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 SE2(3) , the transformed INS mechanization has proven to be group affine, which opens door to log-linearity and filtering consistency .