Forward Kinematic Problem of Three 4-DOF Parallel Mechanisms (4-PRUR1, 4-PRUR2 and 4-PUU) with Identical Limb Structures Performing 3T1R Motion Pattern

This paper investigates the forward kinematic problem of three 4-DOF parallel mechanisms performing three translations and one rotation motion referred to as Schönfliesmotion. The kinematic arrangements of the mechanisms under study in this paper are such that two of them are classified as 4-PRUR and one of them is a 4-PUU. They are respectively special cases of 4-PR′R′R″R″, 4-PR″R″R′R'' and 4-PR″R′R′R″ parallel mechanisms that have originated from thetype 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 mechanism by resultant method. The results obtained from this method show that a set of univariate expressions of degree (72, 64, 64, 82, 4, 28) describe the forward kinematic problem of 4-PRUR1 and 4-PRUR2 parallel mechanisms. As well as a quadratic univariate expression represent 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 homotopy continuation approach which clarifies that the forward kinematic problem of a 4-PRUR1, 4-PRUR2 and 4-PUU parallel mechanisms admit up to 236, 236 and 2 finite solutions, real and complex.