# An Improved Single-Epoch Attitude Determination Method for Low-Cost Single-Frequency GNSS Receivers
## 4 Experiments and Results
### Static Experiment
This may be caused by the limitation of requests for the same observations between baselines. The CLAMBDA-search method performs the best in both the single-system and multi-system solutions, in terms of the highest correct fixing rate. Even for the worst 1.5 m group, CLAMBDA-search can still maintain an 81.90% and 73.55% ambiguity fixing rate, and 100.00% and 96.60% correct fixed rate, which are higher than the CLAMBDA method by 11.76% and 23.61%, respectively. Obviously, the CLAMBDA-search method proposed in this paper is more resistant  
It can be preliminarily concluded that the LAMBDA method has the worst performance for the low-cost receiver SFSE, which is not recommended. The CLAMBDA method cannot eliminate wrong fixing, due to the absence of verification conditions. The MCLAMBDA method could give reliable results, but the number of solutions is small when the observation quality is not so good. The CLAMBDA-search method proposed in this study has the advantages of a high fixing rate, fewer errors, and strong robustness, and thus is more suitable for land applications in complex environments. For the worst data set, group 1.5 m in the degraded environment, the time series of the satellite number, PDOP values, C/N0 values and residuals are shown in Figure 6. Figure 6 shows the number of satellites and PDOP values of both the GPS and GPS/BDS solutions. For baseline one, the blue line is the dual-system solution and the purple dotted line is the single-system solution. For baseline two, the red line is the dual-system solution and the yellow dotted line is the single-system solution. From Figure 6--bottom panel--it can be found that for the baseline one, GPS has a better carrier-to-noise ratio than BDS. For GPS, the C/N0 of the 50% observations are greater than 38 dBHz, whereas the 50% BDS observations are just greater than 36 dBHz. As for baseline two, the carrier-to-noise ratios between GPS and BDS are similar to each other, i.e., the 50% observations are greater than 37 dBHz. Figure 6--bottom panel--also shows the observation residuals that indicate a strong multipath effect. That can explain why the SFSE's LAMBDA cannot fix ambiguities. Table 5 show the Euler angle success rate, Euler angle accuracy, and their average error and accuracy calculated by three methods. The Euler angle success rate is defined as the epoch when baseline one and two can both be fixed. The Euler angle accuracy rate is defined as the epoch when baseline one and two are both correctly fixed. Table 5 also shows that the dual-system attitude error of the CLAMBDA method is also unacceptable. The MCLAMBDA method has a small attitude error, but lower Euler success rate. However, the dual-system attitude error of the proposed CLAMBDA-search method is only 0.15\({}^{\circ}\), 0.57\({}^{\circ}\), 0.68\({}^{\circ}\), which proves its outperformance over the other three methods. Note that we cannot ignore the computational complexity for all the ambiguity searching algorithms. Table 6 shows the calculation time for each test. It should be noted that the maximum number of search times of method I/II/III is set to 10,000, and MCLAMBDA is set to 200,000 in this article. The calculation time consumed by each method only counts the searched results. For epochs that cannot be searched, each epoch ends with maximum number of search times. The elapsed times are measured at every epoch using C functions \"QueryPerformanceCounter()\" at a PC with Intel(r) Core(TM) i7-7500 CPU @ 2.70 GHz and 8192 MB RAM. Figure 6: (a) Observed satellite number, PDOP values. (b) C/N0 cumulative distribution function (CDF), residuals for group 1.5 m in the degraded environment. As shown in Table 6, the searching time of the CLAMBDA method increases with the ascending baseline length, except for the 1.5 m baseline. The 1.5 m baseline requires a much longer searching time, due to poor observation conditions and the resultant unrealistic observation model. As for the MCLAMBDA method, it takes the longest time, except for CLAMBDA-search with \(\eta=2.5^{\circ}\).