## 2 System Model
### Decentralized Cooperative Localization Model  
Considering a team of \(N\) firefighters, every firefighter carries the IMU, magnetometer, barometer, GNSS and UWB sensor. Because the altitude can be obtained by the differential barometer measurements, we only modelled the CL system model in a two-dimensional space. We took the position as the state vector and obtained \(\mathbf{X}_{i}= \begin{bmatrix}p_{i}^{x}&p_{i}^{y}\end{bmatrix} ^{T}\) for firefighter \(i\), where \(p_{i}^{x}\) and \(p_{i}^{y}\) represented the east and north positions. The discrete-time motion model of firefighter \(i\) was formulated as follows:  
\[\mathbf{X}_{i}(k)=\mathbf{F}_{i}(k|k-1)\mathbf{X}_{i}(k-1)+\mathbf{B}_{i}(k-1)\mathbf{U}_{i}(k-1)+ \mathbf{W}_{i}(k-1) \tag{22}\]  
where \(\mathbf{X}_{i}(k-1)\) and \(\mathbf{X}_{i}(k)\) are the positions at time \(k-1\) and \(k\) respectively, \(\mathbf{F}_{i}(k|k-1)=\mathbf{I}_{2\times 2}\) is the state transition matrix, \(\mathbf{B}_{i}(k-1)=\mathbf{I}_{2\times 2}\) is the input matrix, \(\mathbf{U}_{i}(k-1)= \begin{bmatrix}\Delta p_{i}^{x}(k-1)&\Delta p_{i}^{y}(k-1) \end{bmatrix} ^{T}\) is the input vector consisting of position changes at time \(k-1\), and \(\Delta p_{i}^{x}(k-1)\), \(\Delta p_{i}^{y}(k-1)\) are the east and north position changes at time \(k-1\) respectively, \(\mathbf{W}_{i}(k-1)\) is the system noise with a zero mean Gaussian distribution. In the study, two kinds of measurements were considered. One was that the commander used GNSS to obtain the absolute position so as to understand the characteristics of buildings or forests using the map of the Geographic Information System. The absolute measurement equation can be formulated as follows:  
\[\mathbf{Z}_{i}(k)=\mathbf{H}_{i}(k)\mathbf{X}_{i}(k)+\mathbf{V}_{i}(k) \tag{23}\]  
where \(\mathbf{Z}_{i}(k)\) is the position obtained by GNSS at time \(k\), \(\mathbf{H}_{i}(k)=\mathbf{I}_{2\times 2}\) is the identity matrix, and \(\mathbf{V}_{i}(k)\) is the measurement noise with a zero mean Gaussian distribution. Figure 3: The difference between the received power and signal power in the first path. The relative measurement was the UWB ranging among firefighters and the relative measurement model between firefighter \(i\) and \(j\), which can be formulated as follows:  
\[ \begin{array}{ll}\mathbf{Z}_{ij}(k)&=h_{ij}(\mathbf{X}_{i}(k),\mathbf{X}_{j}(k ))+\mathbf{V}_{ij}(k)\\ &=\sqrt{\left(p_{i}^{x}(k)-p_{j}^{x}(k)\right)^{2}+\left(p_{i}^{y}(k)-p_{j}^{y} (k)\right)^{2}}+\mathbf{V}_{ij}(k)\end{array}  \tag{24}\]  
where \(\mathbf{Z}_{ij}(k)\) is the range measurement of firefighter \(i\) to firefighter \(j\) at time \(k\), \(h_{ij}(\mathbf{X}_{i}(k),\mathbf{X}_{j}(k))\) is the relative measurement function, \(p_{j}^{x}(k)\) and \(p_{j}^{y}(k)\) are the east and north positions of firefighter \(j\), and \(\mathbf{V}_{ij}(k)\) is the ranging noise with zero mean Gaussian white distribution. The purpose of this distributed algorithm is to spread the calculation among individual nodes, eliminating the need to maintain a central node. The distributed algorithm using EKF consists of three parts in the CL model, including no measurement update, the absolute measurement update, and the relative measurement update.