姿态的矢量参数化
Vectorial Parameterizations of Pose
摘要：
机器人和计算机视觉问题通常需要处理包括平移和旋转的刚体运动-统称为姿态。在某些情况下，姿态的向量参数化可能是有用的，其中向量空间的元素满射映射到矩阵李群。例如，这些向量表示可以用于优化以及组上的不确定性表示。最常见的映射是矩阵指数，它将李代数的元素映射到相关的李群上。然而，这种选择并不是唯一的。以前已经展示了如何描述SO（3）的所有这些向量参数化，旋转群。一些结果也是已知的组的姿态，其中有可能建立一个家庭的向量映射，包括矩阵指数以及凯莱变换。我们扩展什么是已知的这些姿态映射到机器人中常见的4 × 4表示，并展示了三个不同的例子，建议的姿态映射：（i）姿态插值，（ii）姿态伺服控制，（iii）在pointcloud对齐问题的姿态估计。在点云对齐问题，我们的结果导致一个新的算法的基础上的凯莱变换，我们称之为CayPer。
英文摘要：
Robotics and computer vision problems commonly require handling rigid-body motions comprising translation and rotation - together referred to as pose. In some situations, a vectorial parameterization of pose can be useful, where elements of a vector space are surjectively mapped to a matrix Lie group. For example, these vectorial representations can be employed for optimization as well as uncertainty representation on groups. The most common mapping is the matrix exponential, which maps elements of a Lie algebra onto the associated Lie group. However, this choice is not unique. It has been previously shown how to characterize all such vectorial parameterizations for SO(3), the group of rotations. Some results are also known for the group of poses, where it is possible to build a family of vectorial mappings that includes the matrix exponential as well as the Cayley transformation. We extend what is known for these pose mappings to the 4 x 4 representation common in robotics, and also demonstrate three different examples of the proposed pose mappings: (i) pose interpolation, (ii) pose servoing control, and (iii) pose estimation in a pointcloud alignment problem. In the pointcloud alignment problem our results lead to a new algorithm based on the Cayley transformation, which we call CayPer.
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