For the fast and efficient closed-loop real-time feedback control of 6-UPS parallel robot (6-UPS), a novel high efficiency calculation of the workspace is proposed and investigated. As a typical Nearly General Platform (NGP), 6-UPS has good symmetries. The symmetries effectively reduce computational cost and improve computational efficiency in the kinematics, singularity, dynamics, and optimization. To scrupulously demonstrate the symmetries of workspace, a novel algorithm is proposed. The modified Euler angles (T&T angles) are employed to represent the orientation matrix of 6-UPS, the inverse kinematics is analyzed, and the workspace of 6-UPS is obtained using the discretization algorithm. Meanwhile, the symmetries of the total orientation workspace are also proved. Compared with the traditional methods, the total orientation workspace reduces 5/6 computation cost, which means that the corresponding computation efficiency is increased by 6 times. Through theoretical and numerical calculations, the symmetries of the total orientation workspace of 6-UPS are verified. The proof of the symmetries lays a solid foundation for improving the computational efficiency of kinematics, dynamics, and control of 6-UPS.
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