# Tracking Extended Targets: Novel TPMB Filter Driven by Model and Data Collaboration
## 3 Model-Driven IMM-TPMB Filter
### Implementation of an IMM-TPMB Filter Based on Gate-Control  
#### 3.2.1. Predict  
1. Undetected and newly detected targets: at time step \(k\), the PPP prediction intensity for all undetected and newly detected targets within the sensor-monitored area is as follows: \[\lambda_{k|k-1}(t,x^{1:v}) =\sum\limits_{l=1}^{L_{k|k-1}^{u}}w_{k|k-1}^{0,(l)}\delta_{l_{k|k- 1}^{\lambda_{k|l}}}[t]\delta_{k-\mu^{\delta(l)}_{k|k-1}}+[v]\delta_{x^{0,(l)}_ {k|k-1}}\left[x^{1:v}\right]\] (36) \[+\sum\limits_{l=1}^{L_{k|k-1}^{u}}w_{k}^{b,(l)}\delta_{k}[t]\delta_{1}[v] \delta_{k}^{w_{k|l}^{b,(l)}}[_{x_{k}}]\] \[\widetilde{X}_{k|k-1}^{0,\zeta_{t}^{\prime}(l)}=F(\cdot\left|\widetilde{X }_{k-1|k-1}^{0,\zeta_{t}^{\prime}(l)}\right|)\] (37) \[L_{k|k-1}^{u}=L_{k|k-1}^{u}+L^{b}\] (38) \[w_{k|k-1}^{0,\zeta_{t}^{\prime}(l)}=P^{S}(\widetilde{X}_{k-1|k-1}^{0,\zeta_ {t}^{\prime}(l)})w_{k|k-1}^{0,\zeta_{t}^{\prime}(l)}\] (39) \[w_{k|k-1}^{b,(l)}=\frac{\left\langle\lambda_{k}^{B},1\right\rangle}{L^{b}}\] (40) where the probability density of the single-target particle belief during propagation is \(f(x)=\sum\limits_{l=1}^{L}w^{(l)}\delta_{x^{(l)}}[x]\), the particle weight indexed by \(l\) is \(w^{(l)}\geq 0\), and \(\delta_{i}[u]=1\) only when \(u=i\); otherwise, it is \(0\). \(L_{k}^{b}\) indicates the number of newly detected target particles, and the target component is in the form of particles \(\left\{\left(w^{0,\zeta_{t}(l)},x^{0,\zeta_{t}^{\prime}(l)}\right)\right\}_{l=1 }^{L}\)
2. Surviving targets: for the detected surviving targets, the target state of the \(i^{\text{th}}\) MBM component at time step \(k\) is represented as follows: \[f_{k|k-1}^{i}(t,x^{1:v}) =\sum\limits_{l=1}^{L_{k|k-1}^{d}}w_{k|k-1}^{i,(l)}\delta_{x^{0, \zeta_{t}^{\prime}(l)}_{k|k-1}}[t]\delta_{x^{0,\zeta_{t}^{\prime}(l)}_{k|k-1}} [v]\delta_{x^{0,\zeta_{t}^{\prime}(l)}_{k|k-1}}[x^{1:v}]\] (41) \[+\sum\limits_{l=1}^{L_{k|k-1}^{d}}\left[1-P^{\delta}(\overline{X}_{k-1|k- 1}^{i,(l)})w_{k-1|k-1}^{i,(l)}\delta_{x^{0,\zeta_{t}^{\prime}(l)}_{k-1|k-1}} [t]\right]\] \[\times\ \delta_{k-\mu^{\delta_{t}^{\prime}(l)}_{k-1|k-1}}[v_{k-1|k-1}^{i,(l)}]\delta_{x^{\lambda_{t}^{\prime}(l)}_{k-1|k-1}}[x^{1:v}]\]\[\overline{X}_{k|k-1}^{s^{\prime}(l)}=\sum_{m=1}^{M}\mu_{k}^{s^{2}|\zeta^{\prime}}F _{k}^{s^{\prime}|\zeta^{\prime}}\overline{X}_{k-1|k-1}^{0,\zeta^{\prime},(l)} \tag{42}\]  
\[r_{k|k-1}^{i}=r_{k-1|k-1}^{i}\sum_{l=1}^{L_{k|k-1}^{d}}P^{S}\left(\overline{X}_ {k-1|k-1}^{i,\zeta^{\prime},(l)}\right)w_{k-1|k-1}^{i,\zeta^{\prime},(l)} \tag{43}\]  
\[L_{k|k-1}^{d}=L_{k-1|k-1}^{d} \tag{44}\]  
\[w_{k|k-1}^{i,\zeta^{\prime},(l)}=\frac{P^{S}(\overline{X}_{k-1|k-1}^{i,\zeta^ {\prime},(l)})w_{k-1|k-1}^{i,\zeta^{\prime},(l)}}{\sum\limits_{l=1}^{L_{k|k-1} ^{d}}p^{S}(\overline{X}_{k-1|k-1}^{i,\zeta^{\prime},(l)})w_{k-1|k-1}^{i,\zeta ^{\prime},(l)}} \tag{45}\]  
In the case of undetected and newly detected targets, there is no need to consider the target model transition probability. \(F(\cdot)\) represents the motion model transition function, which performs maneuvering target particle sampling through multi-model mixing prediction.