# 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
(2) The vertical and horizontal alignments of each laser are precisely located according to the manufacturer-provided values set below the accuracy of the range observation. (3) Local coordinate error induced by horizontal and vertical offsets is not linearly dependent in the range observation. Hence, the range offset, horizontal angular offset, and vertical angular offset were considered as APs in this study. Therefore, the point observation model for MBL could finally be determined as Equation (2). The coordinates of \(i^{\text{th}}\) point at scan position \(j\) lying on plane \(k\) from laser \(n\) are related by rigid body transformation as given by:  
\[\left[ \begin{array}{c}X_{ijkn}\\ Y_{ijkn}\\ Z_{ijkn}\end{array} \right]=R_{j}\times\left[\begin{array}{c}\left(\rho_{ijkn} +\Delta\rho_{n}\right)cos(\alpha_{n}+\Delta\alpha_{n})sin\left(\theta_{ijkn}+ \Delta\theta_{n}\right)\\ \left(\rho_{ijkn}+\Delta\rho_{n}\right)cos(\alpha_{n}+\Delta\alpha_{n})cos \left(\theta_{ijkn}+\Delta\theta_  
installation of signalized targets, which is labor-intensive and can decrease the accuracy of point-based self-calibration due to high parameter correlation [42]. Moreover, in the case of multi-beam laser scanners, extracting the exact target points is almost impossible due to the fixed vertical angle. Meanwhile, point coordinates on the surface of planar targets can be used directly instead of center point coordinates of signalized targets. Since signalized targets are not required, plane-based self-calibration is one of the most widely adopted methods. The main advantage of plane-based self-calibration is that the plane parameters within each plane can be estimated in the adjustment model, thereby mitigating the need to measure an accurate reference target and enhancing the method's applicability for in situ calibration. [PERSON] and [PERSON] [43] presented a rigorous approach to bore-sight self-calibration of an airborne laser scanning system by conditioning the geo-referenced LiDAR points to fit into common plane surfaces. However, the objective of their work was more oriented to the estimation of extrinsic parameters between the Inertial Measurement Unit (IMU) and the LiDAR unit, considering only range offset as AP. [PERSON] and [PERSON] [34] conducted plane-based self-calibration with scan data using FARO 880. In their study, self-calibration simulations investigated various scanner configurations, and the results demonstrated that a long baseline between two scan stations enables a more accurate estimation of collimation axis errors. Also, plane-based calibration has been reported to offer almost the same performance as point-based calibration when conducted under a strong network configuration [44]. The self-calibration approach in this study is based on the plane-based functional model of [34]. This model estimates not only exterior orientation parameters (EOPs) and APs, but also plane parameters, simultaneously. The plane-based method can remove the necessity of calibration target set-up and reference target coordinate measurement using additional sensors. The condition equation associated with parameters and observations can be expressed by the plane-based functional model as given by:  
\[[a_{k}\ b_{k}\ c_{k}]\times\left[ \begin{array}{c}X_{ijkn}\\ Y_{ijkn}\\ Z_{ijkn}\end{array} \right]-d_{k}=0 \tag{3}\]  
where \([a_{k}\ b_{k}\ c_{k}]\) are the direction cosines of the normal vector of the plane \(k\), and \(d_{k}\) is the orthogonal distance from the origin of the reference coordinate system to the plane \(k\). The direction cosines must satisfy the unit length constraint:  
\[a_{k}^{2}+b_{k}^{2}+c_{k}^{2}=1 \tag{4}\]