unknowns, the equation system may have no solution. They introduced 4 pre-rotations to solve the problem 4 times to handle this problem. The pre-rotation is to make the no-solution problem happen less likely, however, it can not fully solve this problem in theory.

Minimal solution is essential in the RANSAC framework [@fischler1981random] for robust pose estimation. Any combination of the 3D correspondences resulting in 6 independent constraints forms a minimal configuration. In the literature, specific algorithms have been proposed to solve certain configurations. In [@87340], a minimal solution for three line-to-plane correspondences were proposed. In [@vasconcelos2012minimal; @zhou2014new], they studied the minimal solution of a more specific line-to-plane configuration, , lines are all on a plane. As one line-to-plane correspondence can be treated as two point-to-plane correspondences, these problems can be formulated as a point-to-plane registration problem. Naroditsky [@naroditsky2011automatic] presented a solution to 6 point-to-plane correspondences. Ramalingam [@ramalingam2013theory] solved all the point-to-plane minimal configurations. They designed specific intermediate coordinate systems to simplify each minimal configuration. This method is not convenient in the application, as it needs to implement a bunch of algorithms. 3D registration without correspondence is an related but more complicated problem. Several works [@li20073d; @papazov2011stochastic; @yang2013go; @zhou2016fast] presented globally 