I have a system where an insect starts at origin of an inertial reference frame $(x,y,z)$ at $t_0=0$ and takes flight for say $t=[0,2\pi]$. I know how to model the flight path of the insect where the position vector starts at origin and tracks the flight of the insect. The position vector of the insect is $\vec{r}(t)=t^2\sin(t)\hat{i}+t\cos(t)\hat{j}+t\sin(t)\hat{k}$.
I have a local body-fixed reference frame (Frenet-Serret frame) $(a,b,c)$ on the insect and I want to determine the value of the Euler angles between the inertial $(x,y,z)$ and local $(a,b,c)$ reference frames. How would I calculate the value of the angles throughout flight while $t=[0,2\pi]$?
Is there an equation that can be derrived to show the relationship between the inertial $(x,y,z)$ frame and the $(a,b,c)$ frame?
The coordinates of the $a,b,c$ vectors form the rotation matrix from the inertial frame to the local one. Then you need to identify this matrix with that obtained via the Euler angles.
A row of the Euler matrix has terms like $u,v,w=\cos\alpha,\sin\alpha\cos\beta,\sin\alpha\sin\beta$.
Then
$$\tan\beta=\frac wv,\\ \tan\alpha=\pm\frac{\sqrt{v^2+w^2}}u.$$
A similar column will give you the third angle. You can fill in the details, depending on the exact expression of your Euler system.