The use of finite symmetry groups to enumerate basic kinematic chains has been extended by making use of the subgroup structure of the symmetry groups. This refinement not only makes the enumeration of kinematic chains considerably more efficient, but also permits the enumeration of all distinct inversions of each of the chains found. As examples, all chains with 2, 3, 4, and 5 independent loops and 1, 2, or 3 degrees of freedom and their inversions have been found on a personal computer.
Further Applications of Group Theory to the Enumeration and Structural Analysis of Basic Kinematic Chains
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Tuttle, E. R., Peterson, S. W., and Titus, J. E. (December 1, 1989). "Further Applications of Group Theory to the Enumeration and Structural Analysis of Basic Kinematic Chains." ASME. J. Mech., Trans., and Automation. December 1989; 111(4): 494–497. https://doi.org/10.1115/1.3259027
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