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According to the structural characteristics of the planar 6-UPS parallel robot, the position and orientation of the mobile platform were described by choosing the spatial coordinates of three representative points as parameters. Combination of the constraint conditions during three representative points, nine quadratic polynomial equations with nine parameters were obtained. Finally, six quadratic polynomial equations, including six unknown parameters were obtained by eliminating three parameters of the nine equations. Aiming at the characteristics of the obtained quadratic polynomial equations, the traditional Newton-Raphson numerical iteration algorithm was improved and used to the numerical solution of general six-dimensional quadratic polynomial equations of parallel robots. The iterative algorithm was converged and an unique solution was obtained. The numerical example demonstrated that the time consumptions of the proposed algorithm was 0.14~0.23 ms and the traditional method of rotation matrix was 1.42~2.67 ms respectively under the same conditions. The proposed algorithm of representative points greatly reduced the computational time, improved the convergence speed and computational efficiency and laid a better foundation for the closed-loop realtime control with high-performance of the six DOFs planar parallel robot.