# Modified Uncertainty Error Aware Estimation Model for Tracking the Path of Unmanned Aerial Vehicles
## 2 Tracking of UAVs Using a Kalman Filter Algorithm with Error and Uncertainty Sensitivity
### Kalman Filter Algorithm  
Consider a stochastic UAV tracking environment, which can be described by the following equation  
\[x_{l+1}=O_{l}x_{l}+\delta_{l}y_{l}, \tag{1}\]  
\[z_{l}=Lx_{l}+w_{l}\alpha_{l}, \tag{2}\]  
For \(l\geq 0\), where \(\delta_{l}\in\mathbb{S}^{o\times n}\), \(w_{l}\in\mathbb{S}^{l\times 0}\), \(O_{l}\in\mathbb{S}^{o\times o}\), \(L_{l}\in\mathbb{S}^{q\times o}\) define the exact known matrices, \(x_{l}\in\mathbb{S}^{o}\) describe the state vector, and \(y_{l}\), \(\alpha_{l}\) represent the multidimensional, uncorrelated Gaussian process with zero mean and optimized covariance, \(C_{l}=w_{l}w_{l}^{\prime}\) the corresponding measurements, and \(G_{l}=\delta_{l}\delta_{l}^{\prime}\) the noise covariance of the models.  
The update phase of the Kalman filter can be determined from the corresponding relation described in the following equations.  
\[K\gets D_{l|l-1}^{-1}, \tag{3}\]  
\[M\gets C_{l}^{-1}, \tag{4}\]  
\[w\leftarrow\delta_{|l|}-\delta_{|l|-1}, \tag{5}\]  
\[T\gets L_{l}, \tag{6}\]  
\[e\gets z_{l}-L_{l}\delta_{|l|-1}, \tag{7}\]  
where  
\[\delta_{|l|-1}=O_{l-1}\delta_{l-1|l|-1}, \tag{8}\]  
\[\hat{D}_{|l|-1}=O_{l-1}\hat{I}_{l-1|l-1}\hat{O}_{l-1}+G_{l-1}. \tag{9}\]  
The estimation problem generally depends on the solution of a regularized least square (RLS). Suppose an unknown vector \(x\in\mathbb{S}^{r}\) must be calculated from the measurement \(z=Tx+\alpha\), where \(y\in\mathbb{S}^{q}\) and \(\alpha\) represent the noise. We assume that we have a piece of known information prior \(\overline{x}\in x\) and the configuration \(w=x-\overline{x}\), \(e=z-T\overline{x}\) where  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Description** \\ \hline \(x_{l}\) & State vector \\ \(O_{l}\) & Environment matrices \\ \(\delta_{l}\) & Added noise to the model \\ \(z_{l}\) & States measurement \\ \(L_{l}\) & Measurement matrices-Observed \\ \(w_{l}\) & Estimation variance \\ \(\alpha_{l}\) & Zero mean Gaussian processes with uncorrelated covariance \\ \(y_{l}\) & Zero mean Gaussian processes with optimized covariance \\ \(\mathbb{S}\) & Different state vectors \\ \(C_{l}\) & Corresponding measurement \\ \(G_{l}\) & Noise covariance matrix \\ \(K\) & Dimensional matrices of state noise transition matrix \\ \(D_{l}\) & Measurement covariance matrix \\ \(Y\) & Prior measurement of the covariance matrix \\ \(e\) & Estimate difference \\ \(T\) & State-transition matrix \\ \(l_{l}\) & Equilibrium point \\ \(Q_{l}\), \(M_{l},N_{l},\gamma_{l}\) & Constraint used for measurement updates \\ \(V\) & square diagonal matrix \\ \(F_{l}\) & Expected trajectory \\ \(w^{*}\) & Update this estimate as an argument \\ \(J\) & Maximum added noise \\ \(a_{l+1}^{w}\) & The minimum \((0)\) and the maximum noise \((J)\) can be added to the model \\ \(\tilde{C}_{l},\hat{p}_{l},\mu_{l},\mu_{l}\) & Constraint parameter used for measuring the covariance matrix \\ sequence set \(y_{l}\), \(\alpha_{l}^{*}\), \(\alpha_{l}^{*}\), \(\alpha_{l}^{*}\), \(l\geq 0\) & represent the Gaussian with zero mean and uniform covariance \\ \hline \hline \end{tabular}
\end{table}
Table 1: Table of contents.  
defines the variance of the estimate and \(e\) the difference. The RLS problem [38; 39] can be described by the following equation.  
\[\min_{w}\Bigl{[}w^{\prime}K_{w}+(T_{w-\mathbf{e}})^{\prime}M(T_{w}-\mathbf{e}) \Bigr{]}, \tag{10}\]  
\(T\in\mathbb{S}^{e\times 0}\) and \(K=K^{\prime}\succ 0\), \(M=M^{\prime}\succ 0\) represent dimensional matrices. The optimal strategy for the above equation, when \(T\) and \(e\) are known exactly, is calculated as follows.  
\[w^{*}=\bigl{[}K+T^{\prime}MT\bigr{]}^{-1}T^{\prime}M_{e}. \tag{11}\]  
Therefore, using Equation (11), we can obtain the following KF equation  
\[\hat{x}_{|l|}=\hat{x}_{|l|-1}+Q_{l}\Bigl{(}z_{l}-L_{l}\hat{x}_{|l|-1}\Bigr{)}, \tag{12}\]  
\[Q_{l}=D_{|l|}L_{l}^{\prime}C_{l}^{-1}, \tag{13}\]  
\[D_{l|l}=\Bigl{(}D_{|l|-1}^{-1}+L_{l}^{\prime}C_{l}^{-1}L_{l}\Bigr{)}^{-1} \tag{14}\]