  Stochastic filters for on-line state estimation are a core technology for
autonomous systems. The performance of such filters is one of the key limiting
factors to a system's capability. Both asymptotic behavior (e.g.,~for regular
operation) and transient response (e.g.,~for fast initialization and reset) of
such filters are of crucial importance in guaranteeing robust operation of
autonomous systems.
  This paper introduces a new generic formulation for a gyroscope aided
attitude estimator using N direction measurements including both body-frame and
reference-frame direction type measurements. The approach is based on an
integrated state formulation that incorporates navigation, extrinsic
calibration for all direction sensors, and gyroscope bias states in a single
equivariant geometric structure. This newly proposed symmetry allows modular
addition of different direction measurements and their extrinsic calibration
while maintaining the ability to include bias states in the same symmetry. The
subsequently proposed filter-based estimator using this symmetry noticeably
improves the transient response, and the asymptotic bias and extrinsic
calibration estimation compared to state-of-the-art approaches. The estimator
is verified in statistically representative simulations and is tested in
real-world experiments.
