  Strapdown inertial navigation systems (SINS) are ubiquitious in robotics and
engineering since they can estimate a rigid body pose using onboard kinematic
measurements without knowledge of the dynamics of the vehicle to which they are
attached. While recent work has focused on the closed-form evolution of the
estimation error for SINS, which is critical for Kalman filtering, the
propagation of the kinematics has received less attention. Runge-Kutta
integration approaches have been widely used to solve the initial value
problem; however, we show that leveraging the special structure of the SINS
problem and viewing it as a mixed-invariant vector field on a Lie group, yields
a closed form solution. Our closed form solution is exact given fixed gyroscope
and accelerometer measurements over a sampling period, and it is utilizes 12
times less floating point operations compared to a single integration step of a
4th order Runge-Kutta integrator. We believe the wide applicability of this
work and the efficiency and accuracy gains warrant general adoption of this
algorithm for SINS.
