## 2 Orientation and Localization Equations  
### System Overview  
Figure 4 shows two parallel lines, which together form one phase of a three-phase power transmission system and a UAV flying in close proximity. The transmission lines lie in the \(xy\)-plane of the global coordinate frame \(\{G\}\) positioned exactly between the lines. For the purpose of sensing the magnetic field of the transmission lines, there are magnetometers mounted on the UAV, which are marked with yellow points. Figure 4: System overview. Global frame and UAV frame. Figure 3: Distance separator for transmission lines. The transmission lines of a single phase carry sinusoidal currents of fixed frequency and phase, which can be used to our advantage. They generate an alternating magnetic field, which allows static and low-frequency magnetic interference fields such as the Earth's field to be canceled out. This is realized by a differential measurement of the amplitude of the magnetic signal. The calculation can be implemented in the time domain and is very efficient [24]. In addition, the system can be treated as a static field problem with constant currents. For simplicity, they are chosen as \(I\) in the following. The magnetic field thus generated can be calculated using the law of Biot-Savart for thin conductors. In general, transmission lines are not thin. However, as the distance from the lines increases, the magnetic flux density distribution approaches that of thin conductors relatively quickly:  
\[\vec{B}(\vec{r})=-\frac{\mu_{0}}{4\pi}I\int_{-\infty}^{\infty}\frac{(\vec{r}- \vec{r}^{\,\prime})\times d\vec{s}^{\,\prime}}{|\vec{r}-\vec{r}^{\,\prime}|^{ 3}}. \tag{1}\]  
The integration is executed only along the cables according to the arrangement in Figure 4, i.e., \(d\vec{s}^{\,\prime}=(d\vec{x}^{\,\prime},0,0)^{T}\). Thus, from the cross-product in Equation (1), it immediately follows that:  
\[B_{x}=0,\forall(x,y,z)^{T}, \tag{2}\]  
which means that there is no field component in the direction parallel to the cables. Solving the integral for one cable and using the superposition principle, the remaining field components \(B_{y}\) and \(B_{z}\) can be determined. The total magnetic field equation with the transmission lines at \(\pm y_{0}\) is then:  
\[\vec{B}(y,z)=C \begin{pmatrix}0\\ \frac{-2}{z^{2}+(y+y_{0})^{2}}+\frac{-2}{z^{2}+(y-y_{0})^{2}}\\ \frac{y+y_{0}}{z^{2}+(y+y_{0})^{2}}+\frac{y-y_{0}}{z^{2}+(y-y_{0})^{2}}\end{pmatrix} ,\quad with\quad C=\frac{I\mu_{0}}{2\pi}, \tag{3}\]  
where \(C\) is a prefactor depending on the transmission line current. Figure 5 shows the signal amplitude \(|\vec{B}(y,z)|\) around the transmission lines for \(I=700\) A and \(y_{0}=0.25\) m. Figure 6 shows the locations of equal amplitude in the \(yz\)-plane expressed by isolines. The magnetic field contributions of the other two phases are neglected because the cables are typically at least 2 m apart. Figure 5: Signal power around the transmission lines. The magnetic field-based navigation of a UAV requires that the UAV can sense the magnetic field in all spatial directions. It uses this information to determine its relative position and orientation with respect to the fixed position and orientation of the global coordinate frame \(\{G\}\). The flight capabilities allow the UAV to achieve an arbitrary orientation by rotating the UAV's body about the \(z\)- (blue), \(y\)- (green), and \(x\)-axes (red) of the local drone frame \(L_{u}\) for the respective yaw, pitch, and roll angles \(\alpha\), \(\beta\), and \(\gamma\). The origin position of the local frame \(L_{u}\) in \(\{G\}\), denoted by \(\vec{p}_{D}\), also represents the position of the UAV to be determined. The sensors have predefined positions \(\vec{p}_{i0}\) in \(L_{u}\), where the index \(i\) refers to a particular sensor. Moreover, each magnetometer has its own local coordinate frame \(L_{i}\).