## 2 Materials and Methods
### GNSS/INS Heading Determination Techniques  
The use of a magnetometer for heading determination is the simplest method as the magnetic field of the Earth can be directly measured and used to infer the direction of true north. However, the magnetic vector at the position of the magnetometer will likely have contributions from other sources (e.g., naturally occurring magnetic material in the ground and ferrous materials from the attached vehicle) [11]. Therefore, proper calibration of the magnetometer is required to achieve highest accuracy measurements [18]. Under ideal conditions (e.g., stationary within an open area), a magnetometer can produce heading estimates with an accuracy of \(<\)0.5\({}^{\circ}\), which is suitable for initializing the heading within an INS [19]. Most GNSS/INS sensors used in commercial UAS-lidar systems including the Ap-planix APX-18 UAV utilize MEMS-based IMUs. These types of IMUs often experience gyroscope bias and noise errors that exceed the Earth's rotation rate (i.e., \(>\)15\({}^{\circ}\)/hour); therefore, the stationary gyro-compassing method for determining heading is not feasible [19]. The precision of the heading estimate using the gyro-compassing method is inversely proportional to cosine of latitude, as expressed in Equation (1), and will therefore decrease at higher latitudes [11]. \[\sigma_{\psi}\propto\frac{1}{\cos(\varphi)} \tag{1}\]  
where \(\sigma_{\psi}\) is the precision of the heading estimate and \(\varphi\) is the geodetic latitude. The primary benefit of a dual-antenna GNSS/INS sensor is the capability of using the GNSS compass method for heading determination. A GNSS compass is able to compute the heading angle estimate, measured as the clockwise angle from true North, and the elevation (tilt) angle estimate of the compass vector within a local level (east-north-up) frame (Figure 2). The precision of the tilt angle estimate is dependent on the precision of the vertical component of a GNSS compass vector, which for GNSS positioning is commonly 2-3 times less precise than the horizontal components [20]. As a result, the tilt angle estimates provided by a GNSS compass are not commonly used within the GNSS/INS navigation filter [11]. However, [20] demonstrated that if both receivers used in a GNSS compass utilize a common clock, the precision of the vertical component could be close to those of the horizontal components for a GNSS compass vector. When using a GNSS compass, a few added challenges that must be overcome include maintaining a direct line-of sight to the satellites, observing at least six common satellites between the two antennas, and reducing multipath interference [11]. Figure 1: Applanix APX-18 UAV dual-antenna GNSS/INS sensor. The GNSS compass method employs differential GNSS positioning using carrier phase measurements to precisely determine the relative displacements of the GNSS compass vector within a local frame. For carrier phase measurements to produce precise results, the unknown ambiguity terms must be correctly resolved to their integer values. Once this has been done successfully, the carrier phase measurements will act as very precise pseudo-range data, which makes very precise orientation determination possible [20]. Using this method, the differential position of the secondary GNSS antenna with respect to the primary GNSS antenna will have a precision of between 2 and 10 mm, depending on the local multipath environment [13]. Following the GNSS compass error propagation presented in [20], the precision of the heading estimate is expressed by:  
\[\sigma_{\psi}=\sqrt{\frac{\left(\cos\ \psi\right)^{2}{\sigma_{e}}^{2}+\left(\sin\ \psi\right)^{2}{\sigma_{n}}^{2}}{L^{2}(\cos\ \theta)^{2}}} \tag{2}\]  
where \(\sigma_{e}\) and \(\sigma_{n}\) are the precisions of the relative displacements in the east and north directions, respectively, between the GNSS antennas as determined by differential carrier phase positioning, and \(L\) is the fixed length of the compass baseline.