# Calibration of a Rotating or Revolving Platform with a LiDAR Sensor
## 3 Calibration Method
### Decoupling  
Minimising a function with six free parameters can be computationally expensive. Furthermore, in the case of transformation arguments, there are ambiguities between translation and orientation, i.e., some combinations of translations and rotations can compensate others. To tackle these challenges and especially to reduce the problem's dimensionality, we decouple the rotational part from the full six degrees of freedom transformation. Inspired by [7], we formulate individual error functions for the rotational and translational part of the LiDAR's extrinsic parameters. To do so, it is assumed that the LiDAR is able to see two spheres \(S_{p_{i}}\) and \(S_{q_{i}}\) in a single carrier position \(R_{i}\in\mathbb{R}^{3\times 3}\) and that their respective center points \(p_{i}\) and \(q_{i}\) are estimated. We denote the scanner's extrinsic parameters as translation \(\overline{t}\in\mathbb{R}^{3}\) and orientation \(R\in\mathbb{R}^{3\times 3}\). It will be the final objective to optimize the respective error functions with respect to \(\overline{t}\) and \(R\), separately. The center points of \(S_{p_{i}}\) and \(S_{q_{i}}\) can be represented in the common, rigid frame of the sensing platform's base, by transforming them with the sought sensor transformation \(T_{R^{\prime}S}\in\mathbb{R}^{4\times 4}\), defined by the extrinsic parameters, into the carrier's rotation center and subsequently applying the measured transformation of the sensor carrier \(T_{RR^{\prime}}\in\mathbb{R}^{4\times 4}\). Since the translational component of \(T_{RR^{\prime}}\) is always zero, its orientation is changing over time, it can be reduced to \(R_{i}\in\mathbb{R}^{3\times 3}\). The index \(i\) denotes that the carrier orientation at of the \(i\)-th target pair can be detected:  
\[p^{\prime}_{\ i} =R_{i}*R*(p_{i}+\overline{t}), \tag{1}\] \[q^{\prime}_{\ i} =R_{i}*R*(q_{i}+\overline{t}). \tag{2}\]  
Figure 4 depicts the transformed, estimated center points of a target pair \(p^{\prime}_{\ i}\) and \(q^{\prime}_{\ i}\) can be used to construct a vector \(\vec{d}_{i}\) pointing from \(p^{\prime}_{\ i}\) to \(q^{\prime}_{\ i}\):  
\[\vec{d}_{i}=p^{\prime}_{\ i}-q^{\prime}_{\ i}. \tag{3}\]  
Figure 3: Sphere center points around the revolving LiDAR in the middle of the images. As in Figure 2, the spheres are coloured and denoted with \(p_{i},\ q_{i},\ i=0,1,2,3\) according to their respective frustum-direction and the carrier’s i-th orientation \(R_{i}\). From top-left to bottom-right, the sphere centers are estimated by the LiDAR measurements. (**a**) most clearly reveals the fact that two spheres can be seen by the LiDAR without further movement. (**b**) shows the scanner observing the second pair of spheres, after the carrier has turned by \(90^{\circ}\). In the third image (**c**), the first target pair is detected a second time after another \(90^{\circ}\) rotation. Note that the different colors indicate that it is seen from a different carrier orientation. Furthermore, note that the center points are not aligned. This results from the actual transformation error between the sensor frame and the platforms base frame. Moreover, it clarifies that a redundant measurement of the surroundings begins after a \(180^{\circ}\) carrier rotation. Like (**c**), the last picture (**d**) shows the other redundant sphere detection after the carrier turns further \(90^{\circ}\). In each picture’s center area, the revolving LiDAR is depicted in the coordinate system of the rotating sensor carrier. As can be seen, the LiDAR’s pose with respect to the carrier is static, i.e., it does not change over time. Inserting Equations (1) and (2) in Equation (3) results in  
\[ \begin{split}\vec{d}_{i}&=R_{i}*R*(p_{i}+\vec{t})-(R_{i }*R*(q_{i}+\vec{t}))\\ &=R_{i}*R*p_{i}+R_{i}*R*\vec{t}-(R_{i}*R*q_{i}+R_{i}*R*\vec{t})\\ &=R_{i}*R*p_{i}-R_{i}*R*q_{i}\\ &=R_{i}*R*(p_{i}-q_{i}).\end{split}