## 2 Orientation and Localization Equations
### Position Determination  
In the previous section, it was shown that the UAV orientation and position computation can be decoupled into two independent problems. In doing so, it was shown how the UAV orientation can be computed using the zero field property and without knowledge of the transmission line currents or the magnetic field equation. This section now deals with the computation of the respective sensor positions \(\vec{p}_{i}\), which are needed to finally determine the UAV position \(\vec{p}_{D}\) via Equation (4). With the decoupling presented, the complexity of the equation system (6) can be greatly reduced, since by knowing the angles, the right side can be computed and becomes a simple real-numbered vector. With \(R_{G}^{L_{y}}R_{L_{u}}^{L_{y}}\vec{v}_{i}=\vec{V}_{i}\), the system becomes:  
\[ \begin{pmatrix}\vec{B}(\vec{p}_{1})\\ \vec{B}(\vec{p}_{2})\\ \vec{B}(\vec{p}_{3})\end{pmatrix} = \begin{pmatrix}\vec{V}_{1}\\ \vec{V}_{2}\\ \vec{V}_{3}\end{pmatrix} . \tag{11}\]  
Obviously, the computation of the respective sensor positions is also decoupled. Thus, to obtain \(\vec{p}_{D}\), it is sufficient to solve one of these rows in (11). Then, for a sensor \(i\), the corresponding row is:  
\[\vec{B}(\vec{p}_{i})= \begin{pmatrix}B_{y}(y_{i},z_{i})\\ B_{z}(y_{i},z_{i})\end{pmatrix} = \begin{pmatrix}V_{iy}\\ V_{iz}\end{pmatrix} =\vec{V}_{i}, \tag{12}\]  
which represents a system of two equations. The components \(B_{y}\) and \(B_{z}\) describe the magnetic field in two dimensions in \(\{G\}\) and were obtained by calculating the Biot-Savart integral (1). This means, on the one hand, that the two equations in (12) are linearly independent and theoretically allow the determination of the unknowns \(y_{i}\) and \(z_{i}\). On the other hand, the equations are nonlinear, and the question arises whether it is possible to solve (12) analytically and, if so, how. In this context, there are several approaches, all of which were investigated in the context of this work. Since the following considerations are equally valid for all sensors, the subscript \(i\) is omitted for simplicity. The first row in (12) can be solved for one coordinate \(y\) or \(z\), which is then inserted into the second row to compute the other coordinate; and the same in reverse. However, solving \(B_{y}=V_{y}\) for \(z\) or \(B_{z}=V_{z}\) for \(y\) in the general form leads to extremely large solutions \(z(y,V_{y})\) and \(y(z,V_{z})\), which contain root terms with eighth-order polynomials as arguments. The other two cases, i.e., solving \(B_{y}=V_{y}\) for \(y\) or \(B_{z}=V_{z}\) for \(z\), lead to smaller solutions. For example, from the latter, the four solutions for \(z\) follow:  
\[ \begin{array}{c}z(y,V_{z})=\pm\sqrt{\pm D(y^{2},y)+a(y^{2},y)},\\ \text{with}\quad D=\frac{1}{V_{z}^{2}}\sqrt{4V_{z}^{2}y^{2}_{0}-4 CV_{z}yy^{2}_ {0}+C^{2}y^{2}},\\ a=-y^{2}_{0}-y^{2}+\frac{Cy}{V_{z}}.\end{array}  \tag{13}\]  
The result \(y(z,V_{y})\) has the same structure, but is not used, as the easiest final equation can be found by inserting (13) into \(B_{y}^{2}(y,z)=V_{y}^{2}\). With \(D\) from (13), it becomes:  
\[\frac{V_{z}(Cy\pm D)^{2}(-V_{z}y_{0}^{2}-V_{z}y^{2}+Cy\pm D)}{y^{2}(-2V_{z}y_{ 0}^{2}+Cy\pm D)^{2}}=V_{y}^{2}. \tag{14}\]  
The equation can in principle be solved for \(y\). However, no suitable substitution could be found, so that the root terms disappear, and at the same time, a polynomial of a smaller order than five is obtained. For example, after the substitution with \(E=Cy\pm D\), unwanted \(y^{2}\) expressions still remain in the equation. The square roots can also be eliminated by subsequent isolation and squaring. However, this leads to a high-order polynomial. In summary, it is difficult to impossible to derive an analytical solution only from the equation system (12). In this context, the idea is to consider the signal power as an additional equation.