In order to meet the structure positioning accuracy requirements of the telescope secondary mirror, a fixed length rod end axial translational motion model of hexapod was proposed. From the point of view of differential geometry, the nonlinear kinematics between the input joint space vector and the output workspace vector of the mechanism was studied, and the curvature concept was used to measure the nonlinear bending of the trajectory. Comparing with the Jacobian matrix, it was found that the curvature of the parallel mechanism was consistent with the instantaneous linear property reflected by the Jacobian matrix. The maximum nonlinearity error of the designed secondary mirror parallel adjustment mechanism was about 3.15 μm in the whole motion range. The test results show that after the polynomial error curve fitting correction, the three-dimensional translational repeating positioning accuracy of the secondary mirror adjustment mechanism is less than 2.6 μm, and the two-dimensional rotation repeat positioning accuracy is less than 1.8″, which meet the needs of actual telescope observation. At the same time, the curvature metric method can also provide a new idea for the nonlinear analysis and correction of other parallel mechanisms.