# Kinematic In Situ Self-Calibration of a Backpack-Based Multi-Beam LiDAR System
## 3 Mathematical Models  
### Point Observation and Systematic Error Model of VLP-16  
VLP-16 acquires range and horizontal angle measurements and provides fixed vertical angle for 16 laser channels (described in Section 2.1). The geometric relationship between spherical coordinates (\(\rho\), \(\theta\), \(\alpha\)) and Cartesian coordinates (\(x\), \(y\), \(z\)) is shown in Figure 6. The formulas for converting spherical coordinates (\(\rho\), \(\theta\), \(\alpha\)) to Cartesian coordinates (\(x\), \(y\), \(z\)) are given by Equation (1), where \(\rho\), \(\theta\), and \(\alpha\) are raw distance measurement, encoder angle measurements, and fixed vertical angle, respectively. \[P(x,y,z)=\left[ \begin{array}{c}x\\ y\\ z\end{array} \right]=\left[ \begin{array}{c}\rho\ cos(\alpha)sin(\theta)\\ \rho\ cos(\alpha)cos(\theta)\\ \rho\ sin(\alpha)\end{array} \right] \tag{1}\]  
Sensor modelling is a crucial step in conducting rigorous self-calibration of laser scanners [33; 34]. Numerous researchers have independently defined their own TLS error \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Lens** & **Camera Body** \\ \hline Model & KOWA LM5 JCM & \multicolumn{2}{c}{CM3-U3-50S5C} \\ Projection Model & \multicolumn{2}{c}{Perspective} \\ Image Size (pixel) & \multicolumn{2}{c}{2448 \(\times\) 2048} \\ Pixel Size (mm) & \multicolumn{2}{c}{0.00345} \\ Focal Length (mm) & \multicolumn{2}{c}{5} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Specifications of the stereo camera. Figure 5: Stereo camera: (**a**) KOWA LM5 JCM; (**b**) Chameleon3 USB3 Vision. Figure 6: Conversion from a spherical coordinate system to a Cartesian coordinate system. models. The error model of a laser scanner was estimated, and the accuracies of the estimated parameters were determined by comparing the measurements with an electronic distance measurement (EDM) [35]. A later model consists of about 20 additional parameters (APs) and can be found in [36]. This error model effectively modeled TLS instruments; however, it was defined for the AM-CW TLS, which uses a phase-based distance measuring system. In the case of MBL, laser emitter/receiver sets measure distance by a pulse-based Time of Flight (ToF) system with a fixed vertical angle. Therefore, AP terms should be different with phase-based TLS instruments. For Velodyne LiDAR, systematic error coefficients are defined by six parameters (i.e., range offset, scale error, horizontal angular offset, vertical angular offset, horizontal offset, and vertical offset) given by the manufacturer to model the deviations of measurements. Since the number of APs are multiplied by the number of lasers, the AP terms should be carefully chosen to avoid over-parameterization. Although the manufacturer provides six parameters, some of the APs were neglected in this study. First, the estimation of scale error requires inclusion of an independent scale definition in the self-calibration network [36]. Range scale error cannot be estimated using the calibration approach and must therefore be estimated by other means, and independent baseline testing did not disclose the existence of this error [33]. In performing the tests, therefore, if scale error is included in the adjustment without an independent scale definition, an optimization process is not working properly. Considering there are no a priori known locations of the targets or scanner when performing in situ calibration, the scale error was therefore neglected. Horizontal offset and vertical offset were also fixed to maintain a higher accuracy of adjustment, for the following reasons: (1) Horizontal and vertical offset are highly correlated to the horizontal and vertical rotations, respectively [25; 29]. In the case of VLP-16, the correlation coefficients of the parameters corresponding to the vertical and horizontal rotation corrections were found between 0.92 and 0.98, respectively [31].