四点与五点线问题的奇异性分析
Singularity Analysis for the Perspective-Four and Five-Line Problems
- Sorbonne Université（索邦大学）
- Centre national de la Recherche Scientifique (CNRS)（法国国家科学研究中心）
- Laboratoire des Sciences du Numérique de Nantes (LS2N)（南特数字科学实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过代数几何方法分析四点与五点线视觉伺服问题的奇异性，揭示相机位于单叶双曲面或横截线时引发控制与位姿估计问题，并证明五线情形无奇异，四线存在不可避免的奇异位置。
AI中文摘要：
本文研究了通过观察四条和五条直线进行的基于图像的视觉伺服与位姿估计。我们的主要兴趣在于确定导致控制和稳定性问题的相机与观察直线的相对构型。由于这等价于寻找相应雅可比矩阵的奇异性，我们使用计算代数几何的工具来寻求使得其所有子式同时消失的构型。通过为该矩阵选择合适的基，我们重新审视了三直线情形下的问题，表明其中一种奇异性类型是当相机位于由这些直线唯一确定的单叶双曲面上时。这一结果被进一步利用来证明，在n条直线的情形下，一维奇异性（如果存在）出现在相机位于观察直线的横截线上时。因此，通过强制横截线为复数，我们可以避免四直线情形中上述类型的奇异性，尽管代数表明对于另一种类型的奇异性，相机总是存在多达10个不可避免的奇异位置。对于五条直线，我们发现一般情况下不存在奇异性。同时，对于具有正交性和平行性约束的四条和五条直线，奇异性也得到了表征。此外，使用视觉伺服库进行了一些模拟实验以证实理论结果。正如预期，我们在奇异性附近观察到控制问题以及位姿估计误差的增加。
英文摘要：
This paper deals with image-based visual servoing and pose estimation by observing four and five lines. Our main interest is to determine the relative configurations of the camera and the observed lines that lead to problems in control and stability. Since it is equivalent to finding the singularities of the corresponding Jacobian matrix, we use tools from computational algebraic geometry to seek configurations such that all of its minors vanish simultaneously. By choosing a suitable basis for this matrix, we revisit the problem in the case of three lines to show that one type of the singularities is when the camera lies on the hyperboloid of one sheet uniquely defined by the lines. This result is further exploited to prove that the one-dimensional singularities, if any, in the case of $n$ lines appear when the camera lies on the transversals to the observed lines. Thus, by forcing the transversals to be complex, we can avoid the aforementioned type of singularities in the case of four lines although the algebra shows that there can always be up to 10 inevitable singular locations of the camera for the other type of singularity. For five lines, we find out that there are no singularities in the generic case. The singularities are also characterized for four and five lines with orthogonality and parallelism constraints. Furthermore, a visual servoing library is used to conduct some simulated experiments to substantiate the theoretical results. As expected, we observe problems in control in the vicinity of a singularity as well as increased errors in pose estimation.
