## 3 Ambiguities
### Resolving the Ambiguities  
If no additional electronic means are to be used, the ambiguities can be resolved as suggested in [15], where an electric vehicle approaches the inductive charging coil of a parking lot. In the case of using only two magnetometers, it never knows whether the coil is in front or behind it. The vehicle therefore assumes that the coil is always in front of it or that the driver is always moving ahead of the parking space. In the same way, concrete constraints for the navigation of a UAV must be defined and passed to it as a priori knowledge. These are:  
1. The equivalent static current \(I\) always flows away from the UAV towards that end of the transmission line to which the UAV is looking; i.e., \(I>0\Rightarrow C>0\).
2. During the navigation, the UAV always remains under the transmission lines, i.e., in \(z<0\).
3. The UAV angles never exceed the definition range \(D\), so that \(\alpha\), \(\beta\), \(\gamma\in D=[-\frac{\pi}{2},\frac{\pi}{2}]\). This means that the UAV never makes large turns and never flies on its back.  
Furthermore, a selection criterion for the measured values must be established. This is because, as mentioned earlier, the sign of the measurement \(\vec{v}_{i}\) of the sensor \(i\) changes with each half period. It must now be decided which is the correct one. The selection criterion can be derived from the above constraints. With \(C>0\) and \(z<0\) from Conditions (1) and (2), it follows from the right-hand rule that the magnetic field always has a positive \(y\)-component, i.e., \(B_{y}(y,z)>0\). This can also be shown directly with Equation (3). Moreover, according to Theorem 3, this circumstance never changes as long as the UAV remains in the half plane \(z<0\) and as long as Constraint (3) is not broken. Using Equation (5) and the Cartesian unit vector \(\vec{e}_{y}\) leads to the necessary selection criterion:  
\[(R\,\vec{v}_{i})\cdot\vec{e}_{y}>0. \tag{27}\]  
If \(\vec{v}_{i}\) does not satisfy (27), \(-\vec{v}_{i}\) does, and vice versa. Only one sign can be a match. Direct comparison of the signs of \(\vec{v}_{iy}\) and \(B_{y}(y_{i},z_{i})\) is not allowed, since from \(B_{y}(y_{i},z_{i})>0\), \(\vec{v}_{iy}>0\) does not necessarily follow. The last point to be clarified is the orientation of the UAV. Since the rotation matrix in (27) is used, it must be ensured that the angles can be computed independently of the direction of \(I\).  
Theorem 3.2: _Given two sensors \(i\) and \(k\) with equally aligned local frames and fixed locations \(\vec{p}_{i}\
eq\vec{p}_{k}\), as long as their measurement values \(\vec{v}_{i}\) and \(\vec{v}_{k}\) are obtained from the same signal period, the yaw angle \(\alpha\) and pitch angle \(\beta\) are independent of the direction of the static current \(I\)._Proof.: Referenced to the same time interval containing a whole signal period, the sensors measure \(\vec{v}_{i}\), \(\vec{v}_{k}\
eq\vec{0}\). Half a signal period later, when both measurements are calculated from the same new time interval and the sensors have not moved, the sign of the measurements is reversed. Then, the cross-product in Equation (9) for the yaw angle becomes \(-\vec{v}_{i}\times-\vec{v}_{k}\). Now, by applying the rule \((r\vec{a})\times\vec{b}=\vec{a}\times(r\vec{b})=r(\vec{a}\times\vec{b})\), it follows that:  
\[-\vec{v}_{i}\times-\vec{v}_{k}=(-1)^{2}(\vec{v}_{i}\times\vec{v}_{k})=\vec{v} _{i}\times\vec{v}_{k}.\]  
Since the direction of the static current obviously has no effect on the cross-product, the result for the yaw angle also remains unchanged. Flipping the signs of the field components in Equation (10) for the pitch angle leads to:  
\[\frac{-v_{kx}-v_{ky}A}{-v_{kz}\sqrt{A^{2}+1}}=\frac{v_{kx}+v_{ky}A}{v_{kz} \sqrt{A^{2}+1}},\quad A=tan\,\alpha.\]  
The direction of the static current obviously also has no influence on the pitch angle.  
In summary, the sign of \(I\) does not affect the orientation as long as the measurements are referenced to the same time interval of the corresponding signal.