This paper is concerned with pose estimation and visual servoing from four points. We determine the configurations for which the corresponding Jacobian matrix becomes singular, leading to inaccurate and unstable results. Using an adequate representation and algebraic geometry, it is shown that, for any orientation between the camera and the object, there are always two to six singular locations of the camera in the generic case where the points are not coplanar, corresponding to the intersection of four cylinders. The particular case where the four points are coplanar is also characterized. Furthermore, some realistic example configurations are considered to substantiate the theory and to demonstrate failure cases in pose estimation and image-based visual servoing when the camera approaches a singularity.

本文从四个方面着眼于姿态估计和视觉伺服。我们确定相应的雅可比矩阵变得奇异的配置，从而导致不准确和不稳定的结果。使用适当的表示法和代数几何，可以证明，对于相机和物体之间的任何方向，在一般情况下，相机的点总是有2到6个奇异的位置，其中点不是共面的，对应于四缸。还对四个点共面的特殊情况进行了表征。此外，考虑了一些实际的示例配置，以充实该理论并演示当相机接近奇点时在姿态估计和基于图像的视觉伺服中出现故障的情况。