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This paper involves the investigation of the forward kinematic problem of three 4-DOF parallel robots, named as 4-PRUR1, 4-PRUR2, 4-PUU, performing 3 translations and one rotation, namely Schönflies motion. The foregoing parallel robots are special cases of 3 parallel robots, named as 4-PR′R′R″R″, 4-PR″R″R′R′ and 4-PR″R′R′R″, respectively, arisen from the type synthesis performed for 4-DOF parallel mechanisms with identical kinematic limb structures. Each robot has 4 identical kinematic chains and each chain consists of one prismatic active joint and 4 revolute passive joints. Due to different direction of revolute joints in each limb, 3 different architectures are considered in this paper. The forward kinematic problem is explored in seven-dimensional kinematic space using the so-called Study's parameters and LIA algorithm and eventually, it has been shown that an algebraic expression of degree 4 indicates the forward kinematic of each kinematic chains of parallel robots under study in this paper. Moreover, using homotopy continuation and comparison with resultant method, it reveals that the forward kinematic problem of this robots have up to 236,236 and 2 real solutions, respectively.

Keywords: 4-DOF parallel robot, 3T1R motion pattern, forward kinematic problem, seven-dimensional kinematic space, Study's parameters

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