# Leveraging Hybrid RF-VLP for High-Accuracy Indoor Localization with Sparse Anchors
## 2 Hybrid RF-VLP-Enabled Positioning System
### Integrated Localization Algorithm  
The overview of the localization algorithm is shown in Figure 2. The algorithm takes the RSS of both the optical and Bluetooth signals as input. After a series of computations, it outputs the global coordinates of the receiver. The optical signals are sourced from the LED anchors deployed indoors. We assume that there are M LED anchors installed on the ceiling. Each LED anchor is denoted as \(\text{LED}_{m},(m=1,\cdots,M)\). Meanwhile, N (\(N\geq 3\)) receivers are below each of the LED anchors, sample the optical signal from the LED, and each of the receivers is denoted as \(\text{Rx}_{i},(i=1,\cdots,N*M)\). Each pair of receivers is connected via a BLE link, enabling the estimation of distance between any two receivers through Bluetooth RSS. According to the Friis Equation [26], the received BLE power \(P_{br}\) is equal to  
\[P_{br}=\frac{P_{br}G_{t}G_{r}\lambda^{2}}{(4\pi d_{b})^{2}}, \tag{1}\]  
where \(P_{br}\) is the BLE transmission power, \(G_{t}\) and \(G_{r}\) are the transmitter and receiver antenna gain, and \(\lambda\) is the signal wavelength in meter. By expressing power in decibels and  
Figure 1: Components of the proposed positioning system. performing transformations, \(P_{br}\) can be represented as a logarithmic distributed random variable with a distance-dependent mean value, that is  
\[P_{br}=P_{br0}-10n_{p}\log_{10}(\frac{d_{b}}{d_{b0}})+X_{\sigma}, \tag{2}\]  
where \(P_{br0}\) is a reference power at a known distance \(d_{b0}\) from the transmitter, \(n_{p}\) is the propagation loss factor, and \(X_{\sigma}\) is a zero mean Gaussian distributed random variable with standard deviation \(\sigma\). Therefore, by collecting Bluetooth signal strength data at various distances and substituting it into Equation (2), the value of \(n_{p}\) can be estimated and used for distance mapping in the localization process. The distance can be expressed as  
\[d_{b}=10^{\frac{P_{br0}-P_{br}}{10n_{p}}}d_{b0}. \tag{3}\]  
Similarly, the distance between the LED anchor and the receiver can be estimated using the optical signal's RSS. According to the Lambert emission law [27], the received optical signal power can be expressed as  
\[P_{\sigma r}=\frac{(s+1)A}{2\pi d_{o}^{s}}\cos^{s}(\theta)\cos(\phi)P_{ot}, \tag{4}\]  
where \(A\) represents the optical sensor surface area, \(\theta\) and \(\phi\) are the irradiance and incidence angle of LED and optical sensor, respectively. \(s\) is the Lambertian order of the LED, \(P_{ot}\) is the LED's transmission power. We assume that the light sensor of the receiver is oriented vertically upward, so \(\theta\) is equal to \(\phi\). Then, Equation (4) can be simplified as  
\[P_{\sigma r}=\frac{(s+1)AP_{o}}{2\pi d_{o}^{s+3}}h^{s+1}, \tag{5}\]  
where \(h\) represents the height of the LED anchor. Take a reference power \(P_{\sigma 0}\) at a known distance \(d_{o0}\), then according to Equation (5),  
\[\frac{P_{\sigma r}}{P_{\sigma 0}}=(\frac{d_{o0}}{d_{o}})^{s+3}, \tag{6}\]  
then \(m\) can be estimated and calibrated with the measured optical RSS value at different distance. After measuring the value of \(m\), the distance between the LED and receiver can be expressed as  
\[d_{o}=\frac{d_{o0}}{s+3\sqrt{\frac{P_{br}}{P_{\sigma 0}}}}. \tag{7}\]  
Figure 2: Overview of the proposed localization algorithm. Once the distance between each pair of receivers and the distance between each receiver and the LED are estimated, the coordinates of the receivers can be calculated using the following algorithm. Let \(d_{bij}\) be the Euclidean distance between \(\text{Rx}_{i}\) and \(\text{Rx}_{j}\), then we can define a distance matrix as \(\mathbf{\Delta}=[d_{bij}]_{I\times I}\), where \(I\) represents the total number of receivers, \(d_{bii}=0\), and \(d_{bij}=d_{bji}\).