Approximating Displacements in ${\mathbb {R}}^3$ by Rotations in ${\mathbb {R}}^4$ and Its Application to Pointcloud Registration

Abstract

No proper norm exists to measure the distance between two object poses essentially because a general pose is defined by a rotation and a translation, and thus it involves magnitudes with different units. As a means to solve this dimensionalinhomogeneity problem, the concept of characteristic length has been put forward in the area of kinematics. The idea consists in scaling translations according to this characteristic length and then approximating the corresponding displacement defining the object pose in R 3 by a rotation in R 4 , for which a norm exists. This paper sheds new light on this kind of approximations which permits simplifying optimization problem whose cost functions involve translations and rotations simultaneously. A good example of this kind of problems is the pointcloud registration problem in which the optimal rotation and translation between two sets of corresponding 3D point data, so that they are aligned/registered, have to be found. As a result, a simple closed-form formula for solving this problem is presented which is shown to be an attractive alternative to the previous approaches.

On closed‐form solutions to the 4D nearest rotation matrix problem

“…Recently, it has been found that some problems arising in robotics, computer vision, and computer graphics—such as point‐cloud registration 1 and hand‐eye calibration problems 2 —can be formulated in terms of 4D rotation matrices. Unfortunately, experimental data leads to noisy 4D rotation matrices whose orthonormality must be restored.…”