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This paper deals with the application of Lie Algebra to the mobility analysis of kinematic chains. It develops an algebraic formulation of a group‐theoretic mobility criterion developed recently by two of the authors of this publication. The instantaneous form of the mobility criterion presented here is based on the theory of subspaces and subalgebras of the Lie Algebra of the Euclidean group and their possible intersections. It is shown using this theory that certain results on mobility of over‐constraint linkages derived previously using screw theory are not complete and accurate. The theory presented provides for a computational approach that would allow efficient automation of the new group‐theoretic mobility criterion. The theory is illustrated using several examples. © 2003 Wiley Periodicals, Inc.

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This contribution presents a more general mobility criterion applicable to parallel platforms, unlike previously employed mobility criteria based on the well-known Kutzbach-Grübler criterion that often fails to provide the correct number of degrees of freedom of parallel manipulators, the mobility criterion introduced in this contribution provides the correct number of degrees of freedom for a wider class of parallel manipulators. Furthermore, the analysis provides insight into why the criteria based on the Kutzbach-Grübler criterion often fails. Moreover, the recently developed criterion computes the passive degrees of freedom in parallel platforms and the mobility of some classes of kinematically defficient parallel platforms. Finally, it is important to note that this criterion is based on an analysis of the subalgebras of the Lie algebra, se(3), also known as screw algebra, of the Euclidean group, SE(3). Moreover, it should be emphasized that, unlike other attempts to develop a criterion, the criterion developed in this contribution does not require any consideration of reciprocal screws. As is common in many areas of kinematics, the criterion presented here can be also obtained by an analysis of the subgroups of the Euclidean group.

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This contribution presents a more general mobility criterion applicable to parallel platforms, unlike previously employed mobility criteria based on the well-known Kutzbach-Gru¨bler criterion that often fails to provide the correct number of degrees of freedom of parallel manipulators, the mobility criterion introduced in this contribution provides the correct number of degrees of freedom for a wider class of parallel manipulators. Furthermore, the analysis provides insight into why the criteria based on the Kutzbach-Gru¨bler criterion often fails. Moreover, the newly developed criterion computes the passive degrees of freedom in parallel platforms and the mobility of some classes of kinematically defficient parallel platforms. Finally, it is important to note that this criterion is based on an analysis of the subalgebras of the Lie algebra, se(3), also known as screw algebra, of the Euclidean group, SE(3). As is common in many areas of kinematics, the criterion presented here can be also obtained by an analysis of the subgroups of the Euclidean group.

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