Analytical Determination of Link Interference in Planar Parallel Manipulators Using Elliptical Enclosures

Link interference is one of the issues that reduce the dimensions of the usable workspace in parallel manipulators. Since the links may have complicated shapes, they are often enclosed in convex shapes of simpler geometry for the sake of computational detection of link interference. In this paper, ellipses are used for this purpose, which reduces the problem of detecting interferences between two links to that of identifying the tangency between the corresponding ellipses. First, a condition for tangency of two conics is derived in closed form in terms of the respective coefficients. This condition is then used to obtain the loci of link interference for two parallel manipulators, namely the 5-bar and the 3- \({\underline{\text {R}}\text {RR}}\) planar parallel manipulator.