In the field of robotics, it is now standard to describe the motion of robots relative to a fixed world frame. For example, the position and orientation of the distal link (or robot hand) is described as a rigid-body motion relative to the robot base. However, when it comes to the description of relative rigid-body position and orientation between individual robots in a multi-robot system (e.g., a group of humanoids performing specific tasks), assessing preferred relative rigid-body position and orientation of a robot relative to another would be more appropriate. Furthermore, it would be beneficial to have a "symmetrical" parameterization in which the corresponding parameters for a pair of robots in a group are calculated in the same way for the motion and its inverse. In this paper, we discuss the possible forms of this symmetrical parameterization for the rotations and rigid body motions. This extends and updates our previously presented work. We also present the properties of the symmetrical parameterizations in terms of product formulas. Due to its "symmetrical" property, this type of the parameterization has various potential benefits in the study of humanoid robots.
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