Forward kinematic analysis of the 3-RPRS parallel manipulator

This paper presents a comprehensive study of the forward kinematic problem of a newly introduced spatial parallel manipulator, namely, the 3-RPRS manipulator. The manipulator has three legs with two actuators in each, which connect a moving triangular platform to a fixed base. Three loop-closure equations are formed with the three passive joint angles in the legs, as the unknowns, which are subsequently reduced to a single univariate polynomial equation of degree 16. The coefficients of this equation are obtained as closed-form functions of the architecture parameters of the manipulator and the input joint variables. Furthermore, it is found that the polynomial has only the even-powered terms, leading to 8 pairs of solutions. One solution in each pair represents a pose on one side of the fixed base, and the other, a reflection of the same at the said base plane. The theoretical developments are illustrated via several numerical examples, one of which yields 16 real solutions, thus establishing that the obtained univariate polynomial is minimal in its degree. The results obtained are validated numerically, as well as verified against the inverse kinematics of the manipulator, thereby establishing the correctness of the formulation as well as the results.