  This work deals with error models for trident quaternion framework proposed
in the companion paper "A Trident Quaternion Framework for Inertial-based
Navigation Part I: Motion Representation and Computation" and further uses them
to investigate the static and in-motion alignment for land vehicles.
Specifically, the zero-velocity and odometer velocity measurements are applied
in the static and in-motion alignment process, respectively. By linearizing the
trident quaternion kinematic equation, the right and left trident quaternion
error models are obtained. The resultant models are found to be equivalent to
those derived from profound group affine. Then the two models are used to
design the corresponding extended Kalman filters (EKF), namely, the
left-quaternion EKF (LQEKF) and the right-quaternion EKF (RQEKF). Simulations
and field tests are conducted to evaluate their actual performances. For the
static alignment, owing to their high consistency, the L/RQEKF converge much
faster than the EKF even without any heading information. For the in-motion
alignment, however, the two filters still need the assistance of the
analytical/optimization-based in-motion alignment methods at the very start to
avoid extremely large attitude errors, although they possess much larger
convergence region than the traditional EKF does.
