  This letter re-visits the problem of visual-inertial navigation system (VINS)
and presents a novel filter design we dub the multi state constraint
equivariant filter (MSCEqF, in analogy to the well known MSCKF). We define a
symmetry group and corresponding group action that allow specifically the
design of an equivariant filter for the problem of visual-inertial odometry
(VIO) including IMU bias, and camera intrinsic and extrinsic calibration
states. In contrast to state-of-the-art invariant extended Kalman filter (IEKF)
approaches that simply tack IMU bias and other states onto the
$\mathbf{SE}_2(3)$ group, our filter builds upon a symmetry that properly
includes all the states in the group structure. Thus, we achieve improved
behavior, particularly when linearization points largely deviate from the truth
(i.e., on transients upon state disturbances). Our approach is inherently
consistent even during convergence phases from significant errors without the
need for error uncertainty adaptation, observability constraint, or other
consistency enforcing techniques. This leads to greatly improved estimator
behavior for significant error and unexpected state changes during, e.g.,
long-duration missions. We evaluate our approach with a multitude of different
experiments using three different prominent real-world datasets.
