The article presents a classification scheme based on the number of branches that a linkage may have and a general Grashof type mobility criterion for a group of over sixty types of four-, five-, six-, and seven-bar linkages, including RSSR, RSSP, RPSC, RSCP, RCSP, RCCC, and Duffy’s Groups 1 and 1* spatial linkages (1980). The criterion is derived from the quartic discriminant of the quadratic equation relating the input and output displacements. It has the simplicity compatible to that of the Grashof criterion and includes the Grashof criterion but without its ambiguity on the branch problem. The derivation and the demonstration are carried out through RSSR and RSSP mechanisms. The simplicity and generality of the proposed criterion surpass any mobility criterion available so far for any spatial linkage.

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