# A Robust Tracking Algorithm Based on a Probability Data Association for a Wireless Sensor Network
## 2 Problem Statement  
### Signal Model  
The signal transmission channel between the mobile node and the beacon nodes changes between the LOS condition and NLOS condition, where this transformation is considered a switching mode system, as shown in the two-state Markov process of Figure 1. The state vector of the mobile node is \(x(k)=\left[\begin{array}{cc}x(k)&y(k)&\dot{x}(k)&\dot{y}(k)\end{array} \right]^{T}\), where \((x(k),y(k))\) denotes the position of the mobile node and \(\left(\dot{x}(k),\dot{y}(k)\right)\) denotes the velocity of the mobile node.  
The state vector of the mobile node changes over time according to a force model:  
\[\mathbf{x}(k)=A\mathbf{x}(k-1)+C\omega(k-1);\;k=1,\;\ldots,\;K, \tag{1}\]  
Figure 1: Markov switching model. LOS: line of sight, NLOS: non-LOS.  
where, \(K\) is the number of time steps, and:  
\[A=\left[\begin{array}{cccc}1&0&\Delta t&0\\ 0&1&0&\Delta t\\ 0&0&1&0\\ 0&0&0&1\end{array}\right],C=\left[\begin{array}{cccc}\Delta t^{2}/2&0\\ 0&\Delta t^{2}/2\\ \Delta t&0\\ 0&\Delta t\end{array}\right],\]  
where, \(A\) is the state transition matrix that describes the movement of the mobile node between two consecutive time steps. \(C\) is the interference input matrix describing the mapping of the randomness and the velocity of the mobile node, and \(\Delta t\) is the sampling period. The process driving noise \(\omega(k)\) due to the acceleration of the mobile node is assumed to have a zero-mean, white Gaussian with a covariance matrix \(Q(k)\). Assume that there are \(M\) beacon nodes around the mobile node to detect the signal of the mobile node. We can let \(D(k)=\left[d_{1}(k),d_{2}(k),\ldots,d_{M}(k)\right]^{T}\) denote the range measurements based on the TOA data between the mobile node and \(M\) beacon nodes at time step \(k\). Then:  
\[D(k)=\mathbf{h}(\mathbf{x}(k))+v(k),\;k=1,\;\ldots,\;K, \tag{2}\]  
where \(\mathbf{h}(\mathbf{x}(k))=\left[h_{1}(\mathbf{x}(k)),h_{2}(\mathbf{x}(k)),\ldots,h_{M}(\mathbf{x }(k))\right]^{T}\) and the Euclidean distance between the mobile node and the \(mth\) beacon node with the position \(\left(x_{bm,m},y_{bm,m}\right)\) at time step \(k\) is:  
\[h_{m}(\mathbf{x}(k))=\sqrt{\left(x(k)-x_{bm,m}\right)^{2}+\left(y(k)-y_{bm,m} \right)^{2}},\;m=1,\;\ldots,\;M. \tag{3}\]  
The noise vector \(v(k)=\left[v_{1}(k),v_{2}(k),\ldots,v_{M}(k)\right]^{T}\) contains random variables with a variance describing Gaussian sensor noise due to the NLOS propagation. The measurement covariance matrix is:  
\[R(k)=E\Big{[}[\mathbf{v}(k)-E[\mathbf{v}(k)]]\big{[}\mathbf{v}(k)-E[\mathbf{v}(k)]\big{]}^{T} \Big{]}, \tag{4}\]  
which is defined as:  
\[R(k)=\textit{diag}\Big{[}\sigma_{1}^{2},\sigma_{2}^{2},\ldots,\sigma_{M}^{2} \Big{]}, \tag{5}\]  
where the elements \(\sigma_{m}^{2}\) in \(R(k)\) are defined as:  
\[\sigma_{m}^{2}=\left\{\begin{array}{ll}\sigma_{L}^{2}&\text{if LOS condition}\\ \sigma_{L}^{2}+\sigma_{NLOS}^{2}&\text{if NLOS condition}\end{array}\right.. \tag{6}\]  
We assume that the sensor noise variance \(\sigma_{L}^{2}\), and process covariance matrix \(Q(k)\) are known. \(\sigma_{NLOS}^{2}\) is unknown.