## 3 Relative Position Measurement Calculation Method Using IRA Output
### Conventional Method  
In Section 2.2, the IRA output information about the closest points in the zero Doppler line. Before describing the formula for calculation of the relative position of the closest point, it is necessary to understand the zero Doppler line. To understand the zero Doppler line, it is important to first discuss the Doppler Effect. The Doppler Effect is caused by relative velocity between a source and an observer. For example, when a source is moving, if the observer is in front of the source, the wavelength becomes short and the frequency becomes high. If the observer is at the rear of the source, the wavelength becomes long and the frequency becomes low. The Doppler Effect also appears in the IRA on aircraft. When an aircraft is flying, the IRA transmits radio waves downward, and the Doppler radial velocity changes according to the relative distance from the aircraft to the point on the ground where the radio waves are received. The contour of constant radial velocity is called isodop, which is shown in Figure 8a. The zero Doppler line is a contour with a Doppler radial velocity of zero. The zero Doppler line is perpendicular to the velocity vector of the aircraft as shown in Figure 8b. On the other hand, when we executed the flight test, we found that the attitude and velocity vectors of the aircraft are different. This is due to the angle of attack and the side slip angle, both caused by wind during flight. Figure 9 shows the difference between aircraft attitude and velocity vectors due to wind. Figure 9a is a side view of the aircraft and Figure 9b is a top view. Angle \(\alpha\) is  
Figure 8: Isodop and zero Doppler region (line) of IRA on the aircraft: (**a**) top view; (**b**) side view. formed by the velocity vector of the aircraft with respect to the horizontal plane of the aircraft. Angle \(\beta\) is formed by the velocity vector of the aircraft with respect to the north axis. We will call these angles \"effective observation angles.\" In a simulation environment in which effects of wind are not considered, a zero Doppler line exists relative to the Euler angles (attitude) of the aircraft. However, in an actual flight environment in which the influence of wind exists, the effective observation angles (\(\alpha,\beta\)), not the Euler angle, should be used to calculate the zero Doppler line. The effective observation angles \(\alpha\) and \(\beta\) can be obtained using the following equation. \[\alpha=\tan^{-1}\frac{v_{u}}{\sqrt{{v_{e}}^{2}+{v_{n}}^{2}}} \tag{9}\]  
\[\beta=\left\{ \begin{array}{cc}\tan^{-1}\frac{v_{e}}{v_{n}}&,|v_{n}|\geq|v_{e }|\\ \frac{n}{2}-\tan^{-1}\frac{v_{n}}{v_{e}}&,|v_{n}|<|v_{e}|\end{array} \right. \tag{10}\]  
where \(v_{e}\), \(v_{n}\) and \(v_{u}\) are the velocities measured by the inertial navigation system. The conventional formula used to calculate the relative position of the closest point using effective observation angles is presented in reference [11] as follows. \[p=\left[ \begin{array}{c}p_{e}\\ p_{n}\\ p_{u}\end{array} \right]=\left[ \begin{array}{cc}R\cos\theta\sin\alpha\sin \beta+R\sin\theta\cos\beta\\ R\cos\theta\sin\alpha\cos\beta-R\sin\theta\sin\beta\\ -R\cos\theta\cos\alpha\end{array} \right]=R\left[ \begin{array}{cc}\sin\alpha \sin\beta&\cos\beta\\ \sin\alpha\cos\beta&-\sin\beta\\ -\cos\alpha&0\end{array} \right]\left[ \begin{array}{cc}\cos\theta\\ \sin\theta\end{array} \right]=RS\left[ \begin{array}{c}\cos\theta\\ \sin\theta\end{array} \right] \tag{11}\]  
where  
\[S=\left[ \begin{array}{cc}\sin\alpha\sin\beta&\cos\beta\\ \sin\alpha\cos\beta&-\sin\beta\\ -\cos\alpha&0\end{array} \right] \tag{12}\]  
In the existing paper [11], if the roll angle (\(\gamma\)) is not \(0^{\circ}\), (look angle (\(\theta\)) + roll angle (\(\gamma\))) is substituted for look angle (\(\theta\)). \[p=\left[ \begin{array}{c}R\cos(\theta+\gamma)\sin\alpha\sin\beta+R\sin( \theta+\gamma)\cos\beta\\ R\cos(\theta+\gamma)\sin\alpha\cos\beta-R\sin(\theta+\gamma)\sin\beta\\ -R\cos(\theta+\gamma)\cos\alpha\end{array}