*4.1. Position Accuracy Test*

Figure 3 depicts the available satellites and the relative dilution of the precision (RDOP) on BONE. The other stations will obtain similar indicators for the short baseline. The average available satellites for filtering in the whole day are 6 for GPS, 6 for Galileo, and 10 for GPS+Galileo. The RDOP for GPS+Galileo is 0.7228 and indicates an ideal environment for relative positioning [1]. If the available satellite number of Gallileo is less than four, the position result is obtained only by GPS.

The root mean square (RMS) and standard deviation (STD) of positioning errors on the East (E), North (N), and Up (U) components are shown in Table 2, where the positive and negative values represent improvement and degradation.

For AMC-KF, the RMS on BONE-QCLF, QCLF-ANGS and ANGS-BONE is (0.13417 m, 0.20254 m, 0.30294 m), (0.09421 m, 0.17402 m, 0.27439 m), (0.16395 m, 0.18714 m, 0.26284 m), respectively. While the RMS for DD-KF is (0.28258 m, 0.43671 m, 0.80099 m), (0.23641 m, 0.23035 m, 0.96756 m), (0.53574 m, 0.15190 m, 0.65454 m). The accuracy improvement by (+52.52%, +53.62%, +62.18%), (+60.15%, +24.45%, +71.64%), (+69.40%, −23.20%, +59.84%) is achieved for AMC-KF. Similarly, the STD on each baseline for AMC-KF is (0.10886 m, 0.18606 m, 0.30278 m), (0.08226 m, 0.16659 m, 0.26107 m) and (0.16027 m, 0.16694 m, 0.24867 m) while for DD-KF is (0.10066 m, 0.33062 m, 0.40077 m), (0.11478 m, 0.17340 m, 0.52163 m), (0.23124 m, 0.10958 m, 0.53847 m). AMC-KF ameliorates the performance by (−8.15%, +43.72%, +24.45%), (+28.33%, +3.93%, +49.95%), (+30.69%, −52.35%, +53.82%).

**Figure 3.** RODP and available satellites of BONE.

For BONE-STNY, STNY-NEWH, NEWH-BONE, the AMC-KF possesses an RMS improvement by (+55.90%, +64.82%, +85.24%), (−22.94%, +60.42%, +83.79%), (+53.43%, +82.67%, +67.43%), and an STD improvement by (+31.55%, +46.827%, +78.57%), (−11.22%, +35.33%, +53.71%) and (+47.26%, +73.50%, +66.16%), respectively.

Compared to DD-KF in each direction, the proposed AMC-KF obtains an improvement by (32.24%, 34.48%, 63.07%) on average, despite negative values existing. Thus, AMC-KF is a beneficial scheme for short baseline RTK as it retains a low-level positioning error. Particularly, the positioning errors on U are reduced by more than 60%. One possible reason for the negative values is that AMC-KF also redistributes the positioning residual since errors in different directions are coupled [57].

**Table 2.** RMS and STD improvement on ENU for BONE-QCLF-ANGS-BONE.


#### *4.2. Adaptive Strategy Test*

The variation and statistics of KBW are shown in Figure 4. The AMC-KF is proved to be effective as the KBW increases rapidly after initialization to respond to the input GNSS measurements and varies epoch by epoch. In Figure 4b, the mean and standard deviation found for (STNY-NEWH, QCLF-ANGS, NEWH-BONE, BONE-STNY, BONE- QCLF) are (16.1284, 14.9827, 20.6101, 19.1174, 13.1624, 17.0062) and (4.7120, 5.1061, 6.1030, 5.4980, 3.9002, 5.38731), respectively. Although the KBW is different from each other as all baselines are spatially separated, the similar variation trend verified that the adaptive KBW is sensitive to the environment.

**Figure 4.** KBW time series and statistics for each baseline. (**a**) Time series for each baseline; (**b**) the statistics of KBW time series.

The filter time consumption with the proposed adaptive KBW and the original fixed KBW in MCC is illustrated in Figures 5 and 6. The fixed KBW used here is set to be 1, 5, 25, and 30, as all KBW has shown in Figure 4b fall in [0, 30].

It could be found that the adaptive KBW owns smoother and more stable processing results. It means that the embedded devices and on-chip modules may benefit from power conservation [1]. For adaptive KBW, the average time consumption at each epoch is (0.0683 s, 0.0535 s, 0.0520 s, 0.0674 s, 0.0641 s, 0.0495 s) on ANGS-BONE, BONE-QCLF, BONE- STNY, NEWH-BONE, QCLF-ANGS, STNY-NEWH. While for fixed KBW (1, 5, 25, 30) are (0.0619 s, 0.0599 s, 0.0717 s, 0.0659 s), (0.0528 s, 0.0549 s, 0.0517 s, 0.0544 s), (0.0515 s, 0.0543 s, 0.0581 s, 0.0582 s), (0.0542 s, 0.0681 s, 0.0680 s, 0.0543 s), (0.0658 s, 0.0594 s, 0.0679 s, 0.0662 s) and (0.0581 s, 0.0560 s, 0.0574 s, 0.0499 s), respectively.

**Figure 5.** Time consumption with different KBW strategies on ANGS-BONE, BONE-QCLF, BONE-STNY.

**Figure 6.** Time consumption with different KBW strategies on NEWH-BONE, QCLF-ANGS, STNY-NEWH.

Treating adaptive KBW as the benchmark, the efficiency improvement is demonstrated in Figure 7. For all 24 cases, negative values (shown in nine cases) indicate a longer time consumption than the benchmark, and the positive values (shown in 15 cases) indicate the opposite results. In general, degradation exists in most cases; the calculation load increased by 6.54% in the other 15 cases, and 5 of them take 10% more time. Only three cases achieved more than a 10% improvement, and the remaining six cases averagely improved by 5.41%. The superiority of the adaptive KBW strategy is the most obvious in STNY-NEWH and BONE-STNY. Thus, the proposed AMC-KF and adaptive KBW strategy can generally improve filtering efficiency.

**Figure 7.** Time efficiency improvement for different KBW strategies.

The RMS improvement of the proposed AMC-KF compared to the original fixed KBW is shown in Figure 8. Here, the negative values mean a positioning-accuracy degradation compared to the adaptive KBW strategy.

The RMS increases significantly at least on one direction component while KBW = 1 and 5. Especially, the RMS on U deteriorated by almost six times (−584.261%) compared to the adaptive KBW. However, no significant performance fluctuation appears when the fixed KBW = 25 and 30, except for the −43.13% degradation on U (KBW = 25) and the 23.3% improvement on E (KBW = 30), which both occur on QCLF-ANGS.

Although the large KBW seems better, the improvement is hardly permitted as the increase in KBW amplifies the time consumption and positioning errors. In conclusion, the proposed AMC-KF method takes both efficiency and accuracy into account and is more progressive than the traditional methods with KBW fixed.
