parameters. The camera focal length is 1,000 pixels with a resolution of $640\times480$. Two types of synthetic data were generated. (1) **Ordinary three-dimensional (3D) case**: object points were randomly and uniformly distributed in a cube region $[-2,2]\times[-2,2]\times[4,8]$. (2) **Quasi-singular case**: The distribution cube is $[1,2]\times[1,2]\times[4,8]$. For each experiment result, we report the mean values of 100 trials.

For accuracy evaluation, the rotation error is measured in degrees between the truth rotation $\bf{R}_{\rm{true}}$ and the estimated $\bf{R}$ as ${e_{{\rm{rot}}}}(\deg ) = \left\| {{{\left[ {{\rm{acos}}{{({\bf{r}}_{k,{\rm{true}}}^T \cdot {{\bf{r}}_k})}_{k = 1,2,3}}} \right]}^T}} \right\| \times {\rm{180/}}\pi  $, where ${{\bf{r}}_{k,{\rm{true}}}}$ and ${\bf{r}}_k$ are the $k$th column of $\bf{R}_{\rm{true}}$ and $\bf{R}$ respectively. The translation error is ${e_{{\rm{trans}}}}(\% ) = \left\| {{{\bf{t}}_{{\rm{true}}}} - {\bf{t}}} \right\|/\left\| {\bf{t}} \right\| \times 100\%$.

### Outlier-Free Synthetic Situations

Most P$n$P algorithms do not have the ability to handle outliers, and even a small percentage of outliers will significantly reduce the accuracy, as shown in Fig.\[fig\_pnpcannothandleoutliers\]. Thus, although outlier-free situations are not the main concern of R1PP$n$P, we first conducted comparison experiments between the proposed R1PP$n$P and other P$n$P algorithms in outlier-free situations. The reason for this comparison is that RANSAC+P3P or P4P methods usually need other P$n$P methods as the final refinement step. 