  An Inertial Navigation System (INS) is a system that integrates acceleration
and angular velocity readings from an Inertial Measurement Unit (IMU), along
with other sensors such as GNSS position, GNSS velocity, and magnetometer, to
estimate the attitude, velocity, and position of a vehicle. This paper shows
that the INS problem can be analysed using the automorphism group of the
extended special Euclidean group: a group we term the extended similarity
group. By exploiting this novel geometric framework, we propose an observer
architecture with synchronous error dynamics; that is, the error is stationary
if the observer correction terms are set to zero. In turn, this enables us to
derive a modular, or plug-and-play, observer design for INS that allows
different sensors to be added or removed depending on what is available in the
vehicle sensor suite. We prove both almost-global asymptotic and local
exponential stability of the error dynamics for the common scenario of at least
IMU and GNSS position. To the authors' knowledge, this is the first non-linear
observer design with almost global convergence guarantees or with plug-and-play
modular capability. A simulation with extreme initial error demonstrates the
almost-global robustness of the system. Real-world capability is demonstrated
on data from a fixed-wing UAV, and the solution is compared to the
state-of-the-art ArduPilot INS.
