Algorithm 1 : VB-SRF.

(1) Initialization: ${\overline{\mathbf{x}}}_{0|0}$, ${\overline{\mathbf{P}}}_{0|0}$, ${\mathbf{Q}}_v$, ${\mathbf{Q}}_w$, ${\eta }_0$, ${\alpha }_{1,0}$, ${\alpha }_{2,0}$ (2) Prediction: ${\widehat{\mathbf{x}}}_{k|k-1}={\mathbf{F}}_{k-1}{\overline{\mathbf{x}}}_{k-1|k-1}+{\mathbf{u}}_{k-1}^s$ ${\mathbf{P}}_{k|k-1}={\mathbf{F}}_{k-1}{\overline{\mathbf{P}}}_{k-1|k-1}{\mathbf{F}}_{k-1}^T+{\mathbf{Q}}_v$ ${\mathbf{S}}_k={\mathbf{H}}_k{\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^T+{\mathbf{Q}}_k^m$ ${\eta }_{k|k-1}=\rho {\eta }_{k-1},{\alpha }_{1,k|k-1}=\rho {\alpha }_{1,k-1},{\alpha }_{2,k|k-1}=\rho {\alpha }_{2,k-1}$ where $\rho$ is the scale factor and $0<\rho \le 1$. (3) Update: the update of VB-SRF utilizes iterate filtering framework.  (3.a) First set: ${\overline{\mathbf{x}}}_{k|k}^{\left(0\right)}={\widehat{\mathbf{x}}}_{k|k-1}$, ${\overline{\mathbf{P}}}_{k|k}^{\left(0\right)}={\mathbf{P}}_{k|k-1}$, ${\eta }_k^{\left(0\right)}={\eta }_{k|k-1}$, ${\alpha }_{1,k}^{\left(0\right)}={\alpha }_{1,k|k-1}$, ${\alpha }_{2,k}^{\left(0\right)}={\alpha }_{2,k|k-1}$  (3.b) Calculate state estimation and its covariance using SRF when the measurement is from the target: ${\mathbf{K}}_k={\mathbf{P}}_{k|k-1}{\mathbf{H}}_k^T{\mathbf{S}}_k^{-1}$ ${\epsilon }_k={\left({\mathbf{b}}_k^T{\mathbf{S}}_k^{-1}{\mathbf{b}}_k\right)}^{-1/2}{\mathbf{b}}_k^T{\mathbf{S}}_k^{-1}({\mathbf{H}}_k{\widehat{\mathbf{X}}}_{k|k-1}+{\mathbf{u}}_k^m)$ ${\gamma }_k={\left({\mathbf{b}}_k^T{\mathbf{S}}_k^{-1}{\mathbf{b}}_k\right)}^{-1/2}{\rho }_n\left({\epsilon }_k\right)$ ${\delta }_k={\left({\mathbf{b}}_k^T{\mathbf{S}}_k^{-1}{\mathbf{b}}_k\right)}^{-1/2}[2+{\epsilon }_k{\rho }_2\left({\epsilon }_k\right)-{\rho }_2^2{\epsilon }_k]$ ${\rho }_2\left({\epsilon }_k\right)=\frac{{\epsilon }_k{e}^{-{\epsilon }_k^2/2}+\sqrt{2\pi }({\epsilon }_k^2+1)F_{normal}\left({\epsilon }_k\right)}{e^{-{\epsilon }_k^2/2}+\sqrt{2\pi }\left({\epsilon }_k\right)F_{normal}\left({\epsilon }_k\right)}$ ${\widehat{\mathbf{x}}}_{k|k}=(\mathbf{I}-{\mathbf{K}}_k{\mathbf{H}}_k){\widehat{\mathbf{x}}}_{k|k-1}-{\mathbf{K}}_k{\mathbf{u}}_k^m+{\gamma }_k{\mathbf{K}}_k{\mathbf{b}}_k$ ${\mathbf{P}}_{k|k}=(\mathbf{I}-{\mathbf{K}}_k{\mathbf{H}}_k){\mathbf{P}}_{k|k-1}+{\delta }_k{\mathbf{K}}_k{\mathbf{b}}_k{\mathbf{b}}_k^T{\mathbf{K}}_k^T$  (3.c) For $j=1:N$, iterate the following N (N denotes iterated times) steps: $•$ Calculate the fused state estimation and its covariance: ${\overline{\mathbf{x}}}_{k|k}^{\left(j\right)}=\frac{1}{2\pi c}{\eta }_k^{(j-1)}{\widehat{\mathbf{x}}}_{k|k-1}+\frac{1}{c}(1-{\eta }_k^{(j-1)})f\left({\theta }_k\right|{\mathbf{z}}_{1:k-1}){\widehat{\mathbf{x}}}_{k|k}$ ${\overline{\mathbf{P}}}_{k|k}^{\left(j\right)}=\frac{1}{2\pi c}{\eta }_k^{(j-1)}({\mathbf{P}}_{k|k}+({\widehat{\mathbf{x}}}_{k|k}-{\overline{\mathbf{x}}}_{k|k}){({\widehat{\mathbf{x}}}_{k|k}-{\overline{\mathbf{x}}}_{k|k})}^T)$ $+\frac{1}{c}(1-{\eta }_k^{(j-1)})f\left({\theta }_k\right|{\mathbf{z}}_{1:k-1})({\mathbf{P}}_{k|k-1}+({\widehat{\mathbf{x}}}_{k|k-1}-{\overline{\mathbf{x}}}_{k|k}^{\left(j\right)}){({\widehat{\mathbf{x}}}_{k|k-1}-{\overline{\mathbf{x}}}_{k|k}^{\left(j\right)})}^T)$   where $c=\frac{1}{2\pi }{\eta }_k^{(j-1)}+f\left({\theta }_k\right)(1-{\eta }_k^{(j-1)})$ is a normalization term, and $f\left({\theta }_k\right|{\mathbf{z}}_{1:k-1})$ can be obtained using (A6). $•$ Update parameters: $ln(\frac{{\eta }_k^{\left(j\right)}}{1-{\eta }_k^{(j)}})=\psi ({\alpha }_{1,k}^{(j-1)})-\psi ({\alpha }_{2,k}^{(j-1)})+ln(1/2\pi )-lnf({\theta }_k|{\overline{\mathbf{x}}}_{k|k}^{\left(j\right)})$ ${\alpha }_{1,k}^{\left(j\right)}={\alpha }_{1,k}^{(j-1)}+{\eta }_k^{\left(j\right)}$ ${\alpha }_{2,k}^{\left(j\right)}={\alpha }_{2,k}^{(j-1)}-{\eta }_k^{\left(j\right)}+1$ $•$ End for and set ${\overline{\mathbf{x}}}_{k|k}={\overline{\mathbf{x}}}_{k|k}^{\left(N\right)}$, ${\overline{\mathbf{P}}}_{k|k}={\overline{\mathbf{P}}}_{k|k}^{\left(N\right)}$, ${\eta }_k={\eta }_k^{\left(N\right)}$, ${\alpha }_{1,k}={\alpha }_{1,k}^{\left(N\right)}$, ${\alpha }_{2,k}={\alpha }_{2,k}^{\left(N\right)}$.