Acceleration formula to present in terms of Quaternions

Hello all,

I am applying extended Kalman filter in order to perform sensor fusion of Gyroscope and accelerometer in order to get vehicle orientation.

States are : X = [ q0, q1 ,q2, q3]'

Measurement : Z = [ax, ay, az]'

In EKF prediction is function of X. So in order to write prediction in term of ax ay and az I need formula of acceleration in these direction in term of quaternions.

Measurement include raw signals from accelerometer which are affected by gravity.

Could you provide me formula which include the gravity consideration in order to calculate the acceleration in x,y, and z direction. So I could perform calculation for acceleration.

Thank you in advance!

Kind regards,

Avdhut