Algorithm 1. VB-DVQEKF

Input: $X_{k-1|k-1}$,${\mathit{P}}_{k-1|k-1}$,${\mathit{Z}}_k$,${\widehat{u}}_{k-1|k-1}$,${\widehat{\mathit{U}}}_{k-1|k-1}$,$Q_k$,ρ,N Update time 1.   $X_{K/K-1}={\mathsf{\Phi }}_k{X}_{k-1/k-1}$, 2.   $P_{k/k-1}={\mathsf{\Phi }}_k{P}_{k-1/k-1}{\mathsf{\Phi }}_k^T+Q_k$, 3.   ${\widehat{u}}_{k\mid k-1}={\rho }_R({\widehat{u}}_{k-1|k-1}-n-1)+n+1$, ${\widehat{\mathit{U}}}_{k|k-1}={\rho }_R{\widehat{\mathit{U}}}_{k-1|k-1}$, Iterative Measurement Update. Initialization: ${\widehat{X}}_{k|k}^0=X_{K/K-1}$, ${\widehat{\mathit{R}}}_k^{(0)}={\widehat{\mathit{U}}}_{k|k-1}/\left({\widehat{u}}_{k|k-1}-n-1\right)$, For $i=0:N-1$ update $q^{(i+1)}(X_{k|k})=\mathrm{N}(X_{k|k};{\widehat{X}}_{k|k}^{(i+1)},{\widehat{P}}_{k|k}^{(i+1)})$ 4.   Calculate $h({\widehat{X}}_{k|k}^{(i)})$ using Equations (4) and (24), 5.   $\mathrm{Calculate} {\mathit{H}}_k^{(i)}$ using Equation (5), 6.   ${\mathit{K}}_k^{(i+1)}={\mathit{P}}_{k|k-1}{({\mathit{H}}_k^{(i)})}^{\mathrm{T}}{({\mathit{H}}_k^{(i)}{\mathit{P}}_{k|k-1}{({\mathit{H}}_k^{(i)})}^{\mathrm{T}}+{\widehat{\mathit{R}}}_k^{(i)})}^{-1}$, 7.   ${\widehat{X}}_{k|k}^{(i+1)}={\widehat{\mathit{x}}}_{k|k-1}+{\mathit{K}}_k^{(i+1)}(z_k-h({\widehat{\mathit{x}}}_{k|k}^{(i)})-{\mathit{H}}_k^{(i)}(X_{k|k-1}-{\widehat{X}}_{k|k}^{(i)}))$, 8.   ${\mathit{P}}_{k|k}^{(i+1)}={\mathit{P}}_{k|k-1}-{\mathit{K}}_k^{(i+1)}{\mathit{H}}_k^{(i)}{\mathit{P}}_{k|k-1}$, update $q^{(i+1)}({\mathit{R}}_k)=\mathrm{IW}({\mathit{R}}_k;{\widehat{u}}_{k|k}^{(i+1)},{\widehat{\mathit{U}}}_{k|k}^{(i+1)})$ 9.   Calculate $h({\widehat{X}}_{k|k}^{(i+1)})$ using Equations (4) and (24), 10.   Calculate ${\mathit{H}}_k^{(i+1)}$ using Equation (5), 11.   ${\mathit{B}}_k^{(i+1)}=(Z_k-h({\widehat{X}}_{k|k}^{(i+1)})){(Z_k-h({\widehat{\mathit{X}}}_{k|k}^{(i+1)}))}^{\mathrm{T}}+{\mathit{H}}_k^{(i+1)}{\mathit{P}}_{k|k}^{(i+1)}{({\mathit{H}}_k^{(i+1)})}^{\mathrm{T}}$, 12.   ${\widehat{\mathit{U}}}_{k|k}^{(i+1)}={\widehat{\mathit{U}}}_{k|k-1}+{\mathit{B}}_k^{(i+1)}$$, {\widehat{u}}_{k|k}^{(i+1)}={\widehat{u}}_{k|k-1}+1$, 13.   ${\widehat{\mathit{R}}}_k^{(i+1)}={\widehat{\mathit{U}}}_{k|k}^{(i+1)}/\left({\widehat{u}}_{k|k}^{(i+1)}-n-1\right)$, End 14.   $X_{k|k}={\widehat{X}}_{k|k}^{(N+1)}$, ${\mathit{P}}_{k|k}={\mathit{P}}_{k|k}^{(N+1)}$, ${\widehat{u}}_{k|k}={\widehat{u}}_{k|k}^{(N+1)}$, Output: $X_{k|k}$,${\mathit{P}}_{k|k}$,${\widehat{u}}_{k|k}$,${\widehat{\mathit{U}}}_{k|k}$.