A mathematical model related to stage errors and grid plate errors is constructed according to the different positions of grid plate in the auxiliary measuring device. Based on the least square principle, the error equation is converted into a regular equation. The influence of position scheme on the rank of a relation matrix is investigated to summarize the law between the position and the self-calibration model. With the the conditions that the equation need for the least square solution, in the self-calibration process, the grid plate must be rotated by 90° and simultaneously translated from its initial position, and this position transformation is also simulated. The research results show that it is only the position scheme containing three basic positions that makes the simulation result closer to the true value. These three basic positions are the necessary and sufficient conditions for the achievement of self-calibration by the least-squares method.