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.

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

${x}_{system}{\left[\begin{array}{l}Position\\ Volocity\\ Attitude\\ Bias\text{\hspace{0.17em}}Acc\\ Bias\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}Gyro\\ TACAN\text{\hspace{0.17em}}RANGE\\ TACAN\text{\hspace{0.17em}}Azimuth\end{array}\right]}_{17×1}$          (1)

where $Position={\left[\begin{array}{l}x\\ y\\ z\end{array}\right]}_{3×1}$     (2)

$Velocity={\left[\begin{array}{l}{V}_{N}\\ {V}_{E}\\ {V}_{D}\end{array}\right]}_{3×1}$     (3)

$Attitude={\left[\begin{array}{l}\varphi \\ \theta \\ \psi \end{array}\right]}_{3×1}$      (4)

$Bias\text{\hspace{0.17em}}\text{\hspace{0.17em}}Acc={\left[\begin{array}{l}{\alpha }_{x}\\ {\alpha }_{y}\\ {\alpha }_{z}\end{array}\right]}_{3×1}$     (5)

$Bias\text{\hspace{0.17em}}\text{\hspace{0.17em}}Gyro={\left[\begin{array}{l}{g}_{x}\\ {g}_{x}\\ {g}_{x}\end{array}\right]}_{3×1}$      (6)

$TACAN\text{\hspace{0.17em}}RANGE={\left[\rho \right]}_{1×1}$     (7)

$TACAN\text{\hspace{0.17em}}Azimuth={\left[\theta \right]}_{1×1}$     (8)

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

where

(9)

${Z}_{3}=\left[\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}\right]$     (10)

${l}_{3}=\left[\begin{array}{l}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\end{array}\right]$     (11)

${Z}_{23}=\left[\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}\right]$     (12)

${Z}_{32}=\left[\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}\right]$     (13)

${F}_{66}=\left[\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\end{array}\right]$     (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).

${Q}_{\mathrm{system}}=\left[\begin{array}{cccccc}{z}_{3}& {z}_{3}& {z}_{3}& {z}_{3}& {z}_{3}& {z}_{33}\\ {I}_{3}& {n}_{\mathrm{ra}}^{2}{I}_{3}& {z}_{3}& {z}_{3}& {z}_{3}& {z}_{32}\\ {z}_{3}& {I}_{3}& {n}_{\mathrm{rg}}^{2}{I}_{3}& {z}_{3}& {z}_{3}& {z}_{32}\\ {z}_{3}& {z}_{3}& {I}_{3}& {n}_{\mathrm{bad}}^{2}{I}_{3}& {z}_{3}& {z}_{32}\\ {z}_{3}& {z}_{3}& {z}_{3}& {I}_{3}& {n}_{\mathrm{bgd}}^{2}{I}_{3}& {z}_{32}\\ {z}_{23}& {z}_{23}& {z}_{23}& {z}_{23}& {z}_{23}& {Q}_{66}\end{array}\right]$     (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).

$\begin{array}{l}{Q}_{66}=\left[\begin{array}{l}TACA{N}_{RANGE}{}_{{}_{Errpr}}2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}TACA{N}_{RANGE}{}_{{}_{Errpr}}2\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}\right]\\ \end{array}$     (16)

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

$H={\left[\begin{array}{l}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}..\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}..\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}..\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}..\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}..\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}..\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}\right]}_{6×17}$     (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.

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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)..

${\omega }_{out}={\omega }_{in}+{\Delta }_{B}+{\Delta }_{R}+{\Delta }_{SF}{\omega }_{\mathrm{in}}$     (18)

where $\omega$ out and $\omega$ in are the gyro output and input respectively,${\Delta }_{B}$is the gyro bias${\Delta }_{R}$ , is the gyro random drift rate${\Delta }_{SF}$ ,is the gyro scale factor. Accelerometers are generalised in mathematical from as described by the Eqn (19).

${f}_{out}={f}_{in}+{\epsilon }_{R}+{\Delta }_{SF}{f}_{in}$    (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${\Delta }_{SF\text{\hspace{0.17em}}}$ .is the accelerometer scale factor error.

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

$\frac{d}{\mathrm{dt}}\left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]=0.5\ast \left[\begin{array}{cccc}0& {-\omega }_{n/b,x}^{b}& {-\omega }_{n/b,y}^{b}& {-\omega }_{n/b,z}^{b}\\ {\omega }_{n/b,x}^{b}& 0& {\omega }_{n/b,z}^{b}& {-\omega }_{n/b,y}^{b}\\ {\omega }_{n/b,y}^{b}& {-\omega }_{n/b,z}^{b}& 0& {\omega }_{n/b,x}^{b}\\ {-\omega }_{n/b,z}^{b}& {\omega }_{n/b,y}^{b}& {-\omega }_{n/b,x}^{b}& 0\end{array}\right]+\left[\begin{array}{c}{q}_{0}\\ {q}_{1}\\ {q}_{2}\\ {q}_{3}\end{array}\right]$      (20)

Velocity equation is defined by Eqn (21).

$\Delta vn={C}_{b}^{n}\left({t}_{i}\right)\left[\Delta {v}^{b}+\Delta \theta ×\Delta {v}^{b}\right]$         (21)

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

$\stackrel{.}{{C}_{e}^{n}}=-{\Omega }_{e/n}^{n}{C}_{e}^{n}$     (22)

Attitude Error Equations is defined by Eqn (23).

$\delta {C}_{b}^{n}=-\left(\varphi ×\right){C}_{b}^{n}$     (23)

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

$\delta {\omega }_{e/n}^{n}\approx \left[\begin{array}{l}-\frac{\delta {v}_{y}^{n}}{R}-\frac{\rho x}{R}\delta h\\ -\frac{\delta {v}_{x}^{n}}{R}-\frac{\rho y}{R}\delta h\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\delta \rho z\end{array}\right]$     (24)

Position error equations is defined as

$\delta \stackrel{˙}{\theta }=\delta \rho -{\omega }_{e/n}^{n}×\delta \theta$     (25)

The single DCM result from the rotation of $\Phi$ about x axis, $\theta$ about y axis and $\Psi$ about z axis is defined by Eqn (26).

${C}_{i}^{j}=\left[\begin{array}{l}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\left(\Phi \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{sin}\left(\Phi \right)\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\mathrm{sin}\left(\Phi \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\left(\Phi \right)\text{\hspace{0.17em}}\end{array}\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[\begin{array}{l}\mathrm{cos}\left(\theta \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\mathrm{sin}\left(\theta \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \mathrm{sin}\left(\theta \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\left(\theta \right)\text{\hspace{0.17em}}\end{array}\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[\begin{array}{l}\mathrm{cos}\left(\psi \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{sin}\left(\psi \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ -\mathrm{sin}\left(\psi \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\left(\psi \right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\end{array}\right]=$

$\left[\begin{array}{cc}\mathrm{cos}\left(\theta \right)*\mathrm{cos}\left(\psi \right)& \begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\mathrm{cos}\left(\theta \right)*\mathrm{sin}\left(\psi \right)\\ +\mathrm{sin}\left(\Phi \right)*\mathrm{sin}\left(\theta \right)*\mathrm{cos}\left(\psi \right)\end{array}\\ \begin{array}{l}\mathrm{cos}\left(\theta \right)*\mathrm{sin}\left(\psi \right)\text{\hspace{0.17em}}\\ -\mathrm{sin}\left(\theta \right)\end{array}& \begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\left(\theta \right)*\mathrm{cos}\left(\psi \right)\\ +\mathrm{sin}\left(\Phi \right)*\mathrm{sin}\left(\theta \right)*\mathrm{sin}\left(\psi \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{sin}\left(\Phi \right)*\mathrm{cos}\left(\theta \right)\end{array}\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\begin{array}{c}\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{sin}\left(\Phi \right)*\mathrm{sin}\left(\psi \right)\\ +\mathrm{cos}\left(\Phi \right)*\mathrm{sin}\left(\theta \right)*\mathrm{cos}\left(\psi \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\mathrm{sin}\left(\Phi \right)*\mathrm{cos}\left(\psi \right)\end{array}\\ \begin{array}{l}+\mathrm{cos}\left(\Phi \right)*\mathrm{sin}\left(\theta \right)*\mathrm{sin}\left(\psi \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}\left(\Phi \right)*\mathrm{cos}\left(\theta \right)\end{array}\end{array}\right]$      (26)

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

${x}_{k}^{\text{'}}={r}_{k}\mathrm{cos}{\mu }_{k}$     (27)

${y}_{k}^{\text{'}}={r}_{k}\mathrm{sin}{\mu }_{k}$     (28)

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

$\stackrel{.}{{x}_{k}^{\text{'}}}=-\sqrt{\frac{\mu }{a\left(1-{e}^{2}\right)}}*\mathrm{sin}\text{\hspace{0.17em}}{\mu }_{k}$     (29)

$\stackrel{.}{{y}_{k}^{\text{'}}}=-\sqrt{\frac{\mu }{a\left(1-{e}^{2}\right)}}*\left[\left(\mathrm{cos}\right]\text{\hspace{0.17em}}{\mu }_{k}+e\right)$     (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.

${x}_{k}={x}_{k}^{\text{'}}\mathrm{cos}{\Omega }_{k}-{y}_{k}^{\text{'}}\mathrm{cos}{i}_{k}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{sin}{\Omega }_{k}$     (31)

${y}_{k}={x}_{k}^{\text{'}}\mathrm{sin}{\Omega }_{k}+{y}_{k}^{\text{'}}\mathrm{cos}{i}_{k}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}{\Omega }_{k}$     (32)

${z}_{k}={y}_{k}^{\text{'}}\mathrm{sin}{i}_{k}$     (33)

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

${x}_{k}={x}_{k}^{\text{'}}\mathrm{cos}{\Omega }_{k}-{y}_{k}^{\text{'}}\mathrm{cos}{i}_{k}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{sin}{\Omega }_{k}$     (34)

${y}_{k}^{.}{}_{}=\stackrel{.}{{x}_{k}^{\text{'}}}\mathrm{sin}{\Omega }_{k}+\stackrel{.}{{y}_{k}^{\text{'}}}\mathrm{cos}{i}_{k}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cos}{\Omega }_{k}$     (35)

${z}_{k}={y}_{k}^{\text{'}}\mathrm{sin}{i}_{k}$     (36)

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

$\rho \sqrt{{\left(x-X\right)}^{2}+{\left(y-Y\right)}^{2}+{\left(z-Z\right)}^{2}}\text{\hspace{0.17em}}$     (37)

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

$\Delta \rho ={\left[\begin{array}{l}{\rho }_{1}\\ {\rho }_{2}\\ {\rho }_{3}\\ ...\\ {\rho }_{n}\end{array}\right]}_{n×1}\text{\hspace{0.17em}}$     (38)

$\Delta X=\left[\begin{array}{l}{\rho }_{1}\\ \Delta x\\ \Delta y\\ \Delta z\\ \Delta t\end{array}\right]{\text{\hspace{0.17em}}}_{4×1}$     (39)

$H×\Delta X=\Delta \rho$     (40)

The H matrix is defined by Eq (41).

$H={\left[\begin{array}{l}{c}_{x1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{y1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{z1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ {c}_{x2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{y2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{z2}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ ..\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}..\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}...\\ {c}_{xn}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{yn}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{c}_{zn}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\end{array}\right]}_{n×4}$     (41)

Where ck=(cxk, cyk, czk)is a unit vector from user’s location to kth satellite’s location The offset $\left(\Delta x\right)$is computed by Eqn. (42).

$\Delta X={\left({H}^{T}*H\right)}^{-1\text{\hspace{0.17em}}}{H}^{T}*\rho$     (42)

DOPs are obtained by matrix${\left({H}^{T}*H\right)}^{-1\text{\hspace{0.17em}}}$. The term ${\sigma }_{ij}$represents covariance of error in computed position and time whereas user’s covariance is assumed as one m2.

${\left({H}^{T}*H\right)}^{-1}=\left[\begin{array}{l}{\sigma }_{11}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{12}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{13}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{14}\\ {\sigma }_{21}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{22}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{23}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{24}\\ {\sigma }_{31}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{32}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{33}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{34}\\ {\sigma }_{41}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{42}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{43}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{44}\end{array}\right]$     (43)

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

$GDOP=\sqrt{{\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}+{\sigma }_{44}}$     (44)

$PDOP=\sqrt{{\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}}$     (45)

$HDOP=\sqrt{{\sigma }_{11}+{\sigma }_{22}}$     (46)

$VDOP=\sqrt{{\sigma }_{32}}$     (47)

$TDOP=\sqrt{{\sigma }_{44}}{c}}$     (48)

User velocity $\stackrel{˙}{x},\stackrel{˙}{y}\text{\hspace{0.17em}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\stackrel{˙}{z}$ is calculated from Eqn. (49).

$\left[\begin{array}{l}-{\stackrel{˙}{\rho }}_{1}+\frac{1}{{\rho }_{1}}\left[{\stackrel{˙}{X}}_{1}\left({X}_{1}-x\right)+{\stackrel{˙}{Y}}_{1}\left({Y}_{1}-y\right)+{\stackrel{˙}{Z}}_{1}\left({Z}_{1}-z\right)\right]\\ -{\stackrel{˙}{\rho }}_{2}+\frac{1}{{\rho }_{2}}\left[{\stackrel{˙}{X}}_{2}\left({X}_{2}-x\right)+{\stackrel{˙}{Y}}_{2}\left({Y}_{2}-y\right)+{\stackrel{˙}{Z}}_{2}\left({Z}_{2}-z\right)\right]\\ -{\stackrel{˙}{\rho }}_{3}+\frac{1}{{\rho }_{3}}\left[{\stackrel{˙}{X}}_{3}\left({X}_{3}-x\right)+{\stackrel{˙}{Y}}_{3}\left({Y}_{3}-y\right)+{\stackrel{˙}{Z}}_{3}\left({Z}_{3}-z\right)\right]\end{array}\right]=\left[\begin{array}{ccc}\frac{{X}_{1}-x}{{\rho }_{1}}& \frac{{Y}_{1}-y}{{\rho }_{1}}& \frac{{Z}_{1}-z}{{\rho }_{1}}\\ \frac{{X}_{2}-x}{{\rho }_{2}}& \frac{{Y}_{2}-y}{{\rho }_{2}}& \frac{{Z}_{2}-z}{{\rho }_{2}}\\ \frac{{X}_{3}-x}{{\rho }_{3}}& \frac{{Y}_{3}-y}{{\rho }_{3}}& \frac{{Z}_{3}-z}{{\rho }_{3}}\end{array}\right]\left[\begin{array}{c}\stackrel{˙}{X}\\ \stackrel{˙}{Y}\\ \stackrel{˙}{Z}\end{array}\right]$     (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.

${\rho }_{TACAN}=\sqrt{{\left(x-X\right)}^{2}+{\left(y-Y\right)}^{2}+{\left(z-Z\right)}^{2}}+{b}_{TACAN}+{\upsilon }_{TACAN}$     (50)

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

$\theta ={\mathrm{cos}}^{-1}\left(\frac{y-Y}{{\rho }_{TACAN}}\right)$     (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.

$\sigma =\text{\hspace{0.17em}}\sqrt{\frac{{\sum }_{1}^{N}{\left(measurement-input\right)}^{2}}{N}}\text{\hspace{0.17em}}$     (52)