Quaternions and Direction Cosine Matrix changing in time

[Jared Finneker]

TL;DR

Quaternions and Direction Cosine Matrix changing in time. Need to find the DCM from an inertial to body fixed frame (or vice versa). Quaternion convention is JPL

I've already posted this question on the mathematics website of stack exchange, but I've received more help here in the past so will share it here as well.

I am developing a tool for missile trajectory (currently without guidance). One issue is that the aerodynamic equations on the missile are in the body axis frame. I need to convert them in the inertial axis frame.
Lets make one thing clear: I am using the JPL quaternion convention. This means that it is the last element of the quaternion vector that is scalar, not the first. This is a left handed convention.
I cannot state the details of the paper here as its locked behind a paywall, but I will include the function that is causing most confusion:

$$ C_{{q}^{I}_{B}}(0) = \begin{bmatrix}0&0&1\\0&-1&0\\1&0&0\end{bmatrix} C_{{q}^{E}_{L}}C_{{q}^{I}_{E}}(0)$$

the notation is the following: C refers to the DCM relative to a quaternion q, whose superscript denotes the frame your are transferring from, and subscript the frame you are transferring to. As such, it reads: for the initial state, the DCM for reference frame transformation of the I frame (Inertial) to the B frame (Body fixed) is the matrix above, times the DCM of the ECEF frame (Earth centered) to L frame (launch pad frame) times the DCM from the I frame to ECEF frame at the initial state.

It should be noted that the I and ECEF (E frame) are rotating with respect to one another, in fact it is the ECEF frame that is rotating, and as such any DCM that includes the E frame is a time relation and not a constant value matrix. The same is true for the B frame (body fixed), the rocket is moving in space and therefore $$C_{{q}^{I}_{B}}$$ is a time dependent matrix.

The matrix above should be the matrix of transformation from the Launch pad frame to the Body axis frame, however I do not agree with the result. The paper states that at launch the euler angles are pi, pi/2 and 0.
I have been following Zarchan and other resources that use the Hamilton convention- this is where the quaternion vector has the scalar as its first value. My concern is that the convention is causing the issue because the equation of the DCM to the euler angles is not correct. As such I cannot derive the needed DCM from the I to the B frame. So what is the relation of the DCM from the I frame to the B frame in time?

 

[Kyouran]

How is the launch pad frame defined? NED? I haven't used quaternions in years, so I wouldn't remember how they factor in all this, but usually when transforming from an NED frame to the body frame you use the following DCM:

$\begin{bmatrix} \cos(\phi_2) \cos(\phi_1) & \cos(\phi_2) \sin(\phi_1) & -\sin(\phi_2) \\[0.3em] -\sin(\phi_1) \cos(\phi_3) + \sin(\phi_3) \sin(\phi_2) \cos(\phi_1) & \cos(\phi_3) \cos(\phi_1) + \sin(\phi_3) \sin(\phi_2) \sin(\phi_1) & \sin(\phi_3) \cos(\phi_2) \\[0.3em] \sin(\phi_3)\sin(\phi_1) + \cos(\phi_3) \sin(\phi_2) \cos(\phi_1) & -\sin(\phi_3) \cos(\phi_1) + \cos(\phi_3) \sin(\phi_2) \sin(\phi_1) & \cos(\phi_3) \cos(\phi_2) \end{bmatrix}$

Where $\phi_1$ is yaw, $\phi_2$ is pitch and $\phi_3$ is the roll angle. Then you just need to multiply that with the transformations from ECEF to NED and the transformation from ECI to ECEF.