50.7 ESKF update / Fusing IMU with complementary sensory data

Parent: [IMU index](imu index.md), 50.5-error-state-kalman-filter
Source: Solà 2017 Quaternion kinematics for ESKF

• In the ESKF, the arrival of non-IMU sensor data triggers a correction stage.
• This correction makes the IMU biases observable , allows correct estimation of the biases

The correction stage is three-fold:

1. observe the error state by way of filter correction
2. ‘add’ the observed errors to the nominal state to get the supposed ‘true’ state according to the composition rules in  variables in ESKF using IMUs
3. reset  the error state

Source: Markley 2014

What if several measurements come in without IMU / propagation in between (i.e. without a reset in between)?

• to avoid recalculating the nonlinear function h(x_true), the expectation of the measurement can be computed

which makes the update equation

Residual becomes y_k - delta_theta_k, s. [50.5.1 Observation of the error state (filter correction)](50.5.1 observation of-the-error-state-(filter-correction).md)

But if a reset is done after each measurement update, the equation above simplifies to