Closed form analytical solution and scan pattern shaping theory of three-element Risley prism for LiDAR systems

Abstract

Three-element Risley prism are emerging technologies in LiDAR. It can convert the FoV into a rectangular shape with high coverage rates in specific conditions, making it suitable for applications in autonomous driving scenarios. The addition of a third prism has been proven effective in eliminating blind spots and singularity problems. One challenge facing the use of three-element scanners in autonomous driving is the accurate positioning control of light beams during high-speed motion, especially in specific scan patterns such as near rectangles. This work introduces a closed-form analytical solution and parameters for the scan patterns of the three-element scanner through a non-paraxial tracing method. The closed-form solution includes two groups of coefficients representing all possible configurations of the three-element scanner. The parameters consist of ten degrees of freedom: angular velocity ratio, M1, M2, deflection angle ratio, k1, k1, initial phases, θ0q(q=1,2,3), distance between the two prisms dair1, dair2, and distance from the screen, P. To our knowledge, this is the first study to discuss these degrees of freedom in the context of the closed-form solution. We obtain its scan patterns by MATLAB 2019a and explore a configuration of three-element Risley prism in near-rectangular scan patterns, achieving a wide FoV and high point cloud density while avoiding blind spots and singularity issues. To validate the simulation results, a three-element Risley prism scanner system based on brushless motor is developed. Our study suggests that the three-element rotating prism scanner holds potential superiority in forward-looking LiDAR solutions.