Lie Algebra and the Mobility of Kinematic Chains

Abstract

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.

Mechanism mobility and a local dimension test

“…The 1-bounded local dimension, dim z,1 Z, is the corank of the Jacobian matrix, which as noted earlier was used for mobility analysis in [20,21,28]. Relation 15 shows that this is an upper bound on the mobility; one may establish tighter bounds by considering more derivatives to find thed-bounded local dimension for d > 1.…”

“…Yet another approach, closely related to the current contribution, is to compute mobility as the corank of the Jacobian matrix evaluated at an assembly configuration of the mechanism [20,21,28]. Since these approaches directly use the differentials of the loop equations, they correctly predict the mobility of many paradoxical mechanisms provided that one knows at least one general assembly configuration.…”

Kinematic synthesis using tree topologies

“…The exceptional kinematic chain loop can be substituted for the subalgebra corresponding to the intersection which may not always have a simple equivalence. Fanghella and Galletti [43] find many equivalences of intersections using group theory, and Rico and Ravani [42] do the same using Lie algebras. However, finding the equivalent open kinematic chain is not always a simple task, depending on the mobility class of the loop.…”

“…Let $ J be the screw system corresponding to a given joint or set of joints J, and let $ * J be the smallest subalgebra of se(3) containing all the possible infinitesimal mechanical liaisons [42] between two rigid bodies of a serial chain connected by that joint or set of joints. In general, this is calculated for consecutive rigid bodies separated by a joint.…”

“…Refer to the study of the mobility of singleloop kinematic chains done by Rico and Ravani [42] for a more in depth study of types of kinematic loops.…”