How to find velocity transformation between two coordinate frames?

edit

I am trying to transform a twist message(both linear and angular) from frame A to D(there are intermediate frame B and C).

My understanding is that:

Veloctiy_A_in_D<> = Rot_A_B * Rot_B_C * Rot_C_D * Velocity_A<>

I am not exactly sure if this is right as I am unable to find any proper mathematical reference to validate this. As we are navigating from one reference frame to another, should we also be adding their translational values?

I lookedup at the ROS's commented out

lookUpTwist(..)

function here http://docs.ros.org/hydro/api/tf/html... But, I don't understand why there is a cross product take for linear velocity and was wondering if you guys could help me out understand and formulate it.