Jacobian, Manipulability, Condition Number, and Accuracy of Parallel Robots

TLDR

Jacobian, manipulability, and condition number concepts are foundational in robotics, yet their practical significance remains poorly understood. The paper revisits these concepts for parallel robots to use them as accuracy indices guiding optimal design. The authors analyze local dexterity indices based on Jacobian condition numbers, evaluate global conditioning indices, highlight computational issues, and explore alternative indices for optimal design. The study finds that the standard Jacobian is inadequate for platform positioning error analysis, classical manipulability interpretation is flawed, and condition numbers vary inconsistently, undermining their reliability as accuracy indices.

Abstract

Although the concepts of Jacobian matrix, manipulability, and condition number have existed since the very early beginning of robotics their real significance is not always well understood. In this paper we revisit these concepts for parallel robots as accuracy indices in view of optimal design. We first show that the usual Jacobian matrix derived from the input-output velocities equations may not be sufficient to analyze the positioning errors of the platform. We then examine the concept of manipulability and show that its classical interpretation is erroneous. We then consider various common local dexterity indices, most of which are based on the condition number of the Jacobian matrix. It is emphasized that even for a given robot in a particular pose there are a variety of condition numbers and that their values are not coherent between themselves but also with what we may expect from an accuracy index. Global conditioning indices are then examined. Apart from the problem of being based on the local accuracy indices that are questionable, there is a computational problem in their calculation that is neglected most of the time. Finally, we examine what other indices may be used for optimal design and show that their calculation is most challenging.