Limitation of Rayleigh sky model for bio-inspired polarized skylight navigation in three-dimensional attitude determination

Insects such as desert ants, drosophilae can sense polarized skylight for navigation. Inspired by insects, many researchers have begun to study how to use skylight polarization patterns for attitude determination. Rayleigh sky model has become the most widely used skylight polarization model for bio-inspired polarized skylight navigation, due to its simplicity and practicality. However, this model is an ideal model considering only single Rayleigh scatter event, whose limitation in bio-inspired attitude determination has not been paid much attention and lacks strict inference proof. Aiming at this problem, the rotational symmetry and plane symmetry of Rayleigh sky model are analyzed in detail, and it is theoretically proved that this model contains only single solar vector information, which contains only two independent scalar pieces of attitude information, so it is impossible to determine three Euler angles simultaneously in real-time. To further verify this conclusion, based on a designed hypothetical polarization camera, we discussed what conditions different three-dimensional attitudes must satisfy so that the polarization images taken at different 3D attitudes are the same, which indicates multiple solutions will appear when only using Rayleigh sky model to determine 3D attitude. In conclusion, according to single solar vector information and the existence of multiple solutions, it is fully proved that 3D attitude cannot be determined in real-time based only upon Rayleigh sky model. Code is available at: https://github.com/HuajuLiang/HypotheticalPolarizationCamera.