We measure the long-term systematic component of the astrometric error in the GeMS MCAO system as a function of field radius and Ks magnitude. The experiment uses two epochs of observations of NGC 1851 separated by one month. The systematic component is estimated for each of three field of view cases (15'' radius, 30'' radius, and full field) and each of three distortion correction schemes: 8 DOF/chip + local distortion correction (LDC), 8 DOF/chip with no LDC, and 4 DOF/chip with no LDC. For bright, unsaturated stars with 13 < Ks < 16, the systematic component is < 0.2, 0.3, and 0.4 mas, respectively, for the 15'' radius, 30'' radius, and full field cases, provided that an 8 DOF/chip distortion correction with LDC (for the full-field case) is used to correct distortions. An 8 DOF/chip distortion-correction model always outperforms a 4 DOF/chip model, at all field positions and magnitudes and for all field-of-view cases, indicating the presence of high-order distortion changes. Given the order of the models needed to correct these distortions (~8 DOF/chip or 32 degrees of freedom total), it is expected that at least 25 stars per square arcminute would be needed to keep systematic errors at less than 0.3 milliarcseconds for multi-year programs. We also estimate the short-term astrometric precision of the newly upgraded Shane AO system with undithered M92 observations. Using a 6-parameter linear transformation to register images, the system delivers ~0.3 mas astrometric error over short-term observations of 2-3 minutes.