## 2 Orientation and Localization Equations
### Selecting the Right Solution  
Equation (24) yields four \(z\)-solutions for each \(y\)-solution of Cases (I)-(IV). Thus, depending on the case, there are up to twelve possible positions in the \(yz\)-plane for a given sensor. For navigation, it is necessary to know which of these points is the correct one. Finding it is the task of this subsection. Now, in the first approach, the number of possible positions can be narrowed down by excluding Cases (II)-(IV). The reason for this is that they can only occur in very few locations and for a very short time during navigation, which extremely reduces their probability of occurrence. For example, the solutions of Case (II) all lie on the \(z\)-axis. In Case (III), one solution lies on the \(y\)-axis and the other two lie on a circle with a diameter equal to the distance between the transmission lines, i.e., \(2y_{0}\). This can be shown by computing the radius \(\sqrt{y_{1,2}^{2}+|z^{2}(y_{1,2})|}\), where \(y_{1,2}\) comes from Equation (23) and \(z^{2}\) comes from (16). The case with the lowest probability of occurrence is Case (IV), where the sensor is located directly at the origin. The locations of Cases (II)-(IV) are summarized in Figure 7 and shown as black and red lines. Since they are not relevant for navigation and only increase the implementation effort, they can be ignored. Thus, since only Case (I) is considered, eight possible positions for a single sensor remain, from which the correct one must still be selected. In the next steps, it will be seen that six can be excluded. To do this, some constraints must be met. As explained in Section 2.1, the sinusoidal transmission line current can be considered as an equivalent constant current \(I\), i.e., it never changes its direction. Furthermore, the UAV orientation must not exceed the definition range \(D\) from Section 2.2 with \(\alpha,\beta,\gamma\in D=[-\frac{\pi}{2},\frac{\pi}{2}]\). This would otherwise lead to ambiguities due to the symmetry of the magnetic field. The details concerning ambiguities are the subject of Section 3. Now, the idea is to investigate which of the eight remaining positions lead to a match with the sensor measurement \(\vec{V}\), i.e., which positions satisfy Equation (12). **Theorem 3**: _Given a sensor \(k\) that measures \(\vec{V}_{k}=(V_{ky},V_{kz})^{T}\) and has two \(y\)-solutions according to (21) in Case (I), for each of the two values, there are four associated \(z\)-solutions according to Equation (24). However, for each \(y\), there is only one \(z\)-solution that leads to a match with \(\vec{V}_{k}\), i.e., there are only two possible locations in the \(yz\)-plane where the sensor can lie._  
Let \(y_{k}\) be one of the two \(y\)-solutions. Then, it generates the four \(z\)-solutions, which in simplified notation are \(z_{n}=\pm\sqrt{\pm g+h},g,h\
eq 0\). \(z_{n}\) has two outer and two inner signs. Further, the \(B_{z}\)-component in Equation (3) has a \(z^{2}\)-addend in the nominator as the only \(z\)-dependency, so it can be written as \(B_{z}(y_{k},z^{2})\). Now, if for \(z_{l}=\pm\sqrt{g+h}\), it follows that:  
\[B_{z}(y_{k},z_{l}^{2})=B_{z}(y_{k},g+h)=V_{kz},\]  
then with \(g+h\
eq-g+h\):  
\[V_{kz}=B_{z}(y_{k},g+h)\
eq B_{z}(y_{k},-g+h),\]because \(B_{z}(y_{k},z^{2})\) is strictly monotonous. This means, that only one of the inner signs of \(z_{n}\) leads to a match with \(V_{kz}\). This fact is independent of the outer sign. Once the inner sign is known, similarly, the \(B_{y}\)-component in Equation (3) can be analyzed. Therefore, let \(z_{m}^{2}=g+h\) lead to a match with \(V_{kz}\). Now, if \(B_{y}(y_{k},z_{m})=V_{ky}\), then \(B_{y}(y_{k},-z_{m})\) is:  
\[C\bigg{(}\frac{-(-z_{m})}{z_{m}^{2}+(y_{k}+y_{0})^{2}}+\frac{-(-z_{m})}{z_{m}^ {2}+(y_{k}-y_{0})^{2}}\bigg{)}=-V_{ky}. \tag{26}\]  
As shown in the last step, only one of the outer signs of \(z_{n}\) can be the correct one, leading to a match with \(V_{ky}\). In summary, there are only two possible locations for a single sensor so that Equation (12) is satisfied.