Weighted local conditioning index of a positioning and orienting parallel manipulator

In the presence of positioning and orienting tasks, the singular values of Jacobian matrices have different units, thereby making it impossible to order them and calculate the associated condition numbers. Here, this dimensional in-homogeneity is resolved by introducing a weighting factor. In this method, both the Jacobian and twist vector are made homogeneous, simultaneously. Moreover, relations between the weighting factors used here to the homogeneous Jacobian matrices derived by others are given. This factor should be constant throughout the workspace, while it is pose dependent in the latter methods. As a case study, both methods are applied to a Tricept parallel manipulator with complex degrees of freedom. A local conditioning index, as a dexterity index, is plotted in the workspace. Although both methods lead to homogeneous Jacobian matrices, obvious differences between the plotted local conditioning indices are revealed here. Therefore, those homogeneous Jacobian matrices derived by others, and the associated dexterity indices are unreliable.