## 3 Relative Position Measurement Calculation Method Using IRA Output
### Proposed Method  
Let's see how IRA calculates and outputs the look angle (Figure 11). In the figure, the aircraft was assumed to be flying in a direction to penetrate the paper. The \(X\)-axis of the aircraft is the rightward direction of the aircraft (cross track), the \(Y\)-axis is the nose direction of the aircraft (along track, the direction penetrating the paper plane), and the \(Z\)-axis is the upward direction of the aircraft. The IRA requires two or more antennas, which are installed aligned with the \(X\)-axis (cross track) of the aircraft. The IRA measures the range from each antenna to the closest point (or target). The obtained range values are different, and the difference between the range values is used to calculate the phase difference. See reference [7] for details on how to compute the look angle using the phase difference between the slant ranges. The look angle is calculated using the difference of the phase. Since the IRA antenna is installed aligned with the \(X\)-axis of the aircraft, the look angle (\(\theta\)) refers to the angle relative to the \(X\)-\(Z\) plane of the aircraft (\(-Z\) direction in the figure). In other words, the look angle is determined by the attitude of the flight vehicle, not by the velocity vector. On the other hand, the zero Doppler line is determined by the velocity vector rather than by the attitude of the aircraft. Figure 11: Look angle measurement of IRA [12]. Figure 10: Relative position from aircraft to closest point: (**a**) 3-dimensional graph; (**b**) 2-dimensional graph [12]. To better understand where the zero Doppler line and closest points are located in a real flight environment, Figure 12 is attached. Figure 12a shows the environment in which the roll, pitch, heading, \(\alpha\) (effective pitch), and \(\beta\) (effective heading) angles are all zero. In this case, the pitch and heading angles are the same for \(\alpha\) and \(\beta\), respectively, so that the zero Doppler line (the dotted blue line) is parallel to the \(X\)-axis of the flight, and the center of the zero Doppler line (point C) lies directly below the aircraft. In this case, the look angle is \(\angle\)COT, and can be used to find the location of the closest point. In addition to the condition of Figure 12a, Figure 12b shows a case in which the pitch angle and the \(\alpha\) angle are not \(0^{\circ}\) due to wind blowing from the front. In this case, the center of the zero Doppler line moves forward by angle \(\alpha\), but the zero Doppler line is still parallel to the \(X\)-axis of the flight. The position of the closest point (point T) can be found by rotating \(\theta\) at the center of the shifted zero Doppler line. Figure 12c,d illustrate the case in which the roll angle of the aircraft is not 0 under the conditions of Figure 12a,b. In both cases shown in Figure 12c,d, the zero Doppler line is parallel to the \(X\)-axis of the vehicle, even if the roll angle is not zero. Therefore, instead of using \(\theta\) from Figure 12a,b, we can find the location of the closest point by using the value angle roll + \(\theta\). In addition to the conditions of Figure 12a,b, Figure 12e,f show cases in which the \(\beta\) angle is not 0 due to wind blowing from the side of the vehicle. In this case, the \(X\)-axis and zero Doppler lines of the aircraft are not parallel as shown in Figure 12a-d, but are in a twisted position. To find the closest point, we need the \"effective look angle\" \(\angle\)COT, expressed as angle \(\xi\). However, the look angle is \(\angle\)BOT, which is not the same as \(\angle\)COT. In the case shown in Figure 12f, the center of the zero Doppler line is shifted by angle \(\alpha\). In this case, \(\angle\)COT is required to find the closest point. However, as shown in Figure 12e, because \(\angle\)COT is different from the look angle \(\angle\)BOT, we cannot use the look angle to find the closest point. As shown in Figure 12a-f, there is a difference between roll angle + look angle and \(\xi\) when there is wind blowing from the side.