# An Improved Single-Epoch Attitude Determination Method for Low-Cost Single-Frequency GNSS Receivers
## 4 Experiments and Results
### Static Experiment  
In order to assess the proposed CLAMBDA-search method with low-cost receivers in complex environments, this study utilizes three Ubbox M8T receivers and a patch antenna to form a three-baseline layout. We conducted five tests with baseline lengths of 0.3, 0.5, 1.0, 1.5 and 2.0 m in open and blocked environments. GNSS data collection was carried out near the Huangtaiji Mausoleum in Huanggu District, Shenyang, China. Figure 3 shows the antenna layout for the open-sky and degraded environment, respectively. In the open-sky environment, the antenna is placed at the corner of a flower bed, whereas it is placed under two tall trees in the degraded environment. The GNSS data collection frequency is 1 Hz. The elevation cutoff angle is set to 10\({}^{\circ}\). Each test lasts for about 12 min in the morning of 8 March 2020. Figure 4 demonstrates the antenna setup in the two tests. Base, rover 1, and rover 2 represent three Ublox receivers. \(\overset{\rightarrow}{b}_{1}=\overset{\rightarrow}{BaseRover1}\), \(\overset{\rightarrow}{b}_{2}=\overset{\rightarrow}{BaseRover2}\). \(\alpha\) is the angle between the two baselines. Figure 4 also depicts the number of tracked GPS and BDS satellites and the correspondent PDOP value. The light-green and dark-green line represents the open-sky scenario, while the red and magenta line represents the degraded scenario. We can find that the satellite number is higher with dual satellite systems, and the PDOP value is mostly below 2.5. The active satellite number changes occur in the beginning. This phenomenon is caused by the receiver cold startup on satellite tracking. All the experimental data are solved using LAMBDA (approach I), CLAMBDA (approach II), MCLAMBDA (approach III) and CLAMBDA-search (approach IV) in sequence. The ratio of LAMBDA ambiguity validation is set as three. The baseline variance of CLAMBDA is set as 0.05 m. The step size of CLAMBDA-search is set as 2.5\({}^{\circ}\). We use the average as a reference value, and the ambiguity fixing success rate and correct fixed rate are expressed as follows:  
\[P_{s_{E}}=\frac{\text{\# of fixed epochs}}{\text{total \# of epochs}},P_{s_{E}|c}=\frac{\text{\# of correctly fixed epochs}}{\text{total \# of epochs}} \tag{37}\]  
Tables 1-4 show all the results calculated by the four methods represented in \(P_{s_{E}}(P_{s_{E}|c})\), with the best solution in bold. The meaning of the rows and columns in the tables is the  
Figure 4: (a) Multi-receiver setup. (b) Number of observed satellites and PDOP values with cutoff satellite elevation of 10\({}^{\circ}\). Figure 3: Open-sky (a) and degraded (b) observation environments, recorded with Ublox M8T, patch antenna on 8 March 2020.  
same. It is worth noting that since the checks in chapter validation are common in attitude determination, this method is also used in the MCLAMBDA in this article. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Method**} & **0.3** & **0.5** & **1.0** & **1.5** & **2.0** \\ \hline \multirow{3}{*}{I} & b1 & 1.9(0.0) & 2.8(0.2) & 2.0(0.4) & 3.8(3.0) & 2.9(1.4) \\  & b2 & 3.4(0.1) & 2.8(0.1) & 1.8(0.0) & 6.2(4.5) & 2.9(0.5) \\ \cline{1-1}  & b1 & **66.9**(51.0) & 61.7(58.4) & 74.6(68.8) & 93.1(91.4) & 73.8(71.1) \\ \cline{1-1}  & b2 & **68.1**(54.8) & 42.8(42.2) & 77.6(68.8) & 86.8(78.4) & 75.9(73.3) \\ \cline{1-1}  & b1 & 55.3(**55.3**) & 50.3(50.3) & 79.1(79.1) & 83.0(83.0) & 54.5(54.5) \\ \cline{1-1}  & b2 & 57.0(57.0) & 49.5(49.5) & 78.6(78.6) & 84.6(84.6) & 54.5(54.5) \\ \cline{1-1}  & b1 & 53.7(53.3) & **67.0**(**66.8**) & **81.6**(**81.6**) & **93.8**(**93.4**) & **88.7(**86.2**) \\ \cline{1-1}  & b2 & 61.0**(**60.0**) & **56.0**(**55.8**) & **82.6**(**82.3**) & **94.7**(**94.7**) & **92.4**(**92.3**) \\ \hline \hline \end{tabular}
\end{table}