Zettelkasten - 50.7.3 ESKF reset

Parent: Fusing IMU with complementary sensory data Backlinks: 50.3 Error-State Kalman Filter

moves the rotation error to the global rotation
this keeps the rotation error small and far from any singularities

To update the global state, the reset has to obey unknown_filename.7.png

The reset has to preserve the quaternion norm, therefore an exact unit norm expression must be used, instead of an approximation. Using the Rodrigues parameter , the reset becomes unknown_filename.6.png

which leads to a two step update (1. linear Kalman update, 2. brute force normalisation) of unknown_filename.8.png

Notes:

MEKF using Rodrigues provides a theoretical justification for the brute force update
[+] Avoids the possibility of accumulated errors in the quaternion norm
[+] Rodrigues parameters map the rotation group into three dimensional Euclidian space (180 degree rotation errors are mapped to infinity),
- therefore, probability distributions with infinitely long tails (e.g. Gaussians) are compatible with the Rodrigues parameter space

Source: Solà 2017 Quaternion kinematics for ESKF

Reset of the error state mean after the true state is calculated from the nominal state and the error state
This is important for the orientation, because the new orientation error will be expressed locally w.r.t. orientation frame of the new nominal state
Also, need to update the covariance of the error

Define an error reset function unknown_filename.1.png

The reset operation: unknown_filename.2.png

with unknown_filename.3.png unknown_filename.4.png

Note:

In most cases, can be ignored, making G an 18x18 identity matrix.
For more precise results, i.e. for reducing long term error drift, should not be neglected.