Forward kinematic problem of three 4-DOF parallel mechanisms (4-PRUR1, 4-PRUR2 and 4-PUU) with identical limb structures performing 3T1R motionpattern

This paper investigates the forward kinematic problem of three 4-dofparallel mechanisms performing three translations and one rotation motion, referred toas schonies motion. the kinematic arrangements of the mechanisms under study inthis paper are such that two of them are classied as 4-prur and one of them isa 4-puu. they are, respectively, special cases of 4-pr0r0r00r00, 4-pr00r00r0r0 and 4-pr00r0r0r00 parallel mechanisms that have originated from the type synthesis of 4-dofparallel mechanisms with identical limb structures. the forward kinematic problem isstudied in three-dimensional euclidean space, and a univariate expression describing theforward kinematic problem is obtained for each of the latter parallel mechanisms by theresultant method. the results obtained from this method show that a set of univariateexpressions of degree (72, 64, 64, 82, 4, 28) describes the forward kinematic problem of4-prur1 and 4-prur2 parallel mechanisms. also, a quadratic univariate expressionrepresents the forward kinematic problem of 4-puu parallel mechanisms. in addition,the system of equations corresponding to the forward kinematic problem is solved uponresorting to a homotopy continuation approach, which claries that the forward kinematicproblems of 4-prur1, 4-prur2 and 4-puu parallel mechanisms admit up to 236, 236and 2 nite solutions, real and complex.