Algorithm 1: The adaptive Unscented Kalman filter (AUKF) algorithm.

Input: $f(*),h(*),{\widehat{\mathbf{x}}}_0,{\mathbf{Q}}_0,{\mathbf{R}}_1,{\widehat{\mathbf{P}}}_0,\lambda$.

1: Initialization:

2:   ${\omega }_j=(1-{\omega }_0)/2n_x$; ${\mathbf{c}}^0=\sqrt{\frac{n_x}{1-{\omega }_0}};{\mathbf{c}}^j=\sqrt{\frac{n_x}{1-{\omega }_0}}{\mathbf{r}}^j,j=1,\cdots ,2n_x$.

3: for $k=1\to K$ do

4:   Implement the standard UKF to obtain ${\widehat{\mathbf{x}}}_{k|k}$, ${{\overline{\mathbf{P}}}^{zz}}_{k|k-1}$, ${\mathbf{K}}_k$, ${\widehat{\mathbf{P}}}_{k|k}^{xx}$.

5:    Update the ${\mathbf{Q}}_{k-1}$:

6:     ${\mathit{\mu }}_k={\mathbf{z}}_k^0-h({\overline{\mathbf{x}}}_{k|k-1})$

7:     ${\mathbf{Q}}_{k-1}\leftarrow (1-\lambda ){\mathbf{Q}}_{k-1}+\lambda ({\mathbf{K}}_k{\mathit{\mu }}_k{\mathit{\mu }}_k^T{\mathbf{K}}_k^T)$;

8:    Correct state estimations:

9:     ${\widehat{\mathbf{K}}}_k\stackrel{\mathsf{\Delta }}{=}{\widehat{\mathbf{P}}}_{k|k}^{xz}{({\overline{\mathbf{P}}}_{k|k}^{zz})}^{-1}$;

10:     ${\widehat{\mathbf{x}}}_{k|k}={\widehat{\mathbf{x}}}_{k|k}+{\widehat{\mathbf{K}}}_k({\mathbf{z}}_k^0-{\widehat{\mathbf{Z}}}_{k|k})$;

11:     ${\widehat{\mathbf{P}}}_{k|k}^{xx}={\overline{\mathbf{P}}}_{k|k}^{xx}-{\widehat{\mathbf{K}}}_k{\widehat{\mathbf{P}}}_{k|k}^{xz}{({\mathbf{K}}_k)}^T$;

12:     ${\mathbf{Q}}_k\leftarrow {\mathbf{Q}}_{k-1}$, ${\mathbf{R}}_{k+1}\leftarrow {\mathbf{R}}_k$.

13:     Save the ${\widehat{\mathbf{x}}}_{k|k}$ and ${\widehat{\mathbf{P}}}_{k|k}^{xx}$.

14: end for