The relations among the associated linkages (ALs), the redundant constraints, the passive degree of freedom (DoF), and the degree of freedom are studied systematically for the type synthesis of the closed mechanisms in this paper. First, the kinematic pairs with multidegree of freedoms are formed by combination of the basic joints with single degree of freedom in series, and the formulae are derived for the calculation of the degree of freedoms of the associated linkages, the number of the basic joints, and the valid numbers of the basic links in the associated linkages. Second, many different associated linkages and the number of basic links are derived, and the relations among the associated linkages, the redundant constraints, the passive degree of freedom, and the degree of freedoms of the closed mechanisms are analyzed. Third, some topology graphs are derived and the relative closed mechanisms with the redundant constraints and the passive degree of freedom are synthesized. Finally, the redundant constraints and the degree of freedoms of the closed mechanisms are determined.

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