There exists a duality relationship between a twist and a wrench in the dynamics of a mechanism or a multi-body system. Using this relationship, the coordinate-free expressions for the dynamic analysis of multi-loop mechanisms is derived. In the analysis, a Jacobian matrix expressing the relationship among loops is defined, and it is clarified, by using this matrix, that both twists and wrenches can be easily analyzed based on the duality relationship among them. A parallel mechanism having connecting chains with loops is also analyzed, and it is shown that the same procedure can be applied to a parallel mechanism with connecting chains, each loop of which has a motion space that is different from that of the mechanism.

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