# A Novel Localization Technique Using Luminous Flux
## 4 Adaptive Collaboration Positioning  
As discussed previously, the DoP is used to illustrate the effect of geometry on the relationship between measurement error and position determination error. DoP has various components, such as PDoP and geometric DoP. PDoP is a subset of DoP, where the precursor RSS value for the DoP is analyzed for positional estimates. The proposed positioning algorithm for PDoP at the receiver or photodiode is shown in Figure 3.  
These PDoP factors are defined as the ratio of the statistical value of position errors to the standard deviation (SD) of measurement errors. In this paper, we focus on PDoP, which is defined as:  
\[\mathrm{PDoP}=\frac{\sqrt{\sigma_{\mathrm{x}}^{2}+\sigma_{\mathrm{y}}^{2}+ \sigma_{\mathrm{z}}^{2}}}{\sigma_{\mathrm{UEE}}} \tag{17}\]  
Figure 3: Schematic representation of the positioning algorithm based on positional dilution of precision (PDoP). Note: RSSI = received signal strength indicator. Here, UERE (User-Estimated Range Error) and is used to infer PDoP for RSS-based localization using VLC. Considering m LEDs and one receiver in the localization premises, the measuring equation for the \(i\)th LED is  
\[g_{i}=\frac{k_{i}}{r_{i}}+n_{i} \tag{18}\]  
where \(k\) is equal to  
\[\frac{m+1}{2}RnookPLEDAcos^{m}(\phi)Ts(\Psi)g(\Psi)cos(\Psi) \tag{19}\]  
where \(m\) is the Lambertain order, which is given by \(m=\frac{-ln}{cos\frac{\phi}{2}}\) here, \(\phi\) is the radiation angle with respect to each LED and n is zero-mean Gaussian distribution. Here, \(r\) is the distance between the \(i\)th LED and the receiver, which is given by  
\[r_{i}=\sqrt{\left(x_{i}-x\right)^{2}+\left(y_{i}-y\right)^{2}+\left(z_{i}-z \right)^{2}} \tag{20}\]  
where (x,y,z) denotes the coordinates of the receiver in three-dimensional (3D) space, and (x\({}_{i},\ y_{i},\ z_{i}\)) denotes the coordinates of \(i\)th LED in 3D space. A linear equation can be written in a matrix form as follows:  
\[\Delta g=H\ \Delta X \tag{21}\]  
where  
\[\Delta g=\left( \begin{array}{c}g_{1}-\hat{g}_{1}\\ g_{2}-\hat{g}_{2}\\ \vdots\\ g_{M}-\hat{g}_{M}\end{array} \right)\ \ H=\left[ \begin{array}{ccc}a_{x1}&\cdots&a_{x1} \\ \vdots&\ddots&\vdots\\ a_{xM}&\cdots&a_{zM}\end{array} \right]\ \ \Delta X=\left( \begin{array}{c}\Delta x\\ \Delta y\\ \Delta z\end{array} \right) \tag{22}\]  
When the least squares are used (n \(\geq\) 3), we can obtain the estimation of \(\Delta X\) from the following equation  
\[\Delta X=\left(H^{T}H\right)^{-1}H^{T}\ \Delta g \tag{23}\]  
where H is the RSSI measured from impulse response. H is not square, but \((H^{T}H)^{-1}\) is a non-singular and positive definite matrix. When the measurement noise is taken into consideration, we have  
\[\Delta X+dx=\left(H^{T}H\right)^{-1}H^{T}\ (\Delta g+n) \tag{24}\]  
where \(n\) is the measurement noise vector n = (n\({}_{1}\), n\({}_{2}\),... n\({}_{m}\))T, along with the position error vector, dx = (dx, dy, dz)T. Considering the noise vector, here all 'i' are independent and identically distributed (i.i.d.) random variables, and each n\({}_{i}\) is subject to zero-mean Gaussian distribution. Therefore, each dx\({}_{i}\) is also subject to zero-mean Gaussian distribution. We obtain the covariance of the positional vector and covariance of the noise vector:  
\[\mathrm{cov}(dx)=E\left[dxdx^{T}\right]=E\left[\left(H^{T}H\right)^{-1}H^{T}n \ n^{T}H\left(H^{T}H\right)^{-1}\right]=\left(H^{T}H\right)^{-1}E(nn^{T}) \tag{25}\]  
\[E\left[n_{i}n_{j}\right]=\left\{ \begin{array}{c}\sigma_{n}^{2}\ \mathrm{if}\ i=j\\ 0\ \mathrm{if}\ i\
eq j\end{array} \right.\ \ \mathrm{i},j=1,2\ldots m \tag{26}\]  
where \(\sigma_{n}^{2}\) is the variance of the measured noise. The covariance of position error vector dx can be rewritten as  
\[\mathrm{cov}(dx)=\left(H^{T}H\right)^{-1}I_{mxm}\ \sigma_{n}^{2}=\left(H^{T}H \right)^{-1}\sigma_{n}^{2} \tag{27}\]Considering the left-hand side of the equation  
\[\mathrm{cov}(\mathrm{dx})=\left[