The type/number of operation modes of a parallel manipulator (PM) may vary with the link parameters of the PM. This paper presents a systematic classification of a 3-RER PM based on the type/number of operation modes. The 3-RER PM was proposed as a 4-DOF (degree-of-freedom) 3T1R PM in the literature. Using the proposed method, the classification of a PM based on the type/number of operation modes can be carried out in four steps, including formulation of constraint equations of the PM, preliminary classification of the PM using Gröbner Cover, operation mode analysis of all the types of PMs using primary decomposition of ideals, and identification redundant types of PMs. Classification of the 3-RER PM shows that it has 19 types. Besides the two 4-DOF 3T1R operation modes, different types of 3-RER PMs may have up to two more 3-DOF or other types of 4- DOF operation modes. This work is the first systematic study on the impact of link parameters on the operation modes of the 3-RER PM and provide a solid foundation for further research on the design and control of 3-RER PMs and other multi-mode (or reconfigurable) PMs.

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