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-DOF parallel mechanisms performing three translations and one rotation motion, referred to as Schon ies motion. The kinematic arrangements of the mechanisms under study in this paper are such that two of them are classied as 4-PRUR and one of them is a 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-DOF parallel mechanisms with identical limb structures. The forward kinematic problem is studied in three-dimensional Euclidean space, and a univariate expression describing the forward kinematic problem is obtained for each of the latter parallel mechanisms by the resultant method. The results obtained from this method show that a set of univariate expressions of degree (72, 64, 64, 82, 4, 28) describes the forward kinematic problem of 4-PRUR1 and 4-PRUR2 parallel mechanisms. Also, a quadratic univariate expression represents the forward kinematic problem of 4-PUU parallel mechanisms. In addition, the system of equations corresponding to the forward kinematic problem is solved upon resorting to a homotopy continuation approach, which claries that the forward kinematic problems of 4-PRUR1, 4-PRUR2 and 4-PUU parallel mechanisms admit up to 236, 236 and 2 nite solutions, real and complex.