This paper considers the design of cylindric PRS serial chains. This five degree-of-freedom robot can be designed to reach an arbitrary set of eight spatial positions. However, it is often convenient to choose some of the design parameters and specify a task with fewer positions. For this reason, we study the three through eight position synthesis problems and consider various choices of design parameters for each. A linear product decomposition is used to obtain bounds on the number of solutions to these design problems. For all cases of six or fewer positions, the bound is exact and we give a reduction of the problem to the solution of an eigenvalue problem. For seven and eight position tasks, the linear product decomposition is useful for generating a start system for solving the problems by continuation. The large number of solutions so obtained contraindicates an elimination approach for seven or eight position tasks, hence continuation is the preferred approach.
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