## 7 Sensor Data Fusion Based on TCPF
### Principle of TCPF Algorithm  
The proposed TCPF algorithm first used unscented transformation [27] to integrate the latest observation information into the proposal distribution, so that the proposal distribution was close to the real distribution of the probability density function: \(\widetilde{x}_{i}(k)\)\(\leftarrow\)\(T_{O}(x_{i(a)}(k-1))\). Then, a Gaussian mixture model was constructed and an adaptive threshold was set to reduce the operation steps of clustering similar components in the Gaussian mixture and improve the clustering efficiency. Finally, the stratified sampling proportion capacity was set, the continuous probability density function was layered, and the combination of particle weights from the inferior layer was optimized to improve particle diversity. The specific steps are as follows:  
Step 1 integrate the latest observation information into the proposal distribution. (a) Extract the particle state and covariance matrix processed by the basic particle-filter algorithm, and calculate the sigma point set:  
\[x_{i(a)}(k-1)=\left[\overline{x}_{i(a)}(k-1),\ \overline{x}_{i(a)}(k-1)\pm \sqrt{(n_{a}+\lambda)P_{i(a)}(k-1)}\right] \tag{47}\]  
where \(x_{i(a)}(k-1)\) is the sigma point set, \(\overline{x}_{i(a)}(k-1)\) is the sigma point set after the unscented transformation, \(P_{i(a)}(k-1)\) is the covariance matrix of the sigma point set, \(n_{a}\) is the dimension of the sigma point set, and \(\lambda\) is the scale parameter. (b) Integrate ZigBee-GPS/BDS multi-sensor positioning data into the obtained Sigma sampling point set, and update the system status and covariance. \[\widetilde{x}_{i}(k)=x_{i(a)}(k-1)+k_{k}\mathbf{z}(k-1) \tag{48}\]\[\widetilde{p}_{i}(k)=p_{i(a)}(k-1)-k_{k}p_{i(z(k-1))}(k-1)k_{k}^{T} \tag{49}\]  
where \(\widetilde{x}_{i}(k)\) is the update state of the system, \(\widetilde{p}_{i}(k)\) is the update covariance of the system, \(k_{k}\) is the Kalman gain, and \(p_{i(z(k-1))}(k-1)\) is the covariance matrix of the positioning data. (c) Construct a proposal distribution that is closer to the target probability function using the system state and covariance, and sample from it. \[\widehat{x}_{i}(k)\,\sim q(\boldsymbol{x}(k)\,\boldsymbol{x}(k-1),\, \boldsymbol{z}(k))=N(\widetilde{x}_{i}(k),\widetilde{p}_{i}(k)) \tag{50}\]  
where \(N(\cdot)\) is the Gaussian function. Step 2 construct a Gaussian mixture model. (a) Generate a posterior probability density function \(p_{h}(\boldsymbol{x}(k)|\boldsymbol{z}(k))\) with time step \(k\) according to the Gaussian mixture components. \[p_{h}(\boldsymbol{x}(k)|\boldsymbol{z}(k))=\sum_{i=1}^{C(k)}\xi_{i}N( \boldsymbol{x}(k)|m_{i},v_{i}) \tag{51}\]  
where \(N(\boldsymbol{x}(k)\mid m_{i},v_{i})\) is the \(i\)th component in the mixed Gaussian model, \(C(k)\) is the number of component units of discrete samples, and \(\xi\) is the number of component units for discrete samples, \(k=1,2,\ldots,T,i=1,2,\ldots,n\). (b) Integrate the discrete sampling points sampled by Equation (50) and their corresponding weight \(\{\widehat{x}_{i}(k),\,\widetilde{\omega}_{i}(k)\}\) into the Gaussian mixture component unit of Equation (51), and use the reconstructed continuous posterior probability density function \(p_{g}(\boldsymbol{x}(k)|\boldsymbol{z}(k))\) to resample the discrete particles:  
\[p_{g}(\boldsymbol{x}(k)|\boldsymbol{z}(k))=\sum_{i=1}^{n}\widetilde{\omega}_{ i}(k)N(\widehat{x}_{i}(k)|\widehat{x}_{i}(k-1),h\cdot p(k)) \tag{52}\]  
where  
\[h=0.5N^{-2/n_{x}} \tag{53}\]  
\[\ddot{p}(k)=\sum_{i=1}^{n}\widetilde{\omega}_{i}(k)\big{(}\widehat{x}_{i}(k) -\tilde{x}_{i}(k)\big{)}\big{(}\widehat{x}_{i}(k)-\tilde{x}_{i}(k)\big{)}^{T} \tag{54}\]  
\[\tilde{x}_{i}(k)=\big{(}\sum_{i=1}^{n}\widetilde{\omega}_{i}(k)\,\widehat{x} _{i}(k)\big{)}/n \tag{55}\]  
where \(\ddot{p}(k)\) is the covariance of the discrete particle filter distribution, \(\tilde{x}_{i}(k)\) is the mean value of the discrete particle filter distribution, \(h\) is the normalized constant, and \(n_{x}\) is the particle distribution dimension.