50.7.1.1 H Jacobian matrix in the ESKF filter correction

Parent: Filter correction , eskf-update Source: Solà 2017 Quaternion kinematics for ESKF

Evaluation of the H Jacobian

• In the prediction stage, the filter estimates the error state.
• Therefore, the Jacobian H needs to be defined w.r.t. the error state , and evaluated at the true state estimate 
• However, as the error state mean is zero (not yet observed), the true state is approximated to the nominal state 
• Thus we can use the nominal state as the evaluation point

The first Jacobian

• Depends on the sensor’s particular measurement function

The second Jacobian with

Source: Markley 2014 Measurement sensitivity matrix (Jacobian w.r.t. error states)

Expressing the error quaternion and true quaternion states (making use of the fact that all the error representations are equivalent to first order in  rotation error representation )