is very unstable. For the minimal problem, we run 20000 trails for the 6 point-to-plane configuration. Fig. \[fig:vs\_DMinSol\] shows the results. The DMinSol has a much longer tail than our algorithm.

Table \[table:time\_vs\_orig\] lists the average computational time of different algorithms. For the least-squares problem, we compare the computational time of solving the first order optimality conditions (13). This time is independent of the number of correspondences. As shown in Table \[table:time\_vs\_orig\], our least-squares solution is about 7 times faster than DLSSol, and our minimal solution is about 2 times faster than DMinSol. The above results verify that our intermediate unknowns can increase the numeric stability as well as reduce the computational time.

![Compare our least-squares solution with direct least-squares solution (DLSSol) which solves the polynomial system (14) using the Gröbner basis method [@larsson2017efficient]. []{data-label="fig:vs_DLSSol"}](./figure/simulation/vs_orig/simplify_vs_original_LS_R.png "fig:"){width="23.00000%"} ![Compare our least-squares solution with direct least-squares solution (DLSSol) which solves the polynomial system (14) using the Gröbner basis method [@larsson2017efficient]. []{data-label="fig:vs_DLSSol"}](./figure/simulation/vs_orig/simplify_vs_original_LS_t.png "fig:"){width="23.00000%"}

![Compare our minimal solution with direct minimal solution (DMinSol) that solves the polynomial system (9) using the Gröbner basis method [@larsson2017efficient]. []{data-label="fig:vs_DMinSol"}](./figure/simulation/vs_orig/simplify_vs_original_R.png "fig:"){width="23.00000%"} ![Compare our minimal solution with direct minimal solution (DMinSol) that solves the polynomial system (9) using the Gröbner 