## 3 Bias Impact Analysis
### Bias Impact Analysis for a Spinning Multi-Beam Laser Scanner
These coordinates (\(x^{\prime}\), \(y^{\prime}\), \(z^{\prime}\)) also denote the location of a point with respect to the virtual LiDAR unit frame, which is almost parallel to the IMU body frame. The schematic illustration of these symbolic notations are depicted in Figure 6 for a UAV-based LiDAR system. Substituting Equations (5) and (6) in Equation (4), we get the revised form of the LiDAR point positioning equation, as given in Equation (7), where \(\Delta X\), \(\Delta Y\), \(\Delta Z\) are the lever-arm offset parameters of the LiDAR unit frame relative to the IMU body frame. Now, the impact of the presence of bias in the system mounting parameters can  
Figure 5: Coordinate Systems for UAV-based LiDAR System. Table 1 can now be used to assess the impact of each bias for planar surfaces in different orientations--vertical planes parallel to flight direction, vertical planes perpendicular to flight direction and horizontal planes--thus indicating the impact across flight direction, along flight direction and vertical direction, respectively. **Impact of Bias in Lever-arm component across the flying direction (\(\Delta X\)):**  
The introduced shift _across the flying direction_\((\pm\ \delta\Delta X)\) is flying direction dependent and does not depend on the location of the point in question relative to the virtual laser unit coordinate system. As a result, its impact will be visible in case of having vertical planes parallel to the flying direction scanned from two flight lines in opposite directions. **Impact of Bias in Lever-arm component along the flying direction (\(\Delta Y\)):**  
The introduced shift _along the flying direction_\((\pm\ \delta\Delta Y)\) is flying direction dependent and does not depend on the location of the point in question relative to the virtual laser unit coordinate system. Again, it would impact vertical planes perpendicular to the flying direction scanned from two flight lines in opposite directions. **Impact of Bias in Lever-arm component in the vertical direction (\(\Delta Z\)):**  
The introduced shift in the _vertical direction_\((\delta\Delta Z)\) is flying direction independent and does not depend on the location of the point in question relative to the virtual laser unit coordinate system. As a result, the entire point cloud would be shifted in the vertical direction by the same amount. So, this bias would not affect planes at any orientation for any flight line configuration. **Impact of Bias in Boresight Pitch (\(\Delta\omega\)):**  
A bias in this component \((\delta\Delta\omega)\) will cause shifts along the flying direction as well as in the vertical direction. The impact of boresight pitch bias _along the flying direction_\((\mp\ z^{\prime}\ \delta\Delta\omega=\pm(H\pm\Delta h)\ \delta\Delta\omega)\) is flying direction dependent and its magnitude depends on the height (\(z^{\prime}\)) of the point in question relative to the virtual laser unit coordinate system. This impact would be visible in case of a planar feature perpendicular to the flying direction being scanned by flight lines in opposite directions. The impact of boresight pitch bias in _the vertical direction_\((y^{\prime}\ \delta\Delta\omega)\) would be manifested in horizontal planes. The magnitude of this impact depends on the \(y^{\prime}\)-coordinate of the point in question. So, this impact would be visible even in case of a single flight line capturing a horizontal planar feature as long as there is a significant variation in the \(y^{\prime}\)-coordinate at a given location within such a plane, i.e., if the same portion of the plane is scanned by the laser unit while being at different locations. Since a bias in lever arm \(\Delta Y\) also causes shifts along the flying direction, there is a need to decouple the impacts so as to estimate both these biases accurately.