I am trying to derive a set of initial conditions consisting of a range (ρ), two angles (β and ε) and their derivatives. Calculating ρ β and ε was straightforward ;However, finding $\dot{\rho}$, $\dot{\beta}$ and $\dot{\epsilon}$ is a little tricky.
I am working with three different reference frames, the Inertial frame (I) (O, e1, e2, e3), an intermediate frame A (O', a1, a2, a3) and the final frame B (O', b1, b2, b3). The transformation matrix from I to B is as follows:
\begin{equation} R_{I \rightarrow A} = \begin{bmatrix} \cos(\beta) & \sin(\beta) & 0\\ -\sin(\beta) & \cos(\beta) & 0\\ 0 & 0 & 1 \end{bmatrix} \end{equation}
\begin{equation} R_{A \rightarrow B} = \begin{bmatrix} \cos(-\epsilon) & 0 & -\sin(-\epsilon)\\ 0 & 1 & 0\\ \sin(-\epsilon) & 0 & \cos(-\epsilon) \end{bmatrix} \end{equation}
\begin{equation} R_{I \rightarrow B} = R_{A \rightarrow B}R_{I \rightarrow A} \end{equation}
Now the dynamics can be expressed as follows: \begin{equation} r_{P/O} = r_{O'/O} + r_{P/O'} \end{equation}
and with the help of the transport equation the velocity can be expressed as: \begin{equation} (*) {^I}v_{P/O} = {^I}v_{O'/O} + {^B}v_{P/O'} + {^I}\omega{^B} \times r_{P/O'} {}{}{} \end{equation}
Through the derivation of the dynamics the angular velocity ${^I}\omega{^B}$ can be determined.
\begin{equation} {^I}\omega{^A} = \begin{bmatrix} 0 \ \ \ \ \ a_1 \\ 0 \ \ \ \ \ a_2 \\ \dot{\beta} \ \ \ \ a_3 \end{bmatrix} \end{equation}
\begin{equation} {^A}\omega{^B} = \begin{bmatrix} \ \ \ \ 0 \ \ \ \ \ b_1 \\ -\dot{\epsilon} \ \ \ \ \ b_2 \\ \ \ \ \ 0 \ \ \ \ b_3 \end{bmatrix} \end{equation}
\begin{equation} {^I}\omega{^B} = {^I}\omega{^A} + {^A}\omega{^B} \end{equation}
\begin{equation} {^I}\omega{^B} = \begin{bmatrix} \dot{\beta}\sin(\epsilon) \ \ \ \ \ b_1 \\ -\dot\epsilon \ \ \ \ \ \ \ \ \ \ \ \ b_2 \\ \dot{\beta}\cos(\epsilon) \ \ \ \ b_3 \end{bmatrix} \end{equation}
This is where I get stuck, I know ${^I}v_{P/O}$, ${^I}v_{O'/O}$, $r_{P/O'}$, $r_{P/O}$, $r_{O'/O}$, $\rho$, $\beta$ and $\epsilon$. Another useful piece of information is that the way the frames are defined $r_{P/O'} = [\rho \ b_1; 0 \ b_2; 0 \ b_3]$ always. I'm stuck because I'm trying to solve for the unknown vector ${^B}v_{P/O'}$ in equation (*), which would be straightforward if I knew ${^I}\omega{^B}$, but I do not know $\dot{\beta}$ and $\dot{\epsilon}$. Currently, I have too many unknowns, being the 3x1 vector ${^B}v_{P/O'}$, $\dot{\rho}$, $\dot{\beta}$ and $\dot{\epsilon}$; If I knew $\dot{\beta}$ and $\dot{\epsilon}$ I could calculate ${^I}\omega{^B}$ and consequently ${^B}v_{P/O'}$ which will give me $\dot{\rho}$ as ${^B}v_{P/O'} = [\dot{\rho} \ b_1; v2 \ b_2; v3 \ b_3]$.
I have the correct values of $\dot{\rho}$, $\dot{\beta}$ and $\dot{\epsilon}$ that I can use to check if the values I calculated are correct.
