This paper attempts to establish the unified topological models and corresponding mathematical representations for planar simple joint, multiple joint and geared (cam) kinematic chains. First, the conventional topological representation models of kinematic chains are introduced. Then new topological models of multiple joint and geared (cam) kinematic chains, which are derived from the topological graph of simple joint kinematic chains, are presented. The characteristics of the new topological graphs and their associations with the topological graph of simple joint kinematic chains are also addressed. The most important merit of the new topological graphs is that it makes it much easier to do unified structure synthesis and further establish conceptual design platform for various planar mechanisms of these kinds.

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