The subject of this paper is about the synthesis of the design parameters by considering the prescribed operation modes at the design stage for a parallel manipulator with three RPS legs. The synthesis is based on the Euler parametrization and the results of primary decomposition. The design parameters and the coordinates of one RPS leg are initially defined to formulate the constraint equation associated with this leg. Seven classes of the RPS leg are identified and the geometric properties of each class are highlighted. By selecting three different or identical classes of the RPS leg, a new 3-RPS parallel manipulator is proposed without specific values of the design parameters. The primary decomposition is computed over a set of three constraint equations. One or more Euler parameters in the results of primary decomposition is constrained to be equal to null, which leads to particular type of operation mode. The methodology also provides new archi-tectures of the 3-RPS parallel manipulators based on a classification of the RPS leg, that satisfy the prescribed operation mode.