Limitations of Flight Path Reconstruction techniques

Abstract

The Flight Path Reconstruction (FPR) techniques are performed to verify the data compatibility check and post-flight. This is often achieved by calibrating the onboard sensors such as inertial and airdata sensors. In this paper, the limitations of FPR techniques in terms of Maximum Likelihood Estimation (MLE) and Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) have been reported. To demonstrate the FPR and sensor calibration, kinematic trajectory simulations with wind box type maneuvers have been performed. It is also shown as how a kinematic simulation is valid for the studies carried out in this work.

Appendices

Appendix

The Equations of Motion (EOM) for the simulation is derived from point mass model as follows:

$$ \dot{V}m = T\cos \alpha - D - W\sin \gamma $$

(25)

$$ \dot{\gamma }mV = (L + T\sin \alpha )\cos \mu - W\cos \gamma $$

(26)

$$ \dot{\chi }mV\cos \gamma = (L + T\sin \alpha )\sin \mu $$

(27)

$$ \begin{aligned} \dot{x} & = V\cos \gamma \cos \chi \\ \dot{y} & = V\cos \gamma \sin \chi \\ \dot{h} & = V\sin \gamma \\ \end{aligned} $$

(28)

For constant flight path angle and small angles of attack Eq. (26) reduces to

$$ L\cos \mu = W\cos \gamma = mg\cos \gamma $$

(29)

We know

$$ N_{z} = \frac{L}{mg} $$

(30)

$$ \sqrt {L^{2} \sin^{2} \mu + L^{2} \cos \mu } = \sqrt {L^{2} } = L $$

(31)

$$ \frac{{V\dot{\chi }}}{g} = \frac{L\sin \mu }{mg\cos \gamma } $$

(32)

Substitute Eqs. (29) and (32) in Eq. (31)

$$ L = \sqrt {\left( {\frac{{V^{2} \dot{\chi }}}{{g^{2} }}^{2} } \right)m^{2} g^{2} \cos^{2} \gamma + m^{2} g^{2} \cos^{2} \gamma } $$

(33)

Substitute (33) in (30)

$$ N_{Z} = \sqrt {\frac{{\left( {\frac{{V^{2} \dot{\chi }}}{{g^{2} }}^{2} } \right)m^{2} g^{2} \cos^{2} \gamma }}{{m^{2} g^{2} }} + \frac{{m^{2} g^{2} \cos^{2} \gamma }}{{m^{2} g^{2} }}} $$

Hence

$$ \begin{aligned} N_{Z} & = \sqrt {\left( {\frac{{V^{2} \dot{\chi }}}{{g^{2} }}^{2} } \right)\cos^{2} \gamma + \cos^{2} \gamma } \\ N_{Z} & = \sqrt {1 + \left( {\frac{{V^{2} \dot{\chi }}}{{g^{2} }}^{2} } \right)} \cos \gamma \\ \end{aligned} $$

(34)

Equation (30) can be written as:

$$ \begin{aligned} N_{Z} & = \frac{{0.5\rho V^{2} C_{L\alpha } \alpha }}{mg} \\ \theta & \approx \alpha = \frac{{N_{Z} K}}{{V^{2} }} + \gamma \\ \end{aligned} $$

(35)

Nomenclature

$\vec{X}_{Earth}$ :

Position in the Earth coordinate

$\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {X} }_{Earth} ,V$ :

Velocity in the Earth coordinate

$\ddot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {X} }_{Earth}$ :

Accelerations in the Earth coordinate

$\Phi$ :

Euler angles

$\phi$ :

Bank angle

$\theta$ :

Pitch angle

$\psi$ :

Yaw angle

$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {g}$ :

Gravity vector

$\dot{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {X} }_{AC} ,TAS$ :

Velocity in the body frame relative to wind

W :

Wind vector in the inertial frame

$\alpha$ :

Angle of attack

$\beta$ :

Sideslip angle

TAS :

True Air Speed

$V_{Tbias} ,V_{Tscale}$ :

TAS bias/offset and scale factor

$\alpha_{bias} \,\,\alpha_{scale} \,\,\beta_{bias} \,\,\beta_{scale}$ :

AoA bias/offset, AoA scale factor, AoSS bias/offset, AoSS scale factor

$u,v,w$ :

Velocities in the body frame

$u_{wi} ,v_{wi} ,w_{wi}$ :

Wind velocities in the inertial frame

P:

Position vector

$N_{xb}$ $N_{yb}$ $N_{zb}$ :

Accelerometer bias

$p_{b} \,\,q_{b} \,\,r_{b}$ :

Gyro bias

Suffix ‘m’:

Indicates measurement

ADS:

Air Data System

p q r :

Rates

$N_{x}$ $N_{y}$ $N_{z}$ :

Accelerations