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The research on the development of 6-UPS parallel robot has made great progress in recent years. The 6-UPS parallel robot has been widely applied to the VR, entertainment, medical, aerospace simulator, wave compensation simulator, radio telescope (FAST) and so on. With the development of science and technology, the requirements of high speed, high precision, high rigidity, high dynamic performance, low inertia and small structural size have been proposed for this kind of typical parallel robot. The key point to realize these high performance requirements is that the forward kinematics forms a global real-time feedback accurately and timely. The existing methods for the solution of the forward kinematics of 6-UPS parallel robot can be basically divided into three categories: echelon-form, numerical iterative and extra-sensor approaches (Vertechy et al., 2008). In echelon-form approaches, 40 th , 20 th , 17 th and 14 th univariate polynomial algebraic equations, which obtain all solutions including unreasonable complex and extraneous roots, are obtained by dual quaternion method (Wampler, 1996) (Zhou et al., 2015), special geometric properties (Husty, 1996), Gröbner base algorithm (Cheng et al., 2010), Support Vector Regression (Morell et al., 2013), algebraic elimination method (Cheng et al., 2010) (Huang et al., 2010), and so on. Unfortunately, a variety of assembly modes exist due to the multi-solutions of forward kinematics for 6-UPS parallel robot. Moreover, the calculation of the kinematic model is complicated and time-consuming. Therefore, it is difficult to realize the real-time feedback control with the echelon-form schemes. In numerical iterative approaches, appropriate initial values of the actual position and orientation of 6-UPS parallel robot need to be chosen in advance to guarantee the robustness of the overall system. Yang et al. (2014) (2017) made use of the unit quaternion method to develop a fast numerical solution of forward kinematics. Furthermore, the numerical solution of redundant actuation has been analyzed, in which the singularity of the algorithm has been solved. However, further research is required to study the analytic solution. In summary, the first two

Forward kinematics for 6-UPS parallel robot using extra displacement sensor