Algorithm 1. The specific steps of SVD-ADCKF.

Input:${\mathit{X}}_{\mathit{k}}$, ${\mathit{P}}_{\mathit{k}}$, ${\mathit{Q}}_{\mathit{k}}$

Output:${\mathit{X}}_{\mathit{k}+1}$, ${\mathit{P}}_{\mathit{k}+1}$, ${\mathit{Q}}_{\mathit{k}+1}$

Nonlinear time update

1. Initialize state estimation $X_k$ and covariance estimation $P_k$;

2. Perform SVD decomposition on $P_k$ to obtain $S_k$;

3. Calculate the cubature point $X_k^j$ according to the equation: $\left\{\begin{array}{cc}{X}_k^j=X_k+S_k{I}_j\sqrt{n}& ,j=1,\dots ,n\\ X_k^j=X_k-S_k{I}_j\sqrt{n}& ,j=n+1,\dots ,2n\end{array}\right.$;

4. Calculate the one-step prediction of each cubature point ($X_{k+1|k}^j$) based on the state equation and

the process noise covariance ($Q_k$);

5. Combine each cubature point to obtain the linear approximation time update $X_{k+1|k}$: $$X_{k+1|k}={\sum }_{j=1}^{2n}{\omega }_j{X}_{k+1|k}^j$$

quad where n is system order; ${\omega }_j$ is the weight of each cubature point;

6. Calculate the time update of the linearly approximated error covariance $P_{k+1|k}$: $$P_{k+1|k}={\sum }_{j=1}^{2n}{\omega }_j(X_{k+1|k}^{}-X_{k+1|k}^{}){(X_{k+1|k}^{}-X_{k+1|k}^{})}^T;$$

Linear measurement update

7. Obtain the linear prediction value of the measurement ($Z_{k+1|k}$) based on the time update of

the measurement equation to the linear approximation;

8. Calculate Kalman gain K based on the formula: $K=P_{k+1|k}{H}^T{(H{P}_{k+1|k}{H}^T+R_k)}^{-1}$;

9. Update the system state based on the Kalman gain: $X_{k+1}=X_{k+1|k}+K(Z_{k+1}-Z_{k+1|k})$;

10. Update the system error covariance based on the Kalman gain: $P_{k+1}=P_{k+1|k}-KH{P}_{k+1|k}$;

Process noise covariance update

11. Calculate the residual error: $e_{k+1}=X_{k+1}-X_{k+1|k}$;

12. Calculate the correction matrix: $Q_{\mathrm{k}+1}^{*}=e_{k+1}{e}_{k+1}^T+P_{k+1|k}-P_{k+1}-Q_k$;

13. Calculate the window width M based on Formulas (10) and (11);

14. Correct the process noise covariance: $Q_{k+1}=Q_k+\frac{(Q_{k+1}^{*}-Q_k)}{\mathrm{M}}$;

15. Output $X_{k+1}$, $P_{k+1}$, $Q_{k+1}$ as the initial value for the next moment, and repeat Steps 1–14.

End.