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 $4 \times 4$ 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.

A method to determine the two parameter set of circular cylinders, whosesurfaces contain three given points, is presented in the context of an efficientalgorithm, based on the set of two parameter projections of the points ontoplanar sections, to compute radius and a point where the axes intersect theplane of the given points. The geometry of the surface of points, whose positionvectors represent cylinder radius, r, and axial orientation, isrevealed and described in terms of symmetry and singularity inherent in thetriangle with vertices on the given points. This strongly suggests that, givenone constraint on the axial orientation of the cylinder, there are up to sixcylinders of identical radius on the three given points. A bivariate function,in two of the three line direction Plücker coordinates, is derived to provethis. By specifying r and an axis direction, say, perpendicular to agiven direction, one obtains a sixth order univariate polynomial in one of theline coordinates which yields six axis directions. These ideas are needed in thedesign of parallel manipulators