Generation and forward displacement analysis of RPR-PR-RPR analytic planar parallel manipulators

Analytic manipulators tire manipulators tor which a characteristic, polynomial of fourth degree or lower can be obtained symbolically. Six types of RPR-PR-RPR analytic planar parallel manipulators (APPMs) are first generated using the component approach and the method based on the structure of the univariate equation. Of the six types, four are composed of Assur II kinematic chains while the other two are composed of Assur III kinematic chains. The forward displacement analysis (FDA) of two types of RPR-PR-RPR APPMs composed of Assur III kinematic chains is then performed. The FDA of each of the two types of APPMs composed of Assur III kinematic chains is reduced to the solution of a univariate cubic equation and a quadratic equation in sequence. It is also proven that the maximum number of real solutions to the FDA is 4 for the RPR-PR-RPR planar parallel manipulator with one aligned platform and one orthogonal platform, Examples with 4 real solutions for the RPR-PR-RPR planar parallel manipulator with one aligned platform and one orthogonal platform or 6 real solutions for the RPR-PR-RPR planar parallel manipulator with two aligned platforms are given at the end of this paper.