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

[+] Author and Article Information
J. P. Merlet

 INRIA, BP 93, 06902 Sophia-Antipolis, France

J. Mech. Des 128(1), 199-206 (Jun 20, 2005) (8 pages) doi:10.1115/1.2121740 History: Received May 19, 2005; Revised June 20, 2005

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.

Copyright © 2006 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 2

Mapping between the joint errors space and the generalized coordinates error space induced by J−TJ according to the norm: on top the Euclidean norm and on bottom the infinity norm. The Euclidean norm imposes a relation on the independent joint errors ΔΘ1,ΔΘ2.

Grahic Jump Location
Figure 3

The value of C23(P23)∕C23(P1),CF3(P23)∕CF3(P1) according to the choice of the three reference points on the platform

Grahic Jump Location
Figure 1

The 3−UP̱U robot




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