## 2 Theoretical Background for Spherical Localization Based on Attenuation of EM Waves
### Extended Kalman Filtering  
In the following, an EKF algorithm is derived to estimate the position state vector \(\mathbf{p}\) based on the RSS measurements. This approach allows for tightly-coupled sensor fusion, with, e.g., an IMU or a pressure sensor. We include the observation vector \(\tau\) from (8) directly into the EKF algorithm. During the filter update step, each measured RSS value is processed independently. The loss of one or more RSS values still results in a position update along the remaining single spheres' manifolds. This allows for optimal exploitation of the available information, and it improves the robustness of the system, because glitches in the EM measurement system can be treated by the EKF in a systematic way. The position state update is performed in every time step. In the following, superscripts (\(-\)) and (\(+\)) denote a value gained in the filter prediction or the filter update step, respectively.  
We set the receiver's state dynamics to be a random walk. Thus, the receiver's state at time step \(k\) reads:  
\[\mathbf{p}(k)=\mathbf{p}(k-1)+\mathbf{w}(k). \tag{10}\]  
The process noise vector \(\mathbf{w}\) is assumed to be zero-mean Gaussian white noise with covariance matrix \(\mathbf{Q}\). The non-linear measurement function \(\boldsymbol{\mu}(\mathbf{p})\) (7) defines a vector of RSS mean values as functions of the receiver position. The components \(\text{RSS}_{i}\) introduced by the URSM in (2) are updated by RSS measurements (8). The EKF requires the Jacobian \(\mathbf{J}_{\mu}(k)\) of \(\boldsymbol{\mu}(\mathbf{p})\), which consists of the derivatives \(\mathbf{h}_{i}^{\top}(k)\) of (2):  
\[\mathbf{h}_{i}^{\top}(k)=\
abla_{\mathbf{p}}\,\text{RSS}_{i}(\mathbf{p}(k))= \left[-\frac{20}{\ln 10\cdot\|\mathbf{p}(k)-\mathbf{r}_{i}\|^{2}}\cdot[\mathbf{p}(k)- \mathbf{r}_{i}]^{\top}-20\,\alpha_{i}\,\log_{10}\text{e}\frac{[\mathbf{p}(k)- \mathbf{r}_{i}]\top}{\|\mathbf{p}(k)-\mathbf{r}_{i}\|}\,\right]_{\mathbf{p}= \hat{\mathbf{p}}^{(-)}(k)},\]  
\[\mathbf{J}_{\mu}(k)=\left[\mathbf{h}_{1}^{\top}(k)\;\cdots\;\mathbf{h}_{N}^{ \top}(k)\right]^{\top}.\]  
The predicted position state is:  
\[\mathbf{\hat{p}}^{(-)}(k)=\mathbf{\hat{p}}^{(+)}(k-1)+\mathbf{w}(k).\]  
and its covariance reads:  
\[\mathbf{\hat{p}}^{(-)}(k)=\mathbf{\hat{p}}^{(+)}(k-1)+\mathbf{Q}.\]  
Figure 6: Normalized CRLB plot for a linearly-increasing measurement covariance function. Black markers depict the beacons.  
The innovation and the innovation covariance are computed as:  
\[\kappa(k)=\boldsymbol{\tau}(k)-\boldsymbol{\mu}(\hat{\boldsymbol{p}}^{(-)}(k))\]  
and:  
\[\mathbf{S}(k)=\mathbf{J}_{\mu}(k)\hat{\boldsymbol{P}}^{(-)}(k)\mathbf{J}_{\mu}^ {\top}(k)+\mathbf{C}(k)\]  
respectively.  
The Kalman gain:  
\[\mathbf{K}(k)=\hat{\boldsymbol{P}}^{(-)}(k)\,\mathbf{J}_{\mu}(k)\,\mathbf{S}^{ -1}(k)\]  
allows one to compute the state update:  
\[\hat{\boldsymbol{p}}^{(+)}(k)=\hat{\boldsymbol{p}}^{(-)}(k)+\mathbf{K}(k) \boldsymbol{\kappa}(k)\]  
and the covariance update:  
\[\hat{\boldsymbol{P}}^{(+)}(k)=\left(\mathbf{I}-\mathbf{K}(k)\,\mathbf{J}(k) \right)\hat{\boldsymbol{P}}^{(-)}(k).\]