$n_{\pi}$ point-to-plane correspondences.

When measurements approximate an ambiguous configuration, a prior is required to identify the correct solution. The prior is generally available in real applications. For instance, we generally have a rough estimation of the pose between two sensors in the extrinsic calibration problem [@zhang2004extrinsic; @unnikrishnan2005fast; @naroditsky2011automatic; @gomez2015extrinsic; @zhou2018automatic]. For the pose estimation in SLAM [@zhang2015visual; @zhang2014loam; @newcombe2011kinectfusion; @proencca2018probabilistic; @taguchi2013point], current pose should be consistent with the previous motion trajectory.

Previous works [@olsson2006registration; @olsson2008solving; @olsson2009branch; @briales2017convex] using convex approximation can only find the global minimizer. However, the global minimizer may not be the correct solution of a problem if measurements approximate an ambiguous configuration. Therefore, we can not simply omit local minimizers.

Rotation Parameterization {#sec:Algorithm}
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Solving for the rotation matrix is the crux for pose estimation. Previous works [@olsson2006registration; @olsson2009branch; @briales2017convex; @wientapper2018universal] adopt non-minimal representations for $\bf{R}$, which results in additional quadratic constraints in the minimization problem. This paper adopts the CGR parametrization [@hesch2011direct; @mirzaei2011optimal] which gives a minimal representation for the rotation matrix, removing the quadratic constraints in (\[equ:4\]) . The CGR parametrization expresses a rotation matrix as $$\label{equ:5}
{\mathbf{R}=\frac{{\mathbf{\bar{R}}}}{1+{{\mathbf{s}}^{T}}\mathbf{s}},\ \mathbf{\bar{R}}=\left( \left( 1-{{\mathbf{s}}^{T}}\mathbf{s} \right){{\mathbf{I}}_{3}}+2{{\left[ \mathbf{s} \right]}_{\times }}+2\mathbf{s}{{\mathbf{s}}^{T}} \right)}$$ where $\mathbf{s}=\left[ {{s}_{1}};{{s}_{2}};{{s}_{3}} \right]$ is a 3-dimensional vector, and 