  This paper investigates the problem of inertial navigation system (INS)
filter design through the lens of symmetry. The extended Kalman filter (EKF)
and its variants, have been the staple of INS filtering for 50 years; however,
recent advances in inertial navigation systems have exploited matrix Lie group
structure to design stochastic filters and state observers that have been shown
to display superior performance compared to classical solutions. In this work
we consider the case where a vehicle has an inertial measurement unit (IMU) and
a global navigation satellite system (GNSS) receiver. We show that all the
modern variants of the EKF for these sensors can be interpreted as the recently
proposed Equivariant Filter (EqF) design methodology applied to different
choices of symmetry group for the INS problem. This leads us to propose two new
symmetries for the INS problem that have not been considered in the prior
literature, and provide a discussion of the relative strengths and weaknesses
of all the different algorithms. We believe the collection of symmetries that
we present here capture all the sensible choices of symmetry for this problem
and sensor suite, and that the analysis provided is indicative of the relative
real-world performance potential of the different algorithms.
