Transforming reference frames

relder

I am working on some vector frame transformations for describing the motion of an oil pump jack. I am using these transformations in a Mathematica program that animates the motion equations that I calculate, and the animation is not working correctly. This leads me to believe that my transformations might be incorrect.

Here's a picture of the pump jack.



Here's a simple picture of what I'm using for the reference frames. There's an i,j,k frame, and an a, b, and c frames.


The transformations that I'm currently using are

a[1] -> Cos[\[Theta][t]] i + Sin[\[Theta][t]] j,
a[2] -> -Sin[\[Theta][t]] i + Cos[\[Theta][t]] j,
a[3] -> k,
b[1] -> Cos[\[Beta][t]] a[1] + Sin[\[Beta][t]] a[2],
b[2] -> -Sin[\[Beta][t]] a[1] + Cos[\[Beta][t]] a[2],
b[3] -> a[3],
c[1] -> Cos[\[Gamma][t]] b[1] + Sin[\[Gamma][t]] b[2],
c[2] -> Sin[\[Gamma][t]] b[1] + Cos[\[Gamma][t]] b[2],
c[3] -> b[3]

Unfortunately the program doesn't like this. Any help you have is appreciated.

Ryan