Algorithm 1. ARUKF-MSIF algorithm

$\mathrm{Initiation}: \mathrm{initiate} \mathrm{the} l$$\mathrm{sensors}’ \mathrm{filters} \mathrm{with} {\widehat{X}}^i{}_0$$, P_0^{ii}$$, P_0^{ij}$$, \alpha$$, \mathrm{and} {\chi }_{m,\alpha }^2$;

Step 1: State prediction through (5)–(7) for each sensor.

Step 2: Observation prediction through (7) and (8) for each sensor.

Step 3: Estimate matrix $R^i$ for each sensor through (12) to (17).

Step 4: Process-error judgment through (20) and (54).

Step 5: Abnormal innovation distinguishing.

5.1 If (20) and (54) holds:

5.1.1 Go to step 6.

5.2 Else:

5.2.1 Adapt $Q$ through (28).

Step 6: Calculate Kalman gain and filtering through (9) and (10).

Step 7: MSIF implementation.

7.1 Calculate $P_{}^{ij}$ through (49).

7.2 Calculate matrix weights $\overline{A}$ through (48) or (56), generate the optimal estimation ${\widehat{X}}_{op}$.

Step 8: For the next iteration, repeat steps from 1 to 7.