## 4 Analytical Implementation of VB-TPHD
### Nonlinear Implementation of VB-TPHD Filter
(Update for all trajectories) Input \(p_{D}\),\(l_{2}\),\(l_{3}\),\(\{w_{\gamma_{k}}^{j},\hat{m}_{\gamma_{k}}^{j},\hat{m}_{\gamma_{k}}^{j},\hat{\mu}_{ \omega_{k}}^{j},(\alpha_{\omega_{k}}^{1,j},\beta_{\omega_{k}}^{1,j})_{l=1}^{d _{\gamma_{k}}=l_{\gamma_{k}}+l_{\gamma_{k}}}\)(combine Output of Step 1. \(\{w_{\gamma_{k}}^{j},\hat{m}_{\gamma_{k}}^{j},\hat{\rho}_{\gamma_{k}}^{j},( \alpha_{\gamma_{k}}^{1,j},\beta_{\gamma_{k}}^{1,j})_{l=1}^{d}\}_{j=1}^{l_{\gamma _{k}}=(l_{\gamma_{k}}+1)L_{\gamma_{k}}}\). (Update for undetected trajectories)  for \(j=1\), \(\ldots,l_{\omega_{k}}\) \(w_{\gamma_{k}}^{j}=(1-p_{D})w_{\omega_{k}^{j}}^{j}\), \(\hat{m}_{\gamma_{k}}^{j}=\hat{m}_{\omega_{k}}^{j}\), \(\hat{p}_{\gamma_{k}}^{j}=\hat{p}_{\omega_{k}}^{j}\), \(\hat{a}_{\gamma_{k}}^{1,j}=\hat{a}_{\omega_{k}}^{1,j}\), \(\hat{p}_{\gamma_{k}}^{1,j}=\hat{p}_{\omega_{k}}^{


```
Set \(\hat{m}_{\tau_{k}}^{j}=\hat{m}_{\tau_{k}}^{j,N}\), \(\hat{\rho}_{\tau_{k}}^{j}=\hat{\rho}_{m}^{j,N}\), \(\hat{\rho}_{\tau_{k}}^{Lj}=\hat{\rho}_{\tau_{k}}^{j,N}\). \(Q^{j}(z_{k})=N\left(z_{k};\hat{H}_{\omega_{k}}^{j}\hat{m}_{k^{\prime}}^{j},\hat{S }_{k}^{j,n}\right)\), \(w_{\tau_{k}}^{j+l_{\omega_{k}}+j}=p_{D}w_{\omega_{k}}^{j}Q^{j}(z_{k})\). end for \(w_{\tau_{k}}^{l_{\omega_{k}}+j}=\frac{w_{\tau_{k}}^{l_{\omega_{k}}+j}}{\lambda _{\omega}V+\sum_{k=1}^{k}w_{\tau_{k}}^{l_{\omega_{k}}+j}}\), for \(j=1,\ldots,l_{\omega_{k}}\) end for
```
**Algorithm 1**\(Cont\). ```
Step 1. (Prediction for newborn targets)  follow Step 1 of **Algorithm 1**. Step 2. (Prediction for existing trajectories)
```
Input:\(p_{S}\), \(\Phi_{k^{\prime}}^{j}\), \(v_{k^{\prime}}^{j}\), \(\theta\), \(\kappa\), \(\theta\), \(\
u\), \(\{w_{\tau_{k-1}}^{j},\pi_{\tau_{k-1}}^{j},\rho_{\tau_{k-1}}^{j},(\alpha_{\tau_{k -1}}^{1,j},\xi_{\tau_{k-1}}^{1,j})_{l=1}^{l_{\eta_{k-1}}}\}_{j=1}^{l_{\eta_{k-1}}}\). Output:\(\{w_{\xi_{k}}^{j},\hat{m}_{\xi_{k}}^{j},\hat{p}_{\xi_{k}}^{j},(\alpha_{\xi_{k}}^{1,j}, \xi_{\tau_{k}}^{1,j})_{l=1}^{l_{\xi_{k}}}\}_{j=1}^{l_{\xi_{k}}}\) for \(j=1,\ldots,l_{\xi_{k}}\) \(w_{\xi_{k}}^{j}=p_{\eta}w_{\xi_{k}}^{j},\hat{m}_{\xi_{k-1}}^{j}=\left[ \begin{matrix} m_{\tau_{k-1}}^{j}\\ 0\end{matrix} \end{matrix}\right]\), \(\hat{p}_{\xi_{k-1}}^{j}= \begin{bmatrix}p_{\eta_{k-1}}^{j}&0\\ 0&\mathcal{Q}_{k-1}\end{bmatrix} \) \(\left\{\hat{m}_{\xi_{k-1}}^{j},\hat{p}_{\xi_{k-1}}^{j},\,\alpha,\,\beta,\,k\right\}\). for \(s=0,\ldots,2 NUM\). \(\hat{m}_{\xi_{k-1}}^{j}=\sum_{s=0}^{2n}\mu_{\xi_{k-1}}^{j}\Phi_{k}(\hat{m}_{ \xi_{k-1}}^{j},v_{k})\), \(\hat{X}_{\xi_{k-1}}^{j}=\sum_{s=0}^{2n}\mu_{\xi_{k-1}}^{j}\hat{m}_{\xi_{k-1}}^{ j}-\Phi_{k}^{j}(\hat{m}_{\xi_{k-1}}^{j},v_{k})\), \(\hat{m}_{\xi_{k-1}}^{j}=\left[\left(\hat{m}_{\mu_{k-1}}^{j}\right)^{T},\left( \hat{m}_{\xi_{k-1}}^{j}\right)^{T}\right]^{T}\), \(\hat{p}_{\xi_{k}}^{j}=\left[\hat{p}_{\tau_{k-1}}^{j}\hat{p}_{\tau_{k-1}}^{j} \hat{x}_{\xi_{k-1}}^{j}\text{diag}(u_{\xi_{k-1}}^{j})\left(\hat{X}_{\xi_{k-1}} ^{j}\right)^{T}\right]\), where \(\hat{F}_{k-1}^{j}=\left[0_{1_{\xi_{k-1}}^{j}-1},1\right]\otimes\Phi_{k^{\prime}}\). use Equation (41) to obtain \((\alpha_{\xi_{k}}^{Lj},\hat{p}_{\xi_{k}}^{Lj})_{l=1}^{d}\). end for Step 3. (Update for all trajectories)
```
**Algorithm 2**Prediction and Update for the AUK-VB-TPHD Filter. ```
Step 1. (Prediction for newborn targets)  follow Step 1 of **Algorithm 1**. Step 2. (Prediction for existing trajectories)
```  
**Algorithm  
**Remark 1**. : _Noted that AUK-VB-TPHD is more suitable to non-differentiable nonlinear models than AEK-VB-TPHD, which avoids the complicated and error-prone Jacobian matrix calculation._