  This paper proposes a unified mathematical framework for inertial measurement
unit (IMU) preintegration in inertial-aided navigation system in different
frames under different motion condition. The navigation state is precisely
discretized as three parts: local increment, global state, and global
increment. The global increment can be calculated in different frames such as
local geodetic navigation frame and earth-centered-earth-fixed frame. The local
increment which is referred as the IMU preintegration can be calculated under
different assumptions according to the motion of the agent and the grade of the
IMU. Thus, it more accurate and more convenient for online state estimation of
inertial-integrated navigation system under different environment. Furthermore,
the covariance propagation based on left perturbation is proposed for the first
time, which is independent of the inputs of the gyroscope and accelerometer.
Finally, we show the monotonicity of the uncertainty for determinant optimality
criteria and R\'enyi entropy optimality criteria.
