## 2 Materials and Methods
### Effects of Heading Precision on the Positional Precision of a UAS-Lidar Point Cloud  
UAS-lidar mapping systems that directly georeference lidar observations require high-accuracy trajectory data to produce high-accuracy point clouds. Understanding how trajectory errors propagate to the positional accuracy of a UAS-lidar point cloud is an important consideration for hardware selection and mission planning. Most GNSS/INS sensor manufacturers report the performance specifications of their products in terms of trajectory errors as shown in Table 2 for the APX-18 performance specifications. Unlike trajectory position errors which directly propagate into positional errors in a point cloud (i.e., coordinate translations), orientation errors propagate into a point cloud in non-linear ways based on the range and direction from the lidar sensor to an observed point. As a result, evaluating the mapping accuracy of a GNSS/INS sensor based on the manufacturer's specifications can be difficult. Figure 4: Misalignment of the forward axis of the navigation reference frame and the GNSS velocity vector for (**a**) fixed-wing UAS and (**b**) multirotor UAS. (**c**) Theoretical heading precision as a function of GNSS-derived velocity. Several published works have rigorously analyzed the effects of trajectory errors on point cloud coordinate accuracies for both UAS-based (e.g., [14]) and occupied aircraft-based laser scanning systems (e.g., [22; 23; 24]). Thus, a full error analysis is not repeated here; only the effect of heading precision on the positional precision of a point cloud is discussed in order to provide the theoretical foundation for the experiment of this study. Equation (8) expresses the 3D coordinates of a point (vectors with the subscript \(p\)) observed by a UAS-lidar system within a north-east-down local level reference frame (superscript \(l\)). The lidar sensor reference frame is denoted by the superscript \(s\) and the GNSS/INS sensor reference frame is denoted by the superscript \(n\). \[\left[ \begin{array}{c}X\\ Y\\ Z\end{array} \right]_{p}^{l}=\mathbf{R_{3}}(\psi)\mathbf{R_{2}}(p)\mathbf{R_{1}}(r)\left( \mathbf{R_{s}^{n}}\cdot\rho\left[ \begin{array}{c}\sin(\alpha)\\ \sin(\theta)\cos(\alpha)\\ \cos(\theta)\cos(\alpha)\end{array} \right]_{p}^{s}+\left[ \begin{array}{c}l_ {x}\\ l_{y}\\ l_{z}\end{array} \right]^{n}\right)+\left[ \begin{array}{c}X\\ Y\\ Z\end{array} \right]_{IMU}^{l} \tag{8}\]  
where \(\mathbf{R_{1}}\), \(\mathbf{R_{2}}\), and \(\mathbf{R_{3}}\) represent the fundamental rotations around the \(x\), \(y\), and \(z\)-axes of a frame, respectively; \(r\), \(p\), and \(\psi\) are the roll, pitch, and heading angles, respectively, of the \(n\) reference frame relative to the \(l\) frame; \(\mathbf{R_{s}^{n}}\) represents the boresight rotations from the \(s\) frame to the \(n\) frame; \(\rho\), \(\theta\), and \(\alpha\) are the range, horizontal angle, and vertical angle, respectively, from the lidar sensor to the observed point in the \(s\) frame; \(l_{x}\), \(l_{y}\), and \(l_{z}\) are the \(x\), \(y\), \(z\) lever-arm offsets, respectively, from the center of the IMU to the origin of the lidar sensor measured in the \(n\) frame; and \(X_{IMU}^{l}\), \(Y_{IMU}^{l}\), and \(Z_{IMU}^{l}\) are the \(X\), \(Y\), and \(Z\) coordinates of the center of the IMU in the \(l\) frame, respectively. To isolate the effects of the heading precision, it is assumed that the lidar sensor frame (\(s\)) and the navigation reference frame (\(n\)) are perfectly aligned (i.e., \(\mathbf{R_{s}^{n}}=\mathbf{\mathrm{I}}\)), and that the remaining trajectory estimates (i.e., \(r\), \(p\), \(X_{nav}^{l}\), \(Y_{nav}^{l}\), and \(Z_{nav}^{l}\)), calibration parameters (i.e., \(l_{x}\), \(l_{y}\), and \(l_{z}\)), and lidar observations (i.e., \(\rho\), \(\theta\), \(\alpha\)) are all free of error. Using uncorrelated variance propagation, the optimistic coordinate precisions for an observed point in the local level frame are expressed by:  
\[\sigma_{X_{p}}=\sigma_{\psi}\left|\left[