## 3 Pre-Processing of Multi-Sensor Data from the Unmanned Ship  
### System Modeling  
#### 3.1.1 Measurement Equation of GPS/BDS  
The measurement equation of GPS/BDS consists of nonlinear pseudorange measurement equations [14]. The measurement equations of the BDS pseudorange \(p^{\text{B}}\) and the GPS pseudorange \(p^{\text{G}}\) are expressed as follows:  
\[\begin{array}{l}p_{i}^{\text{B}}(k)=\left|r(k)-r_{i}^{\text{B}}(k)\right|+c \delta t_{\text{B}}+\varepsilon_{i}(k)+v_{i}(k)\\ =\sqrt{\left(X^{i}(k)-X_{i}^{\text{B}}(k)\right)^{2}+\left(Y^{i}(k)-Y_{i}^{ \text{B}}(k)\right)^{2}+\left(Z^{j}(k)-Z_{i}^{\text{B}}(k)\right)^{2}}\\ +\delta t_{\text{B}}+\varepsilon_{i}(k)+v_{i}(k)\end{array} \tag{1}\]  
\[\begin{array}{l}p_{j}^{\text{G}}(k)=\left|r(k)-r_{j}^{\text{G}}(k)\right|+c \delta t_{\text{G}}+\varepsilon_{j}(k)+v_{j}(k)\\ =\sqrt{\left(X^{j}(k)-X_{j}^{\text{G}}(k)\right)^{2}+\left(Y^{j}(k)-Y_{j}^{ \text{G}}(k)\right)^{2}+\left(Z^{j}(k)-Z_{j}^{\text{G}}(k)\right)^{2}}\\ +\delta t_{\text{G}}+\varepsilon_{i}(k)+v_{i}(k)\end{array} \tag{2}\]  
where \(X^{j}(k)\), \(Y^{i}(k)\) and \(Z^{j}(k)\) are the position coordinates of the receiver at time \(k\); \(X_{i}^{\text{B}}(k)\), \(Y_{i}^{\text{B}}(k)\) and \(Z_{i}^{\text{B}}(k)\) are the position coordinates of the \(i\)th BDS satellite at time \(k\); \(X_{j}^{\text{G}}(k)\), \(Y_{j}^{\text{G}}(k)\) and \(Z_{j}^{\text{G}}(k)\) are the position coordinates of the \(j\)th GPS satellite at time \(k\); \(r(k)\) is the position vector of the receiver; \(r_{i}^{\text{B}}(k)\) and \(r_{j}^{\text{G}}(k)\) are the position vectors of the \(i\)th BDS and the \(j\)th GPS satellite at time \(k\), respectively; \(p_{i}^{\text{B}}(k)\) and \(p_{j}^{\text{G}}(k)\) are the pseudoranges of the \(i\)th BDS and the \(j\)th GPS satellite at time \(k\), respectively; \(\delta t_{\text{B}}\), \(\delta t_{\text{G}}\) is the error of the BDS- and GPS-receiving clock, respectively; \(\varepsilon_{i}(k)\) is the non-white noise error of channel \(i\) at time \(k\); \(v_{i}(k)\) is the measurement  
Figure 3: Particle filter-based multi-sensor data fusion positioning framework for unmanned ships.  
noise of channel \(i\) at time \(k\), \(i=1\), \(2\), \(\ldots\), \(n_{1}\), \(j=1\), \(2\), \(\ldots\), \(n_{2}\), \(k=1\), \(2\), \(\ldots\), \(T\). \(n_{1}\) and \(n_{2}\) are the number of BDS and GPS satellites received, respectively. \(T\) is the maximum duration.  
When the signals of \(n\) satellites (\(n=n_{1}+n_{2}\geq 5\)) are received, the observation variables of the system are  
\[\mathbf{z}^{\text{B,G}}(k)=\left[p_{1}^{\text{B}}(k),p_{2}^{\text{B}}(k),\cdot\cdot \cdot,p_{n_{1}}^{\text{B}}(k),\ p_{1}^{\text{G}}(k),p_{2}^{\text{G}}(k),\cdot \cdot\cdot,p_{n_{2}}^{\text{G}}(k)\right]^{T} \tag{3}\]  
Assuming that the number of GPS and BDS satellites observed at time \(k\) are equal (namely \(n_{1}=n_{2}\)), the observation equation of the GPS/BDS satellite positioning system is established as  
\[\mathbf{z}^{\text{B,G}}(k)=\left[\begin{array}{c}\mathbf{p}^{\text{B}}(k)\\ \mathbf{p}^{\text{G}}(k)\end{array}\right]+R(k) \tag{4}\]  
where \(\mathbf{p}^{\text{B}}=\left\{p_{1}^{\text{B}},p_{2}^{\text{B}},\cdot\cdot,p_{n1}^{ \text{B}}\right\}\) is the BDS pseudorange, \(\mathbf{p}^{\text{G}}=\left\{p_{1}^{\text{G}},p_{2}^{\text{G}},\cdot\cdot\cdot,p_ {n2}^{\text{G}}\right\}\) is the GPS pseudorange, and \(R\) is the covariance matrix of observation noise.