The instantaneous velocity analysis of a 2R1T (R denotes rotation

T denotes translation) spherical parallel mechanism was performed. As the center of the moving platform of the mechanism and the center of the base are plane-symmetric about the three spherical joints

the position solution of the moving platform can be easily obtained by using geometric method

but the calculation of instantaneous velocity is relatively complicated. In particularly

the calculation of angular velocity is heavily dependent on the correct solution of the rotation matrix or the complete Jacobian matrix

and the calculation efficiency is low and error-prone. For this reason

the Riemann symmetric space theory method is used in this paper. Starting from the symmetry of the subchain screw system

the constraints on joint movement of each subchain are established and the calculation of instantaneous velocity is simplified. Instantaneous velocity forms a linear space of which the base just explains the instantaneous degree of freedom of the mechanism. Finally

an example is given to verify the correctness and effectiveness of the method. By comparing ADAMS (Automatic Dynamic Analysis of Mechanical Systems) software simulation with theoretical calculation

the angular velocity calculation error is within a range of -0.004 rad/s~0.006 rad/s; and the linear velocity calculation error is within a range of -0.01 mm/s~0.015 mm/s

the calculation time is 5.187 seconds by using the symmetric space theory.