd, 2005, 549p.

22.   Robert, M.R. Applied mathematics in integrated navigation systems. AIAA. Ed 3rd, 2007, 340p.

23.   Piotr, K. Integration of INS with TACAN and Altimeter. Military university of technology, 2 Gen. S. Kaliski, Poland, 2007.

Mr Ravi Shankar

Mr Ravi Shankar has received BSc(Eng) (Comp. Sc. Eng.) from B.I.T Sindri, Jharkand and MSc (Aerospace Vehicle Design (Avionics)) from Cranfield University, U.K., in 2002 and 2011 respectively. He is presently working at Hindustan Aeronautics Limited, Banglore. He worked in the area of development of controller for ground based radio, GPS receiver, ground-based voice recorder, VoIP integration, communication system integration and testing on military aircraft. Presently working on control software development for software defined radio..

Gyros are generalised in mathematical from as described by the Eqn (18)..



ω out = ω in + Δ B + Δ R + Δ SF ω in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaad+gacaWG1bGaamiDaaqabaGccqGH9aqpcqaHjpWDdaWg aaWcbaGaamyAaiaad6gaaeqaaOGaey4kaSIaeuiLdq0aaSbaaSqaai aadkeaaeqaaOGaey4kaSIaeuiLdq0aaSbaaSqaaiaadkfaaeqaaOGa ey4kaSIaeuiLdq0aaSbaaSqaaiaadofacaWGgbaabeaakiabeM8a3j aadMgacaWGUbaaaa@4E11@      (18)

where ω out and ω in are the gyro output and input respectively, Δ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadkeaaeqaaaaa@382F@ is the gyro bias Δ R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadkfaaeqaaaaa@383F@ , is the gyro random drift rate Δ SF MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadofacaWGgbaabeaaaaa@390B@ ,is the gyro scale factor. Accelerometers are generalised in mathematical from as described by the Eqn (19).



f out = f in + ε R + Δ SF f in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamOzamaaBa aaleaacaWGVbGaamyDaiaadshaaeqaaOGaeyypa0JaamOzamaaBaaa leaacaWGPbGaamOBaaqabaGccqGHRaWkcqaH1oqzdaWgaaWcbaGaam OuaaqabaGccqGHRaWkcqqHuoardaWgaaWcbaGaam4uaiaadAeaaeqa aOGaamOzamaaBaaaleaacaWGPbGaamOBaaqabaaaaa@4891@     (19)

where fout and fin are the output and input of an accelerometer, ϵ B is the accelerometer bias, ϵ R is an accelerometer time dependent random bias Δ SF MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aaS baaSqaaiaadofacaWGgbGaaGPaVdqabaaaaa@3A96@ .is the accelerometer scale factor error.

Quaternion differential equation (attitude) in NED frame is defined by Eqn (20)



d dt q 0 q 1 q 2 q 3 = 0.5 0 ω n / b , x b ω n / b , y b ω n / b , z b ω n / b , x b 0 ω n / b , z b ω n / b , y b ω n / b , y b ω n / b , z b 0 ω n / b , x b ω n / b , z b ω n / b , y b ω n / b , x b 0 + q 0 q 1 q 2 q 3      (20)

Velocity equation is defined by Eqn (21).



Δvn= C b n ( t i )[Δ v b +Δθ×Δ v b ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam ODaiaad6gacqGH9aqpcaWGdbWaa0baaSqaaiaadkgaaeaacaWGUbaa aOGaaiikaiaadshadaWgaaWcbaGaamyAaaqabaGccaGGPaGaai4wai abfs5aejaadAhadaahaaWcbeqaaiaadkgaaaGccqGHRaWkcqqHuoar cqaH4oqCcqGHxdaTcqqHuoarcaWG2bWaaWbaaSqabeaacaWGIbaaaO Gaaiyxaaaa@4F4E@         (21)

The position is expressed by direction cosine matrix differential Eqn (22).



C e n . = Ω e/n n C e n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaybyaeqale qabaGaaiOlaaqdbaGaam4qamaaDaaaoeaacaWGLbaabaGaamOBaaaa aaGccqGH9aqpcqGHsislcqqHPoWvdaqhaaWcbaGaamyzaiaac+caca WGUbaabaGaamOBaaaakiaadoeadaqhaaWcbaGaamyzaaqaaiaad6ga aaaaaa@4407@      (22)

Attitude Error Equations is defined by Eqn (23).



δ C b n =( ϕ× ) C b n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaam 4qamaaDaaaleaacaWGIbaabaGaamOBaaaakiabg2da9iabgkHiTmaa bmaabaGaeqy1dyMaey41aqlacaGLOaGaayzkaaGaam4qamaaDaaale aacaWGIbaabaGaamOBaaaaaaa@447A@      (23)

Velocity error equation with respect to velocity and earth radius yields is defined by Eqn (24).



δ ω e/n n [ δ v y n R ρx R δh δ v x n R ρy R δh δρz ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaeq yYdC3aa0baaSqaaiaadwgacaGGVaGaamOBaaqaaiaad6gaaaGccqGH ijYUdaWadaabaeqabaGaeyOeI0YaaSaaaeaacqaH0oazcaWG2bWaa0 baaSqaaiaadMhaaeaacaWGUbaaaaGcbaGaamOuaaaacqGHsisldaWc aaqaaiabeg8aYjaadIhaaeaacaWGsbaaaiabes7aKjaadIgaaeaacq GHsisldaWcaaqaaiabes7aKjaadAhadaqhaaWcbaGaamiEaaqaaiaa d6gaaaaakeaacaWGsbaaaiabgkHiTmaalaaabaGaeqyWdiNaamyEaa qaaiaadkfaaaGaeqiTdqMaamiAaaqaaiaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlabes7aKjabeg8aYjaadQhaaaGaay5waiaaw2faaaaa@730F@      (24)

Position error equations is defined as



δ θ ˙ =δρ ω e/n n ×δθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaeqiTdqMafq iUdeNbaiaacqGH9aqpcqaH0oazcqaHbpGCcqGHsislcqaHjpWDdaqh aaWcbaGaamyzaiaac+cacaWGUbaabaGaamOBaaaakiabgEna0kabes 7aKjabeI7aXbaa@49A7@      (25)

The single DCM result from the rotation of Φ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyeaaa@376E@ about x axis, θ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AA@ about y axis and Ψ MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiQdKfaaa@3783@ about z axis is defined by Eqn (26).



C i j =[ 100 0cos(Φ)sin(Φ) 0sin(Φ)cos(Φ) ][ cos(θ)0sin(θ) 010 sin(θ)0cos(θ) ][ cos(ψ)sin(ψ)0 sin(ψ)cos(ψ)0 001 ]= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaam4qamaaDa aaleaacaWGPbaabaGaamOAaaaakiabg2da9maadmaaeaqabeaacaaI XaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaicdaaeaacaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7ciGGJbGaai4BaiaacohacaGGOaGaeuOPdyKaaiykaiaaykW7ca aMc8UaaGPaVlGacohacaGGPbGaaiOBaiaacIcacqqHMoGrcaGGPaaa baGaaGimaiaaykW7caaMc8UaaGPaVlabgkHiTiaaykW7ciGGZbGaai yAaiaac6gacaGGOaGaeuOPdyKaaiykaiaaykW7caaMc8Uaci4yaiaa c+gacaGGZbGaaiikaiabfA6agjaacMcacaaMc8oaaiaawUfacaGLDb aacaaMc8UaaGPaVpaadmaaeaqabeaaciGGJbGaai4BaiaacohacaGG OaGaeqiUdeNaaiykaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlabgkHiTiGa cohacaGGPbGaaiOBaiaacIcacqaH4oqCcaGGPaGaaGPaVlaaykW7ae aacaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGymaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicdacaaMc8Ua aGPaVdqaaiGacohacaGGPbGaaiOBaiaacIcacqaH4oqCcaGGPaGaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaI WaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7ciGGJbGaai4BaiaacohacaGGOaGaeqiUdeNaaiykaiaaykW7aaGa ay5waiaaw2faaiaaykW7caaMc8+aamWaaqaabeqaaiGacogacaGGVb Gaai4CaiaacIcacqaHipqEcaGGPaGaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlGacohacaGGPbGaaiOBaiaacIcacqaHip qEcaGGPaGaaGPaVlaaykW7caaMc8UaaGPaVlaaicdaaeaacqGHsisl ciGGZbGaaiyAaiaac6gacaGGOaGaeqiYdKNaaiykaiaaykW7caaMc8 UaaGPaVlaaykW7ciGGJbGaai4BaiaacohacaGGOaGaeqiYdKNaaiyk aiaaykW7caaMc8UaaGPaVlaaicdacaaMc8UaaGPaVdqaaiaaykW7ca aMc8UaaGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaicdacaaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaIXaaaaiaawUfacaGLDbaacqGH9aqp aaa@94C0@

[ cos(θ)*cos(ψ) cos(θ)*sin(ψ) +sin(Φ)*sin(θ)*cos(ψ) cos(θ)*sin(ψ) sin(θ) cos(θ)*cos(ψ) +sin(Φ)*sin(θ)*sin(ψ) sin(Φ)*cos(θ) sin(Φ)*sin(ψ) +cos(Φ)*sin(θ)*cos(ψ) sin(Φ)*cos(ψ) +cos(Φ)*sin(θ)*sin(ψ) cos(Φ)*cos(θ) ] MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5DBF@      (26)

Position in orbital plane is defined by Eqns. (27) and (28), where and are corrected argument of latitude and radius respectively.



x k ' = r k cos μ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaDa aaleaacaWGRbaabaGaai4jaaaakiabg2da9iaadkhadaWgaaWcbaGa am4AaaqabaGcciGGJbGaai4BaiaacohacqaH8oqBdaWgaaWcbaGaam 4Aaaqabaaaaa@4169@      (27)

y k ' = r k sin μ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaDa aaleaacaWGRbaabaGaai4jaaaakiabg2da9iaadkhadaWgaaWcbaGa am4AaaqabaGcciGGZbGaaiyAaiaac6gacqaH8oqBdaWgaaWcbaGaam 4Aaaqabaaaaa@416F@      (28)

Velocity in orbital plane is defined by Eqns. (29) and (30).



x k ' . = μ a(1 e 2 ) *sin μ k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaca WG4bWaa0baaSqaaiaadUgaaeaacaGGNaaaaaqabeaacaGGUaaaaOGa eyypa0JaeyOeI0YaaOaaaeaadaWcaaqaaiabeY7aTbqaaiaadggaca GGOaGaaGymaiabgkHiTiaadwgadaahaaWcbeqaaiaaikdaaaGccaGG PaaaaaWcbeaakiaacQcaciGGZbGaaiyAaiaac6gacaaMc8UaeqiVd0 2aaSbaaSqaaiaadUgaaeqaaaaa@4B16@      (29)


y k ' . = μ a(1 e 2 ) *[(cos] μ k +e) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaCbiaeaaca WG5bWaa0baaSqaaiaadUgaaeaacaGGNaaaaaqabeaacaGGUaaaaOGa eyypa0JaeyOeI0YaaOaaaeaadaWcaaqaaiabeY7aTbqaaiaadggaca GGOaGaaGymaiabgkHiTiaadwgadaahaaWcbeqaaiaaikdaaaGccaGG PaaaaaWcbeaakiaacQcacaGGBbGaaiikaiGacogacaGGVbGaai4Cai aac2facaaMc8UaeqiVd02aaSbaaSqaaiaadUgaaeqaaOGaey4kaSIa amyzaiaacMcaaaa@5001@      (30)

Position of Kth satellite in ECEF coordinate is defined from Eqns. (31) to (33), where ik is inclination between inclination plan and orbital plan.



x k = x k ' cos Ω k y k ' cos i k sin Ω k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGRbaabeaakiabg2da9iaadIhadaqhaaWcbaGaam4Aaaqa aiaacEcaaaGcciGGJbGaai4BaiaacohacqqHPoWvdaWgaaWcbaGaam 4AaaqabaGccqGHsislcaWG5bWaa0baaSqaaiaadUgaaeaacaGGNaaa aOGaci4yaiaac+gacaGGZbGaamyAamaaBaaaleaacaWGRbaabeaaki aaykW7caaMc8Uaci4CaiaacMgacaGGUbGaeuyQdC1aaSbaaSqaaiaa dUgaaeqaaaaa@528D@      (31)

y k = x k ' sin Ω k + y k ' cos i k cos Ω k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbaabeaakiabg2da9iaadIhadaqhaaWcbaGaam4Aaaqa aiaacEcaaaGcciGGZbGaaiyAaiaac6gacqqHPoWvdaWgaaWcbaGaam 4AaaqabaGccqGHRaWkcaWG5bWaa0baaSqaaiaadUgaaeaacaGGNaaa aOGaci4yaiaac+gacaGGZbGaamyAamaaBaaaleaacaWGRbaabeaaki aaykW7caaMc8Uaci4yaiaac+gacaGGZbGaeuyQdC1aaSbaaSqaaiaa dUgaaeqaaaaa@5283@      (32)

z k = y k ' sin i k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaacaWGRbaabeaakiabg2da9iaadMhadaqhaaWcbaGaam4Aaaqa aiaacEcaaaGcciGGZbGaaiyAaiaac6gacaWGPbWaaSbaaSqaaiaadU gaaeqaaaaa@40AF@      (33)

Velocity of Kth satellite in ECEF coordinate is defined by Eqns (34) to (36)



x k = x k ' cos Ω k y k ' cos i k sin Ω k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGRbaabeaakiabg2da9iaadIhadaqhaaWcbaGaam4Aaaqa aiaacEcaaaGcciGGJbGaai4BaiaacohacqqHPoWvdaWgaaWcbaGaam 4AaaqabaGccqGHsislcaWG5bWaa0baaSqaaiaadUgaaeaacaGGNaaa aOGaci4yaiaac+gacaGGZbGaamyAamaaBaaaleaacaWGRbaabeaaki aaykW7caaMc8Uaci4CaiaacMgacaGGUbGaeuyQdC1aaSbaaSqaaiaa dUgaaeqaaaaa@528D@      (34)

y k . = x k ' . sin Ω k + y k ' . cos i k cos Ω k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaDa aaleaacaWGRbaabaGaaiOlaaaakmaaBaaaleaaaeqaaOGaeyypa0Za aCbiaeaacaWG4bWaa0baaSqaaiaadUgaaeaacaGGNaaaaaqabeaaca GGUaaaaOGaci4CaiaacMgacaGGUbGaeuyQdC1aaSbaaSqaaiaadUga aeqaaOGaey4kaSYaaCbiaeaacaWG5bWaa0baaSqaaiaadUgaaeaaca GGNaaaaaqabeaacaGGUaaaaOGaci4yaiaac+gacaGGZbGaamyAamaa BaaaleaacaWGRbaabeaakiaaykW7caaMc8Uaci4yaiaac+gacaGGZb GaeuyQdC1aaSbaaSqaaiaadUgaaeqaaaaa@554C@      (35)

z k = y k ' sin i k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaacaWGRbaabeaakiabg2da9iaadMhadaqhaaWcbaGaam4Aaaqa aiaacEcaaaGcciGGZbGaaiyAaiaac6gacaWGPbWaaSbaaSqaaiaadU gaaeqaaaaa@40AF@      (36)

Pseudo range is computed from Eqn. (37), where x, y and z are users coordinates and X, Y and Z are satellite coordinates.



ρ (xX) 2 + (yY) 2 + (zZ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaO aaaeaacaGGOaGaamiEaiabgkHiTiaadIfacaGGPaWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaaiikaiaadMhacqGHsislcaWGzbGaaiykam aaCaaaleqabaGaaGOmaaaakiabgUcaRiaacIcacaWG6bGaeyOeI0Ia amOwaiaacMcadaahaaWcbeqaaiaaikdaaaaabeaakiaaykW7aaa@4A30@      (37)


The matrix is derived from Eqn. (37) and n is the number of visible satellites as defined by Eqn. (38)



Δρ= [ ρ 1 ρ 2 ρ 3 ... ρ n ] n×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq yWdiNaeyypa0ZaamWaaqaabeqaaiabeg8aYnaaBaaaleaacaaIXaaa beaaaOqaaiabeg8aYnaaBaaaleaacaaIYaaabeaaaOqaaiabeg8aYn aaBaaaleaacaaIZaaabeaaaOqaaiaac6cacaGGUaGaaiOlaaqaaiab eg8aYnaaBaaaleaacaWGUbaabeaaaaGccaGLBbGaayzxaaWaaSbaaS qaaiaad6gacqGHxdaTcaaIXaaabeaakiaaykW7aaa@4E97@      (38)


ΔX=[ ρ 1 Δx Δy Δz Δt ] 4×1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iwaiabg2da9maadmaaeaqabeaacqaHbpGCdaWgaaWcbaGaaGymaaqa baaakeaacqqHuoarcaWG4baabaGaeuiLdqKaamyEaaqaaiabfs5aej aadQhaaeaacqqHuoarcaWG0baaaiaawUfacaGLDbaacaaMc8+aaSba aSqaaiaaisdacqGHxdaTcaaIXaaabeaaaaa@4C9A@      (39)

H×ΔX=Δρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamisaiabgE na0kabfs5aejaadIfacqGH9aqpcqqHuoarcqaHbpGCaaa@3F25@      (40)

The H matrix is defined by Eq (41).



H= [ c x1 c y1 c z1 1 c x2 c y2 c z2 1 ....... c xn c yn c zn 1 ] n×4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamisaiabg2 da9maadmaaeaqabeaacaWGJbWaaSbaaSqaaiaadIhacaaIXaaabeaa kiaaykW7caaMc8Uaam4yamaaBaaaleaacaWG5bGaaGymaaqabaGcca aMc8UaaGPaVlaadogadaWgaaWcbaGaamOEaiaaigdaaeqaaOGaaGPa VlaaykW7caaMc8UaaGymaaqaaiaadogadaWgaaWcbaGaamiEaiaaik daaeqaaOGaaGPaVlaaykW7caWGJbWaaSbaaSqaaiaadMhacaaIYaaa beaakiaaykW7caaMc8Uaam4yamaaBaaaleaacaWG6bGaaGOmaaqaba GccaaMc8UaaGPaVlaaykW7caaIXaaabaGaaiOlaiaac6cacaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaac6caca GGUaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaiOlaiaa c6cacaGGUaaabaGaam4yamaaBaaaleaacaWG4bGaamOBaaqabaGcca aMc8UaaGPaVlaadogadaWgaaWcbaGaamyEaiaad6gaaeqaaOGaaGPa VlaaykW7caWGJbWaaSbaaSqaaiaadQhacaWGUbaabeaakiaaykW7ca aMc8UaaGPaVlaaigdacaaMc8UaaGPaVlaaykW7aaGaay5waiaaw2fa amaaBaaaleaacaWGUbGaey41aqRaaGinaaqabaaaaa@997F@      (41)

Where ck=(cxk, cyk, czk)is a unit vector from user’s location to kth satellite’s location The offset (Δx) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiabfs 5aejaadIhacaGGPaaaaa@3992@ is computed by Eqn. (42).



ΔX= ( H T *H) 1 H T *ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaam iwaiabg2da9iaacIcacaWGibWaaWbaaSqabeaacaWGubaaaOGaaiOk aiaadIeacaGGPaWaaWbaaSqabeaacqGHsislcaaIXaGaaGPaVdaaki aadIeadaahaaWcbeqaaiaadsfaaaGccaGGQaGaeqyWdihaaa@4581@      (42)

DOPs are obtained by matrix ( H T *H) 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaaiikaiaadI eadaahaaWcbeqaaiaadsfaaaGccaGGQaGaamisaiaacMcadaahaaWc beqaaiabgkHiTiaaigdacaaMc8oaaaaa@3DE3@ . The term σ ij MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMgacaWGQbaabeaaaaa@399E@ represents covariance of error in computed position and time whereas user’s covariance is assumed as one m2.



( H T *H) 1 =[ σ 11 σ 12 σ 13 σ 14 σ 21 σ 22 σ 23 σ 24 σ 31 σ 32 σ 33 σ 34 σ 41 σ 42 σ 43 σ 44 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaaiikaiaadI eadaahaaWcbeqaaiaadsfaaaGccaGGQaGaamisaiaacMcadaahaaWc beqaaiabgkHiTiaaigdaaaGccqGH9aqpdaWadaabaeqabaGaeq4Wdm 3aaSbaaSqaaiaaigdacaaIXaaabeaakiaaykW7caaMc8Uaeq4Wdm3a aSbaaSqaaiaaigdacaaIYaaabeaakiaaykW7caaMc8Uaeq4Wdm3aaS baaSqaaiaaigdacaaIZaaabeaakiaaykW7caaMc8Uaeq4Wdm3aaSba aSqaaiaaigdacaaI0aaabeaaaOqaaiabeo8aZnaaBaaaleaacaaIYa GaaGymaaqabaGccaaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIYaGa aGOmaaqabaGccaaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIYaGaaG 4maaqabaGccaaMc8UaaGPaVlabeo8aZnaaBaaaleaacaaIYaGaaGin aaqabaaakeaacqaHdpWCdaWgaaWcbaGaaG4maiaaigdaaeqaaOGaaG PaVlaaykW7cqaHdpWCdaWgaaWcbaGaaG4maiaaikdaaeqaaOGaaGPa VlaaykW7cqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqaaOGaaGPaVl aaykW7cqaHdpWCdaWgaaWcbaGaaG4maiaaisdaaeqaaaGcbaGaeq4W dm3aaSbaaSqaaiaaisdacaaIXaaabeaakiaaykW7caaMc8Uaeq4Wdm 3aaSbaaSqaaiaaisdacaaIYaaabeaakiaaykW7caaMc8Uaeq4Wdm3a aSbaaSqaaiaaisdacaaIZaaabeaakiaaykW7caaMc8Uaeq4Wdm3aaS baaSqaaiaaisdacaaI0aaabeaaaaGccaGLBbGaayzxaaaaaa@9B8B@      (43)

DOPs are defined by Eqs (44) to (48).



GDOP= σ 11 + σ 22 + σ 33 + σ 44 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaam4raiaads eacaWGpbGaamiuaiabg2da9maakaaabaGaeq4Wdm3aaSbaaSqaaiaa igdacaaIXaaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaaIYaGaaG OmaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqa aOGaey4kaSIaeq4Wdm3aaSbaaSqaaiaaisdacaaI0aaabeaaaeqaaa aa@4A8A@      (44)

PDOP= σ 11 + σ 22 + σ 33 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamiuaiaads eacaWGpbGaamiuaiabg2da9maakaaabaGaeq4Wdm3aaSbaaSqaaiaa igdacaaIXaaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaaIYaGaaG OmaaqabaGccqGHRaWkcqaHdpWCdaWgaaWcbaGaaG4maiaaiodaaeqa aaqabaaaaa@463C@      (45)

HDOP= σ 11 + σ 22 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamisaiaads eacaWGpbGaamiuaiabg2da9maakaaabaGaeq4Wdm3aaSbaaSqaaiaa igdacaaIXaaabeaakiabgUcaRiabeo8aZnaaBaaaleaacaaIYaGaaG Omaaqabaaabeaaaaa@41DF@      (46)

VDOP= σ 32 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamOvaiaads eacaWGpbGaamiuaiabg2da9maakaaabaGaeq4Wdm3aaSbaaSqaaiaa iodacaaIYaaabeaaaeqaaaaa@3D9D@      (47)

TDOP= σ 44 c MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaamivaiaads eacaWGpbGaamiuaiabg2da9maakaaabaWaaSGaaeaacqaHdpWCdaWg aaWcbaGaaGinaiaaisdaaeqaaaGcbaGaam4yaaaaaSqabaaaaa@3EAD@      (48)

User velocity x ˙ , y ˙ and z ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaaca GaaiilaiqadMhagaGaaiaaykW7caaMc8Uaamyyaiaad6gacaWGKbGa aGPaVlqadQhagaGaaaaa@40FE@ is calculated from Eqn. (49).



[ ρ ˙ 1 + 1 ρ 1 [ X ˙ 1 ( X 1 x )+ Y ˙ 1 ( Y 1 y )+ Z ˙ 1 ( Z 1 z ) ] ρ ˙ 2 + 1 ρ 2 [ X ˙ 2 ( X 2 x )+ Y ˙ 2 ( Y 2 y )+ Z ˙ 2 ( Z 2 z ) ] ρ ˙ 3 + 1 ρ 3 [ X ˙ 3 ( X 3 x )+ Y ˙ 3 ( Y 3 y )+ Z ˙ 3 ( Z 3 z ) ] ]=[ X 1 x ρ 1 Y 1 y ρ 1 Z 1 z ρ 1 X 2 x ρ 2 Y 2 y ρ 2 Z 2 z ρ 2 X 3 x ρ 3 Y 3 y ρ 3 Z 3 z ρ 3 ][ X ˙ Y ˙ Z ˙ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaamWaaqaabe qaaiabgkHiTiqbeg8aYzaacaWaaSbaaSqaaiaaigdaaeqaaOGaey4k aSYaaSaaaeaacaaIXaaabaGaeqyWdi3aaSbaaSqaaiaaigdaaeqaaa aakmaadmaabaGabmiwayaacaWaaSbaaSqaaiaaigdaaeqaaOWaaeWa aeaacaWGybWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0IaamiEaaGaay jkaiaawMcaaiabgUcaRiqadMfagaGaamaaBaaaleaacaaIXaaabeaa kmaabmaabaGaamywamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadM haaiaawIcacaGLPaaacqGHRaWkceWGAbGbaiaadaWgaaWcbaGaaGym aaqabaGcdaqadaqaaiaadQfadaWgaaWcbaGaaGymaaqabaGccqGHsi slcaWG6baacaGLOaGaayzkaaaacaGLBbGaayzxaaaabaGaeyOeI0Ia fqyWdiNbaiaadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkdaWcaaqaai aaigdaaeaacqaHbpGCdaWgaaWcbaGaaGOmaaqabaaaaOWaamWaaeaa ceWGybGbaiaadaWgaaWcbaGaaGOmaaqabaGcdaqadaqaaiaadIfada WgaaWcbaGaaGOmaaqabaGccqGHsislcaWG4baacaGLOaGaayzkaaGa ey4kaSIabmywayaacaWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaaca WGzbWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0IaamyEaaGaayjkaiaa wMcaaiabgUcaRiqadQfagaGaamaaBaaaleaacaaIYaaabeaakmaabm aabaGaamOwamaaBaaaleaacaaIYaaabeaakiabgkHiTiaadQhaaiaa wIcacaGLPaaaaiaawUfacaGLDbaaaeaacqGHsislcuaHbpGCgaGaam aaBaaaleaacaaIZaaabeaakiabgUcaRmaalaaabaGaaGymaaqaaiab eg8aYnaaBaaaleaacaaIZaaabeaaaaGcdaWadaqaaiqadIfagaGaam aaBaaaleaacaaIZaaabeaakmaabmaabaGaamiwamaaBaaaleaacaaI ZaaabeaakiabgkHiTiaadIhaaiaawIcacaGLPaaacqGHRaWkceWGzb GbaiaadaWgaaWcbaGaaG4maaqabaGcdaqadaqaaiaadMfadaWgaaWc baGaaG4maaqabaGccqGHsislcaWG5baacaGLOaGaayzkaaGaey4kaS IabmOwayaacaWaaSbaaSqaaiaaiodaaeqaaOWaaeWaaeaacaWGAbWa aSbaaSqaaiaaiodaaeqaaOGaeyOeI0IaamOEaaGaayjkaiaawMcaaa Gaay5waiaaw2faaaaacaGLBbGaayzxaaGaeyypa0ZaamWaaeaafaqa beWadaaabaWaaSaaaeaacaWGybWaaSbaaSqaaiaaigdaaeqaaOGaey OeI0IaamiEaaqaaiabeg8aYnaaBaaaleaacaaIXaaabeaaaaaakeaa daWcaaqaaiaadMfadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWG5b aabaGaeqyWdi3aaSbaaSqaaiaaigdaaeqaaaaaaOqaamaalaaabaGa amOwamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadQhaaeaacqaHbp GCdaWgaaWcbaGaaGymaaqabaaaaaGcbaWaaSaaaeaacaWGybWaaSba aSqaaiaaikdaaeqaaOGaeyOeI0IaamiEaaqaaiabeg8aYnaaBaaale aacaaIYaaabeaaaaaakeaadaWcaaqaaiaadMfadaWgaaWcbaGaaGOm aaqabaGccqGHsislcaWG5baabaGaeqyWdi3aaSbaaSqaaiaaikdaae qaaaaaaOqaamaalaaabaGaamOwamaaBaaaleaacaaIYaaabeaakiab gkHiTiaadQhaaeaacqaHbpGCdaWgaaWcbaGaaGOmaaqabaaaaaGcba WaaSaaaeaacaWGybWaaSbaaSqaaiaaiodaaeqaaOGaeyOeI0IaamiE aaqaaiabeg8aYnaaBaaaleaacaaIZaaabeaaaaaakeaadaWcaaqaai aadMfadaWgaaWcbaGaaG4maaqabaGccqGHsislcaWG5baabaGaeqyW di3aaSbaaSqaaiaaiodaaeqaaaaaaOqaamaalaaabaGaamOwamaaBa aaleaacaaIZaaabeaakiabgkHiTiaadQhaaeaacqaHbpGCdaWgaaWc baGaaG4maaqabaaaaaaaaOGaay5waiaaw2faamaadmaabaqbaeqabm qaaaqaaiqadIfagaGaaaqaaiqadMfagaGaaaqaaiqadQfagaGaaaaa aiaawUfacaGLDbaaaaa@DD32@      (49)

TACAN range and bearing are calculated from Eqn.(50) and Eqn. (51) respectively23 where (x, y, z) are aircraft position and (X,Y,Z) are TACAN ground station location.



ρ TACAN = (xX) 2 + (yY) 2 + (zZ) 2 + b TACAN + υ TACAN MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaadsfacaWGbbGaam4qaiaadgeacaWGobaabeaakiabg2da 9maakaaabaGaaiikaiaadIhacqGHsislcaWGybGaaiykamaaCaaale qabaGaaGOmaaaakiabgUcaRiaacIcacaWG5bGaeyOeI0Iaamywaiaa cMcadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaGGOaGaamOEaiabgk HiTiaadQfacaGGPaWaaWbaaSqabeaacaaIYaaaaaqabaGccqGHRaWk caWGIbWaaSbaaSqaaiaadsfacaWGbbGaam4qaiaadgeacaWGobaabe aakiabgUcaRiabew8a1naaBaaaleaacaWGubGaamyqaiaadoeacaWG bbGaamOtaaqabaaaaa@5AB5@      (50)

Where bTACAN is a corrupting bias and uTACAN is a measurement noise.



θ= cos 1 ( yY ρ TACAN ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiabeI7aXjabg2 da9iGacogacaGGVbGaai4CamaaCaaaleqabaGaeyOeI0IaaGymaaaa kmaabmaabaWaaSaaaeaacaWG5bGaeyOeI0Iaamywaaqaaiabeg8aYn aaBaaaleaacaWGubGaamyqaiaadoeacaWGbbGaamOtaaqabaaaaaGc caGLOaGaayzkaaaaaa@478D@      (51)

Sigma error is defined by Eqn. (52), where N is the number of samples, measurement is the measured data and input is the expected value.



σ= 1 N (measurementinput) 2 N MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0JaaGPaVpaakaaabaWaaSaaaeaacqGHris5daqhaaWcbaGaaGym aaqaaiaad6eaaaGccaGGOaGaamyBaiaadwgacaWGHbGaam4Caiaadw hacaWGYbGaamyzaiaad2gacaWGLbGaamOBaiaadshacqGHsislcaWG PbGaamOBaiaadchacaWG1bGaamiDaiaacMcadaahaaWcbeqaaiaaik daaaaakeaacaWGobaaaaWcbeaakiaaykW7aaa@5283@      (52)