Algorithm 1 Multi-Vehicle Cooperative Target Tracking

Initialization:${\widehat{x}}_{1|1}$,$P_{1|1}$,${\left\{{\alpha }_{1|1,t},{\beta }_{1|1,t}\right\}}_{t=1}^{m_2}$

Time outer loop

For $k=2,3,\cdots$

Prediction:

${\widehat{x}}_{k|k-1}=F_k{\widehat{x}}_{k-1|k-1}$

$P_{k|k-1}=F_k{P}_{k-1|k-1}{F}_k^T+Q$

${\alpha }_{k|k-1,t}=\rho {\alpha }_{k-1|k-1,t} t=1,2,\dots ,m_2$

${\beta }_{k|k-1,t}=\rho {\beta }_{k-1|k-1,t} t=1,2,\dots ,m_2$

Correction:  $J{H}_k={\frac{\partial h\left(x\right)}{\partial x}|}_{x={\widehat{x}}_{k|k-1}}$

${\overline{H}}_k={\left[J{H}_k,H\right]}^T$

$y_k={\left[y_k^1,y_k^2\right]}^T$

${\alpha }_{k|k,t}^0={\alpha }_{k|k-1,t}$

${\beta }_{k|k,t}^0={\beta }_{k|k-1,t}$

Variational inner loop

For $i=0,1,2,\cdots$

${\langle {\mathrm{\Sigma }}_k^2\rangle }^{i+1}=\mathrm{diag}\left(\frac{{\beta }_{k|k,1}^i}{{\alpha }_{k|k,1}^i},\dots ,\frac{{\beta }_{k|k,m_2}^i}{{\alpha }_{k|k,m_2}^i}\right)$

${\mathrm{\Sigma }}_k^{i+1}=\mathrm{diag}\left({\mathrm{\Sigma }}_k^1,{\langle {\mathrm{\Sigma }}_k^2\rangle }^{i+1}\right)$

$S_k^{i+1}={\overline{H}}_k{P}_{k|k-1}{\overline{H}}_k^T+{\mathrm{\Sigma }}_k^{i+1}$

$K_k^{i+1}=P_{k|k-1}{\overline{H}}_k^T{\left(S_k^{i+1}\right)}^{-1}$

${\widehat{x}}_{k|k}^{i+1}={\widehat{x}}_{k|k-1}+K_k^{i+1}\left(y_k-{\overline{H}}_k{\widehat{x}}_{k|k-1}\right)$

$P_{k|k}^{i+1}=\left(I-K_k^{i+1}{\overline{H}}_k\right)P_{k|k-1}$

${\alpha }_{k|k,t}^{i+1}={\alpha }_{k|k-1,t}+\frac{1}{2} t=1,2,\dots ,m_2$

${\beta }_{k|k,t}^{i+1}={\beta }_{k|k-1,t}+\frac{1}{2}{(H{P}_{k|k}^{i+1}{H}^T+(y_k^2-H{\widehat{x}}_{k|k}^{i+1}){(y_k^2-H{\widehat{x}}_{k|k}^{i+1})}^T)}_{tt} t=1,2,\dots ,m_2$

Convergence judgment

Condition 1: $\frac{\Vert {\widehat{x}}_{k|k}^{i+1}-{\widehat{x}}_{k|k}^i{\Vert }_2}{\Vert {\widehat{x}}_{k|k}^i{\Vert }_2}<\epsilon$   Condition 2: $i>c$  End the inner loop

${\widehat{x}}_{k|k}={\widehat{x}}_{k|k}^{I+1},P_{k|k}=P_{k|k}^{I+1}$, $I$ is the number of VB iteration

End the outer loop

Outputs:${\left\{{\widehat{x}}_{k|k},P_{k|k}\right\}}_{k=2,3,\cdots }$,