Construction of Confidence Regions for Uncertain Spatial Displacements With Dual Rodrigues Parameters

This article follows our recent work on the computation of kinematic confidence regions from a given set of uncertain spatial displacements with specified confidence levels. Dual quaternion algebra is used to compute the mean displacement as well as relative displacements from the mean. In constructing a 6D confidence ellipsoid, however, we use dual Rodrigues parameters resulting from dual quaternions. The advantages of using dual quaternions and dual Rodrigues parameters are discussed in comparison with those of three translation parameters and three Euler angles, which were used for the development of the so-called rotational and translational confidence limit (RTCL) method. The set of six dual Rodrigues parameters are used to define a parametric space in which a $6×6$ covariance matrix and a 6D confidence ellipsoid are obtained. An inverse operation is then applied to first obtain dual quaternions and then to recover the rotation matrix and translation vector for each point on the 6D ellipsoid. Through examples, we demonstrate the efficacy of our approach by comparing it with the RTCL method known in literature. Our findings indicate that our method, based on the dual Rodrigues formulation, yields more compact and effective swept volumes than the RTCL method, particularly in cases involving screw displacements.