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In this paper the algebraic input-output (IO) equations for planar RRRP and PRRP linkages are derived by mapping the linkage displacement constraints into Study's soma coordinates and then using tangent-half angle substitutions to transform the trigonometric into algebraic expressions. Both equations are found to be equivalent to the one that has already been derived for RRRR linkages, giving exciting new insight into kinematic analysis and synthesis of planar four-bar linkages. The algebraic properties of the IO curve equations yield information regarding the topology of the linkage, such as the sliding position limits of the prismatic joints and/or the angle limits of the rotational joints. Additionally, the utility of the equations is successfully demonstrated with two approximate synthesis examples.

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