Prove spatial velocity identity - screw theory This question involves a proof regarding coordinate transformations of velocities of screw motions. This comes from "A Mathematical Introduction to Robotic Manipulation" (the text is available for free here: http://www.cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf) 
The identity is shown below, and is on pg. 59 of the above link. Here's the image:

I know how to get $V_{a,c}^c$ (with respect to frame c). That comes directly from the chain rule, switching the order of $\dot{g}$ and $g^{-1}$. But I don't know how to get $V_{a,c}^b$ (with respect to frame b), i.e. the left-hand side. 
Any help much appreciated.
 A: The proof of 2.15 is more or less the same as proof of 2.14 as stated.
Let's start with definition of $V_{ac}^b$
$$
V^b_{ac} = g^{-1}_{ac}\dot g_{ac}
$$
Now substitute $g_{ac} = g_{ab}g_{bc}$ and $g^{-1}_{ac}= g^{-1}_{bc}g^{-1}_{ab}$.
\begin{align}
V^b_{ac} &= g^{-1}_{bc}g^{-1}_{ab} \dot{\overline{ g_{ab}g_{bc}}}  \\
&= g^{-1}_{bc}g^{-1}_{ab} \left( \dot g_{ab}g_{bc} + g_{ab}\dot g_{bc}\right) \\
&=  g^{-1}_{bc}\left(g^{-1}_{ab} \dot g_{ab}\right)g_{bc} + g^{-1}_{bc} \dot g_{bc} \\
&= Ad_{g^{-1}_{bc}} V^b_{ab} + V^b_{bc}
\end{align}
I hope it is clear. 
(Just in case: The overdot overline notation means $\dot{\overline{ g_{ab}g_{bc}}}= \frac{d}{dt}\left(g_{ab}g_{bc}\right)$. The bar is useful to indicate what has to be differentiated, only the dot would be confusing.)

In the text there is no meaning given to the symbol $V^c$ but only to $V^b$(velocity in body frame) and $V^s$ (velocity in global frame).

I'll try to explain how to think about screw motions and what is the diffference between . We can explain this quite easily with rotations. 
Let's have two coordinate frames $A,B \in SO(3)$. The fastest way to transform $A$ to $B$ is by rotation $g_{ab}$ around some great circle. You might be familiar with slerp, that is the way how to calculate this rotation. Velocity is just vector perpendicular to that great circle. You can express it the spatial(world) frame or in the rotating(body) frame, which is rotating from frame $A$ to the frame $B$. What is quite peculiar is that when you express velocity in the body frame, which is moving, the coordinates of velocity stays the same. In fact only vectors parallel to the velocity vector has constant coordinates in body frame. All other vectors, not parallel to velocity, expressed in body frame change in time.
Rotations around great circles are the same to the $SO(3)$ as screw motions to the $SE(3)$. So screw motions are just generalization of above so except changing orientation you can change position too.
