Ryan i'm not sure what you mean by motivation. From my end, my motivation is very clear: i'm doing research in nonlinear control & robotics and i'm investigating a very specific thing (the application of a specific map $q=h(y)$). However, tensorial analysis is not strong point and i'm not really sure about the last three equations. I could rephrase the question with a very specific argument: does the inertia tensor and the Coriolis vector transform this way? However in order to provide a background, i've elaborated on the whole thing!

Well, it is a problem in control theory and particularly in tracking control. In a nutshell, given a tracking controller $f(x,g(x))$ for a nonlinear system, which tracks arbitrary reference paths $g(x)$, if the reference is a straight line, can i simplify the controller?

Ok let me just be a little more formal on the math. The vectors x,y∈Rn (column vectors), thus differentiating gives $\dot x=DΨ \dot y$ ($D\Psi \in \mathbb{R}^{n\times n}$- a square matrix). It is indeed the change of variables formula for the derivative.

Jacque, if you're referring to $\dot x=D\Psi \dot y$, then this is correct. You can calculate it using basic analysis.