The translational displacement computational algorithm base on a novel phase-shift model is proposed eliminating the geometric error of the grating interferometer for precision positioning of a multi-degree-of-freedom motion stage. Firstly, the mechanism of the geometric error of the grating interferometer is analyzed, and the novel phase-shift model of the grating interferometer is constructed based on rigid body kinematics and affine geometry transformation. High accuracy of the model is demonstrated by ZEMAX simulation. Then, according to Taylor series expansion, the phase-shift model is simplified by polynomial regression to solve the problems of a large amount of computational effort and inability to derive the translational displacement computational algorithm. The availability and accuracy of the translational displacement computational algorithm are verified by ZEMAX simulation and experiment.