# Calibration of a Rotating or Revolving Platform with a LiDAR Sensor
## 2 Related Work
### Calibration Target Required  
A representative of the first method type is described by [PERSON] et al. [5]. It estimates the extrinsic parameters of a LiDAR scanner with respect to another LiDAR's frame. Both are mounted on the top of the same car. Highly reflective patches are attached to pillars at the edge of a street track. While driving along this calibration track, the patches are detected by the reflection intensity of their LiDAR measurements and then reduced to a single feature point in both sensor frames. Using the known vehicle pose in a global reference frame, the error between patches from both sensors in a common frame is minimised. This is done by solving a second-order cone program which reformulates a more cost effective least-square problem. While the problem formulation is interesting, due to the reduction of computational costs, the proposed method requires a positioning system, and driving along a previously manipulated track. Both is not always feasible and may be error prone. Furthermore, the calibration results in knowing a sensor's pose with respect to another sensor's frame and not with respect to a part of the platform construction. This makes knowledge of the exact pose of the other LiDAR relative to the car necessary.  
In contrast to more common numerical minimisation algorithms like in [5], a simple analytical method is proposed by [PERSON] et al. [8]. The platform's architecture obtains 3D point clouds from a 2D laser range finder by tilting it to different angles. Since the rotation center differs from the sensor center, it is important to know the radius exactly. [PERSON] et al. [8] propose to measure points on a planar target, e.g., a wall, in different known tilt angles of the sensor. Using the vertical distances of those points, the radius can be inferred using angle and distance information in a triangle spanned by the rotation center and two of the LiDAR measurements. This method ignores all other dimensions of the sensor pose except the radius of the revolving LiDAR. However, these are important for many platforms and thus the proposed method is strongly bound to this specific architecture.  
The method by [PERSON] et al. [7] uses a target plane to estimate the pose of a 2D laser range finder orbiting a rotation center. The 3D scanline direction on the target is inferred to construct a cost function scoring the scanner's orientation separately from its translation, which is optimised by a second cost function. These are used as objectives to minimize the distance between points on neighbouring scanlines of different platform orientations. The decoupling removes some ambiguities between translation and rotation of the scanner's frame. Thus, rotations which compensate translations are removed from the search space of the objective functions. The decoupling is a clever approach not only to remove the ambiguities, but also to split up the optimisation problem into two functions with lower dimensionality. Some of our main ideas are based on this work. However, it is expected to depend on the device architecture of the LRF used on this platform.