## 2 Materials and Methods
### Effects of Heading Precision on the Positional Precision of a UAS-Lidar Point Cloud
\begin{array}{c}\sin(\psi)\cos(p)\\ \sin(\psi)\sin(p)\sin(r)+\cos(\psi)\cos(r)\\ \sin(\psi)\sin(p)\cos(r)-\cos(\psi)\sin(r)\end{array} \right]^{T}\cdot\left[ \begin{array}{c}\rho\cdot\sin(\alpha)+l_{x}\\ \rho\cdot\sin(\theta)\cos(\alpha)+l_{y}\\ \rho\cdot\cos(\theta)\cos(\alpha)+l_{z}\end{array} \right]\right| \tag{9}\]  
\[\sigma_{Y_{p}}=\sigma_{\psi}\left|\left[ \begin{array}{c}\cos(\psi)\cos(p) \\ \cos(\psi)\sin(p)\sin(r)-\sin(\psi)\cos(r)\\ \cos(\psi)\sin(p)\cos(r)+\sin(\psi)\sin(r)\end{array} \right]^{T}\cdot\left[ \begin{array}{c}\rho\cdot\sin(\alpha)+l_{x}\\ \rho\cdot\sin(\theta)\cos(\alpha)+l_{y}\\ \rho\cdot\cos(\theta)\cos(\alpha)+l_{z}\end{array} \right]\right| \tag{10}\]  
\[\sigma_{Z_{p}}=0 \tag{11}\]  
where \(\sigma_{X_{p}}\), \(\sigma_{Y_{p}}\), and \(\sigma_{Z_{p}}\) are the precisions in the \(X\), \(Y\), and \(Z\) coordinates, respectively, for an observed point within the north-east-down local level frame based only on the precision of the heading estimate. Equation (11) illustrates that heading precision has no effect on the precision of the \(Z\) (vertical) coordinate of an observed point. If it is assumed that the \begin{table}
\begin{tabular}{c c c c c} \hline  & **SPS 1** & **RTK** & **PP-RTX 2** & **Post-Processed** \\ \hline Position (m) & 1.5–3.0 & 0.02–0.05 & 0.03–0.06 & 0.02–0.05 \\ Velocity (m/s) & 0.05 & 0.02 & 0.015 & 0.015 \\ Roll and Pitch (deg) & 0.04 & 0.03 & 0.025 & 0.025 \\ True Heading (deg) & 0.15 & 0.10 & 0.08 & 0.080 \\ \hline \end{tabular}
\end{table} 
Table 2: Applanix APX-18 UAV performance specifications and root mean square error (Source: [16]). UAS-lidar system is operated while nominally level (i.e., \(r\approx 0^{\circ},\ p\approx 0^{\circ}\)), Equations (9) and (10) simplify and are now expressed by Equations (12) and (13), respectively. \[\sigma_{X_{p}}=\sigma_{\psi}\left|\sin(\psi)(\rho\cdot\sin(\alpha)+l_{x})+\cos (\psi)\left(\rho\cdot\sin(\theta)\cos(\alpha)+l_{y}\right)\right| \tag{12}\]  
\[\sigma_{Y_{p}}=\sigma_{\psi}\left|\cos(\psi)(\rho\cdot\sin(\alpha)+l_{x})-\sin (\psi)\left(\rho\cdot\sin(\theta)\cos(\alpha)+l_{y}\right)\right| \tag{13}\]  
Equations (12) and (13) illustrate that the \(l_{z}\) lever-arm offset, under the assumption of nominally level operations, has no effect on the positional precision of an observed point based only on the precision of the heading estimate. As a result, it is commonly recommended to position the origin of the lidar sensor in line with the vertical axis of the navigation reference frame of the GNSS/INS sensor (i.e., \(l_{x}=0\) and \(l_{y}=0\)). A similar argument can be made for the alignment of the primary GNSS antenna with respect to the GNSS/INS sensor. To represent the horizontal precision of an observed point, the distance root mean square measure (\(dRMS\)) can be used. Contrary to one-dimensional statistics, there is no fixed probability level for this measure as the probability depends on the ratio of the \(X\) and \(Y\) coordinate standard deviations [25]. The \(dRMS\) measure is expressed by:  
\[dRMS_{\psi}=\sqrt{{\sigma_{X_{p}}}^{2}+{\sigma_{Y_{p}}}^{2}} \tag{14}\]  
where \(dRMS_{\psi}\) is the horizontal precision of an observed point with respect to the precision of the heading estimate. If it is assumed that the \(l_{x}\) and \(l_{y}\) lever-arm offsets are zero and Equations (12) and (13) are substituted into Equation (14), the \(dRMS_{\psi}\) measure is then expressed by:  
\[dRMS_{\psi}=\sigma_{\psi}\cdot\rho\cdot\sqrt{1-\cos^{2}(\alpha)\cos^{2}(\theta)} \tag{15}\]  
Equation (15) illustrates the relationship between lidar observations, GNSS/INS heading precision, and the horizontal precision of an observed point under level operating conditions (Figure 5). The heading precision and lidar range observation are the most significant terms within the relationship and are directly proportional to the horizontal precision of an observed point.