## 3 Relative Position Measurement Calculation Method Using IRA Output
### Proposed Method
In addition to the condition of Figure 12e, Figure 12g shows a case in which the roll angle is not \(0^{\circ}\). In this case, the roll angle is \(\angle\)COA and the look angle is \(\angle\)BOT. In the figure, we can see that the roll angle and the look angle are on different planes. It is not appropriate to add the roll angle and look angle arithmetically in this situation. Figure 12: _Cont._  
In conclusion, to calculate the relative position from the IRA antenna to the closest point, an effective look angle \(\xi\) is required instead of an angle roll angle + look angle. The formula for calculating \(\alpha\) and \(\beta\) presented in the previous paper [11] was intuitively easy to derive. However, calculating the effective look angle, \(\xi\), is not simple because this value must be obtained by synthesizing the velocity vector and the look angle calculated based on the attitude of the flight body. In this paper, we consider the above assumptions and derive the equation to calculate the effective look angle. The position of the left antenna in the body frame is:  
\[x_{l}^{b}=\left[ \begin{array}{ccc}-L_{1}&0&0\end{array} \right]^{T} \tag{14}\]  
Figure 12: 3-dimensional graph of relative position from aircraft to closest point in real flight condition: (a) case in which roll, pitch, heading, \(\alpha\), and \(\beta\) angles are all zero; (b) pitch and \(\alpha\) angles are not zero; (c) roll angle is not zero; (d) roll, pitch and \(\alpha\) angles are not zero; (e) \(\beta\) angle is not zero; (f) pitch, \(\alpha\) and \(\beta\) angles are not zero; (g) roll and \(\beta\) angles are not zero. The position of the right antenna in the body frame is:  
\[x_{r}^{b}=\left[ \begin{array}{ccc}+L_{2}&0&0\end{array} \right]^{T} \tag{15}\]  
where  
\[x_{l}^{n}= \begin{array}{c}C_{b}^{n}x_{l}^{b}\end{array}  \tag{16}\]  
\[x_{r}^{n}= \begin{array}{c}C_{b}^{n}x_{r}^{b}\end{array}  \tag{17}\]  
According to the IRA angle calculation algorithm (Figure 11), the range difference between the left antenna and the center antenna and the range difference between the right antenna and the center antenna are expressed as functions of the look angle (\(\theta\)) and the antenna baseline (\(L_{1},L_{2}\)) as follows. \[R- \begin{array}{c}\left\|x_{l}^{n}-p\right\|=L_{1}\sin\theta\end{array}  \tag{18}\]  
\[ \begin{array}{c}\left\|x_{r}^{n}-p\right\|-R=L_{2}\sin\theta\end{array}  \tag{19}\]  
\[ \begin{array}{c}\sqrt{\left(x_{l}^{n}-p\right)^{T}}\left(x_{l}^{n}-p\right) =R-L_{1}\sin\theta\end{array}  \tag{20}\]  
\[ \begin{array}{c}\sqrt{\left(x_{r}^{n}-p\right)^{T}}\left(x_{r}^{n}-p\right) =R+L_{2}\sin\theta\end{array}  \tag{21}\]  
The squares of both sides of Equations (20) and (21) are as follows,  
\[L_{1}^{2}-2x_{l}^{n^{T}}p+R^{2}=R^{2}-2L_{1}R\sin\theta+L_{1}^{2}\sin^{2}\theta \tag{22}\]  
\[L_{2}^{2}-2x_{r}^{n^{T}}p+R^{2}=R^{2}+2L_{2}R\sin\theta+L_{2}^{2}\sin^{2}\theta \tag{23}\]  
After removing \(R^{2}\) from both sides and dividing by \(-2R\),  
\[\frac{L_{1}^{2}}{-2R}+\frac{1}{R}x_{l}^{n^{T}}p=L_{1}\sin\theta-\frac{L_{1}^{2 }\sin^{2}\theta}{2R} \tag{24}\]  
\[\frac{L_{1}^{2}}{-2R}+\frac{1}{R}x_{r}^{n^{T}}p=-L_{2}\sin\theta-\frac{L_{1}^{ 2}\sin^{2}\theta}{2R} \tag{25}\]  
Equations (24) and (25) are approximated as follows by \(L_{1}\ll R,\ L_{2}\ll R\)  
\[\frac{1}{R}x_{l}^{n^{T}}p=L_{1}\sin\theta \tag{26}\]  
\[\frac{1}{R}x_{r}^{n^{T}}p=-L_{2}\sin\theta. \tag{27}\]  
Substituting Equations (14)-(17) into the above two equations,  
\[\frac{1}{R}x_{l}^{n^{T}}p=\frac{1}{R}L_{1}\left[ \begin{array}{ccc}-1&0&0 \end{array} \right]C_{n}^{b}RS\left[ \begin{array}{c}\cos\xi\\ \sin\xi\end{array} \right]=L_{1}\left[ \begin{array}{ccc}-1&0&0\end{array}  \right]C_{n}^{b}S\left[ \begin{array}{c}\cos\xi\\ \sin\xi\end{array} \right]=L_{1}\sin\theta. \tag{28}\]  
\[\frac{1}{R}x_{r}^{n^{T}}p=\frac{1}{R}L_{2}\left[ \begin{array}{ccc}+1&0&0 \end{array} \right]C_{n}^{b}RS\left[ \begin{array}{c}\cos\xi\\ \sin\xi\end{array} \right]=L_{2}\left[ \begin{array}{ccc}+1&0&0\end{array}  \right]C_{n}^{b}S\left[