Before revision:  Through formulating the attitude, velocity and position as one group state of group of double direct spatial isometries  , the transformed INS mechanization has proven to be group affine, which opens door to log-linearity and filtering consistency .
After revision:  Through formulating the attitude, velocity and position as one group state of group of double direct spatial isometries , the transformed INS mechanization has proven to be group affine, which means that the corresponding vector error state model will be trajectory-independent .