  The task of strapdown inertial navigation system (SINS) initial alignment is
to calculate the attitude transformation matrix from body frame to navigation
frame. In this paper, such attitude transformation matrix is divided into two
parts through introducing the initial inertially fixed navigation frame as
inertial frame. The attitude changes of the navigation frame corresponding to
the defined inertial frame can be exactly calculated with known velocity and
position provided by GNSS. The attitude from body frame to the defined inertial
frame is estimated based on the SINS mechanization in inertial frame. The
attitude, velocity and position in inertial frame are formulated together as
element of the group of double direct spatial isometries.It is proven that the
group state model in inertial frame satisfies a particular "group affine"
property and the corresponding error model satisfies a "log linear" autonomous
differential equation on the Lie algebra. Based on such striking property, the
attitude from body frame to the defined inertial frame can be estimated based
on the linear error model with even extreme large misalignments. Two different
error state vectors are extracted based on right and left matrix
multiplications and the detailed linear state space models are derived based on
the right and left errors for SINS mechanization in inertial frame. With the
derived linear state space models, the explicit initial alignment procedures
have been presented. Extensive simulation and field tests indicate that the
initial alignment based on the left error model can perform quite well within a
wide range of initial attitude errors, although the used filter is still a type
of linear Kalman filter. This method is promising in practical products
abandoning the traditional coarse alignment stage.
