## 3 Bias Impact Analysis
### Bias Impact Analysis for a Spinning Multi-Beam Laser Scanner  
The objective of this section is to derive a mathematical formulation that shows the impact of biases in the mounting parameters on the coordinates of points along planar features with different orientations. This analysis can further aid the development of an optimal/minimal flight and target configuration for calibration. Note that planar features are used specifically for this analysis as they facilitate the observation of positional deformations in one direction at a time, i.e., the effect in the direction normal to the plane. For simplifying the bias impact analysis without any loss of generality, we make several assumptions. Firstly, the IMU is mounted on the UAV with its \(X\)- and \(Y\)-axes aligned along the starboard and flight line directions, respectively. The IMU is assumed to be perfectly vertical, i.e., the \(Z\)-axis of the IMU body frame is assumed to be perfectly aligned with the vertical direction of the mapping frame. Also, we assume that the flight line directions are either from South-to-North (\(\kappa=0^{\circ}\)) or from North-to-South (\(\kappa=180^{\circ}\)). These assumptions facilitate the decision as to whether the impact is along/across the flight line and vertical directions. As a result, the rotation matrix \(R_{b}^{m}(t)\) would be given by Equation (3), where the top and bottom signs are for S-N and N-S flight line directions, respectively. In order to generalize the analysis regardless of the orientation of  
Figure 4: Illustration of a multi-beam spinning LiDAR unit (VLP-16 Puck Hi-Res). Figure 3: Illustration of point positioning of a UAV-based Light Detection and Ranging (LiDAR) system. the LiDAR unit relative to the IMU body frame, Equation (1) is slightly modified by introducing a virtual LiDAR unit frame, \(Lu^{\prime}\), which is almost aligned with the IMU body frame. This facilitates the decision as to whether the impact is along/across the flight line and vertical directions. Moreover, the use of a virtual LiDAR unit frame also prevents gimbal lock in the mounting parameter estimation. This modification is implemented by expressing the term \(R_{Lu}^{b}\) in Equation (1) as: \(R_{Lu}^{b}=R_{Lu^{\prime}}^{b}\,R_{Lu}^{Lu^{\prime}}\), where \(R_{Lu}^{Lu^{\prime}}\) is defined according to the laser scanner unit alignment relative to the IMU. The modified LiDAR point positioning is given by Equation (4). The laser unit coordinate system for the LiDAR unit alignment on the UAV platform used in this system and the assumed IMU body frame coordinate system (with \(X\), \(Y\), \(Z\)-axes along starboard, forward and up directions, respectively) are shown in Figure 5.  
\[R_{b}^{m}(t)=\left[ \begin{array}{ccc}\pm 1&0&0\\ 0&\pm 1&0\\ 0&0&1\end{array} \right] \tag{3}\]  
\[r_{I}^{m}=r_{b}^{m}(t)+R_{b}^{m}(t)\;r_{Lu}^{b}+R_{b}^{m}(t)\;R_{Lu^{\prime}} ^{b}\;R_{Lu}^{Lu^{\prime}}\;r_{I}^{Lu}(t) \tag{4}\]  
Since the virtual LiDAR unit frame is almost aligned with the IMU body frame, it results in small values for the differential angular boresight parameters (\(\Delta\omega\), \(\Delta\phi\), \(\Delta\kappa\)) relating the two frames. So, the matrix, \(R_{Lu^{\prime}}^{b}\) can be written as shown in Equation (5), using the small angle approximations. Here, \(\Delta\omega\), \(\Delta\phi\), \(\Delta\kappa\) denote the rotation around the \(X\), \(Y\), \(Z\)-axes of the IMU body frame (i.e., across flight, along flight and vertical directions), respectively. Hence, these parameters denote the boresight pitch, roll and heading angles, respectively. The point coordinates relative to the virtual LiDAR unit frame, as derived using the nominal value for \(R_{Lu}^{Lu^{\prime}}=R_{\omega}(90^{\circ})\;R_{\phi}(0^{\circ})\;R_{\kappa}(0 ^{\circ})\) according to Figure 5, are given by Equations (6a) and (6b) in terms of the flying height, \(H\pm\Delta h\) and scan angles (\(\alpha\) and \(\beta\)).