## 2 Materials and Methods
### GNSS/INS Heading Determination Techniques
If it is assumed that the precisions of the relative horizontal displacements are equal (i.e., \(\sigma_{e}=\sigma_{n}\)) and the GNSS compass is operated while nominally level (i.e., \(\theta\approx 0^{\circ}\)), Equation (2) simplifies and is now expressed by:  
\[\sigma_{\psi}=\frac{\sigma_{h}}{L} \tag{3}\]  
where \(\sigma_{h}\) is the precision of the relative displacement between the GNSS antennas in the local horizontal plane. Figure 3 illustrates the theoretical heading precisions of the GNSS compass method for different compass baseline lengths. Figure 2: GNSS compass heading (\(\psi\)) and tilt (\(\theta\)) angle estimates expressed within a local level (east–north–up) frame. Both single-antenna and dual-antenna GNSS/INS sensors utilize the dynamic alignment method for heading determination during UAS-lidar mapping operations. Dynamic heading alignment works by comparing the accelerometer measurements from the INS solution and the position and velocity measurements from the GNSS solution to estimate the internal dynamics of the system [11; 13]. The method works best when the sensor experiences rapid changes of direction and varying forces of acceleration [13]. The complex computations utilized by a GNSS/INS navigation filter to estimate heading and its associated precision are beyond the scope of this study; see [9] for further information. A simpler concept that illustrates how sensor dynamics can be used to determine heading is the GNSS velocity method, which is sometimes used to initialize heading within the dynamic alignment method [19]. The GNSS velocity method uses a trigonometric relationship based on GNSS-derived velocity estimates within a local reference frame to determine heading. It is expressed by:  
\[\psi=\text{atan}\bigg{(}\frac{v_{e}}{v_{n}}\bigg{)}+c \tag{4}\]  
where \(v_{e}\) and \(v_{n}\) are the east and north components of the GNSS-derived velocity, respectively, and \(c\) is the misalignment angle between the GNSS velocity vector and the forward axis of the navigation reference frame. The precision of an optimistic heading estimate can be calculated using uncorrelated variance propagation and is expressed by:  
\[\sigma_{\psi}=\sqrt{\frac{{v_{n}}^{2}}{{({v_{n}}^{2}+{v_{e}}^{2})}^{2}}\cdot {\sigma_{v_{e}}}^{2}+\frac{{v_{e}}^{2}}{{({v_{n}}^{2}+{v_{e}}^{2})}^{2}}\cdot {\sigma_{v_{n}}}^{2}+{\sigma_{c}}^{2}} \tag{5}\]  
where \(\sigma_{v_{n}}\) and \(\sigma_{v_{n}}\) are the precisions of the GNSS-derived velocity components in the east and north directions, respectively, and \(\sigma_{c}\) is the precision in the misalignment angle that represents the achievable limit for the heading precision (i.e., best case) using the GNSS velocity method. If it is assumed that the precisions of the horizontal velocity components are equal (i.e., \(\sigma_{v_{e}}=\sigma_{v_{n}}\)), Equation (5) simplifies and is now expressed by:  
\[\sigma_{\psi}=\sqrt{\frac{{\sigma_{v_{n}}}^{2}}{{v_{h}}^{2}}+{\sigma_{c}}^{2}} \tag{6}\]  
Figure 3: Theoretical GNSS compass heading precisions for different GNSS compass baseline lengths. where \(v_{h}\) is the GNSS-derived horizontal velocity (i.e., \(v_{h}=\sqrt{{v_{e}}^{2}+{v_{n}}^{2}}\)) and \(\sigma_{v_{h}}\) is its associated precision. If it is also assumed that the misalignment angle is known (i.e., \(\sigma_{c}=0^{\circ}\)), Equation (6) simplifies and is now expressed by:  
\[\sigma_{\psi}=\frac{\sigma_{v_{h}}}{v_{h}} \tag{7}\]  
Initializing the heading from a GNSS-derived velocity is only applicable if the forward axis of the navigation reference frame is parallel to the velocity vector, which is a reasonable assumption for most land vehicle navigation applications [21], or if the misalignment angle can be determined. Within UAS-lidar applications, it is common for the forward axis of the navigation reference frame and the GNSS-derived velocity vector to be in different directions. Fixed-wing UAS are affected by crosswind, which can misalign the forward axis and the velocity vector (Figure 4a).