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

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 dimensional-inhomogeneity problem, the concept of
characteristic length
has been put forward in the area of kinematics. The idea consists of scaling translations according to this characteristic length and then approximating the corresponding displacement defining the object pose in
${\mathbb {R}}^3$
by a rotation in
${\mathbb {R}}^4$
, for which a norm exists. This article 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 3-D 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.