# Calibration of a Rotating or Revolving Platform with a LiDAR Sensor
## 2 Related Work
### Targetless Calibration  
The method proposed by [PERSON] et al. [9] calibrates two sensor properties without any calibration targets: intrinsic LiDAR parameters, i.e., the internal laser arrangement, external LiDAR parameters specifying its pose relative to the robots' base frame. To determine the internal and external parameters, a single energy function is formulated. This function rewards points of neighbouring laser beams if they lie on the same surface, i.e., having the same normal, while applying a penalty, if they do not. During a movement of the vehicle with tracked trajectory, the measured points can be transformed into a common frame. The external parameters minimising the energy function should be the true scanner pose. To avoid ambiguities and local minima, [PERSON] et al. [9] use a grid search instead of a Newton-based method, though the energy function would be contentious. They state that the search space is tractable. They also state not to make strong assumptions about their environment. However, the error formulation of their objective function implies planar surfaces. This is a relatively strong assumption, e.g., when thinking about urban rescue, forest, or underground environments. In addition, the requirement of a local positioning system is not always practical. [PERSON] et al. [10] constructed a sensor platform assembled from three independent 2D laser range finders mounted on a rotating plate. To calibrate the extrinsic parameters of all three scanners, they formulated a _crispness_ function representing the sharpness of the resulting 3D point cloud, i.e., the neighbourhood distribution of the measured points. This measure consists of a Gaussian mixture model (GMM) of each individual point's neighbourhood. The GMM gets more peaky as the deviation of nearby points in the neighbourhood of a point decreases, which happens if the transformation to the scanners approximate towards the true ones. To quantify this measure in a score that then can be maximised, the model is put into the Renyi Quadratic Entropy function [11]. The scanner transformations maximising the model's entropy should be the ones sought. [PERSON]'s calibration method is tailored to their platform and only calibrates the relative position of the sensors to each other. [PERSON] et al. [12] extend [PERSON]'s method by the calibration of several other LiDAR or LRF sensors mounted on a vehicle. They require to drive a path, so that all sensors cover as much of the environment as possible. Utilising a positioning system with known covariance matrix, i.e., known measurement uncertainty, the sensor data can be transformed into a common coordinate system. This allows a comparison between the sensor data of the different sensors, especially between [PERSON]'s platform and the other sensor data. To do so, [PERSON] et al. introduced another entropy based cross-sensor similarity measure. It integrates the Jensen-Renyi Divergence [13] in the crispness function of [PERSON] et al. [10]. The new measure will reach its maximum with a transformation that results in the most similar sensor measures in the common frame. The requirement of driving and a working positioning system makes this method inapplicable in many situations. The necessary conditions prevail over the improvements made compared to [PERSON]'s method. While driving along a V-shaped track, the point clouds recorded by the LiDARs mounted on a vehicle should be aligned when they are transformed using the correct transformation \(M_{L}\) between LiDARs and vehicle frame. To measure the alignment, [PERSON] et al. [14] use the definition of a fitted plane and the pairwise distance between those planes, introduced in in [PERSON]'s previous publication [15]. This allows [PERSON] et al. [14] to formulate an error function depending on trajectory and LiDAR pose to transform the plane representations into a common frame and minimize its distance.