## 4 Analytical Implementation of VB-TPHD
### Nonlinear Implementation of VB-TPHD Filter  
The TPHD filter can provide closed solution under iteration, which can be regarded as a very effective multi-trajectory filtering method under the linear Gaussian model and can obtain the ideal effect through the Gaussian mixture form based on the Kalman filter. Now assume that both the state process and the measurement process are nonlinear Gaussian models [16]:  
\[x_{k}=\;\varphi_{k}(x_{k-1},v_{k}), \tag{57}\]  
\[z_{k}=h_{k}(x_{k},v_{k}), \tag{58}\]  
where \(\varphi_{k}(\cdot)\) and \(h_{k}(\cdot)\) are known nonlinear dynamic and measurement equations, \(v_{k}\) and \(v_{k}\) are state noise and the measurement noise of Gaussian distribution, and \(Q_{k}\) and \(R_{k}\) are covariances of state noise and measurement noise, respectively. It is not feasible to use the Gaussian mixture VB-TPHD filter to process nonlinear Gaussian models directly, but it can still be applied to nonlinear systems through an improved Kalman filter, such as extended Kalman (EK) filter for linearing locally nonlinear Gaussian systems [36] and an unscented Kalman (UK) filter that approximates nonlinear Gaussian systems using precisely selected \(\sigma\) points [37; 38]. Then, the realization of AEK-VB-TPHD and AUK-VB-TPHD is given in detail. The key steps of AEK-VB-TPHD and AUK-VB-TPHD are shown in Algorithms 1 and 2. These two filters not only adopt the L-scan method to avoid including the whole trajectories into the calculation like AGM-VB-TPHD, but also execute the operation of pruning, absorption and state extraction according to Equations (53)-(56). It is worth mentioning that because TPHD is the trajectory involved in the iterative cycle and the filtering state is multi-dimensional, the weight \(u\) of \(\sigma\) points will be negative, which leads to the negative determination of the estimated covariance matrix and the next Cholesky decomposition cannot be carried out. Therefore, SVD [38] is used for the decomposition of the matrix in AUK-VB-TPHD. ```
Step 1. (Prediction for newborn targets) Input \(z_{k}\in Z_{k}\), \(K\), \(h_{k}^{-1}\). Output :\(\{w_{\gamma_{k}}^{j},m_{\gamma_{k}}^{j},P_{\gamma_{k}}^{i}(\alpha_{\gamma_{k}} ^{1,j},\beta_{\gamma_{k}}^{1,j})_{l=1}^{d}\}_{j=1}^{l_{\gamma_{k}}}\). for \(j=1\): \(J_{\gamma_{k}}\) (number of measurements)  use Equations (32)-(34). end for Step 2. (Prediction for existing trajectories) Input \(p_{s}\), \(\Phi_{k^{\prime}}^{j}\), \(v_{k^{\prime}}^{j}\), \(\rho\), \(\{w_{\gamma_{k-1}}^{j},\hat{m}_{\gamma_{k-1}}^{j},\hat{P}_{\gamma_{k-1}}^{i},( \alpha_{\gamma_{k-1}}^{1,j},\beta_{\gamma_{k-1}}^{1,j})_{l=1}^{d}\}_{j=1}^{l_ {\gamma_{k-1}}}\). Output :\(\{w_{\gamma_{k}}^{j},\hat{m}_{\gamma_{k}}^{j},\hat{P}_{\gamma_{k}}^{i}(\alpha_ {\gamma_{k}}^{1,j},\beta_{\gamma_{k}}^{1,j})_{l=1}^{d}\}_{j=1}^{l_{\gamma_{k}}}\). for \(j=1\), \(\ldots,l_{\xi_{k}}\)  use Equations (37)-(38) to obtain \(w_{\xi_{k}}^{j}\), \(\hat{m}_{\xi_{k}}^{j}\). \(\hat{P}_{\xi_{k}}^{j}= \begin{bmatrix}\hat{p}_{\gamma_{k-1}}^{j}&\hat{p}_{\gamma _{k-1}}^{j}&\hat{F}_{k-1}^{j}\\ \hat{F}_{k-1}^{j}\hat{p}_{\gamma_{k-1}}^{j}&\hat{F}_{k-1}^{j}\hat{p}_{\gamma_{ k-1}}^{j}&\hat{F}_{k-1}^{j}\end{bmatrix} ^{T}+\hat{G}_{k-1}^{j}\mathcal{Q}_{k-1} \left(\hat{G}_{k-1}^{j}\right)^{T}\end{bmatrix}\),  Where \(\hat{F}_{k-1}^{j}= \begin{bmatrix}0_{1,\hat{\mu}_{k-1}^{j},1},1\end{bmatrix}  \otimes F_{k-1}^{j}\), \(F_{k-1}^{j}=\frac{\partial\Phi_{k}^{j}(\alpha_{\gamma_{k-1}}^{j})}{\partial \hat{\mu}_{k-1}^{j}}\bigg{|}_{\alpha_{\gamma_{k-1}}^{j}=\hat{m}_{k-1}^{j}}\), \(G_{k-1}^{j}=\frac{\partial\Phi_{k}^{j}(\alpha_{\gamma_{k-1}}^{j},\alpha_{\gamma _{k}}^{j})}{\partial\hat{\
u}_{k}^{j}}\bigg{|}_{\alpha_{\gamma_{k}}^{j}=0}\). use Equation (41) to obtain \((\alpha_{\delta_{\delta_{\delta}}}^{1,j},\beta_{\xi_{k}}^{j})_{l=1}^{d}\). Step 3.