Forward displacement analysis of third-class analytic 3-RPR planar parallel manipulators

Analytic manipulators are manipulators with characteristic polynomial of fourth degree or lower. This paper presents some new results on the forward displacement analysis of third-class analytic 3-RPR planar parallel manipulators. Analytic 3-RPR planar parallel manipulators are first classified into four types. Of the four types, three are third-class analytic planar parallel manipulators. A third-class mechanism here refers to a mechanism which contains one third-class Assur kinematic chain. The forward displacement analysis of different types of third-class analytic planar parallel manipulators is then performed. The forward displacement analysis of planar parallel manipulators with non-similar aligned platforms is reduced to the solution of a cubic equation in x and a quadratic equation, in sequence – instead of the polynomial equation of degree 6 in x derived previously – while that of each of the other two types of third-class analytic planar parallel manipulators is reduced to the solution of two quadratic equations in sequence.