I am analyzing a program which transforms vector fields by action of diffeomorphisms and feedbacks.
Here are the operation that I don't understand (it's Mathematica code)
dyn=u F + v G;
adXY=LieBracket[F,G,{x,y,z}];
Phi=Transpose[{F,G,-adXY}/.{x->0,y->0,z->0}];
tmp=Phi.{x,y,z};
newdyn = Inverse[T1].(dyn/.{x->tmp[[1]],y->tmp[[2]],z->tmp[[3]]});
$F,G$ are two (smooth) vector fields on $\mathbb{R}^3\ni (x,y,z)=q$ and we have a control system of the form $$ \dot q = u F(q) + v G(q) $$ where $u,v$ are scalar control functions. The components of $F,G$ are polynomials in $q=(x,y,z)^\intercal$ (they are obtained by Taylor expansions).
Consider:
$$\varphi(Q) = \overbrace{(F,G,-[F,G])_{\mid x=y=z=0}^\intercal}^{A}\,
\, Q=
q.$$
So $\varphi$ is a linear transformation and $\varphi^{-1}(q) = A^{-1} q$ and $\frac{\partial \varphi}{\partial Q}(Q) = A$.
Then they perform the pushforward of $F,G$ by $\varphi$ so that $$ \dot Q = A^{-1} \dot q = A^{-1}\left(u\, F(AQ)+v\, G(AQ)\right). $$
Also, what could be the use of such transformation $\varphi$?
On many examples, I observe that $\varphi$ removes the constant terms in the components $F_2$ and $G_1$.
Edit: Source of the Program : Aloui thesis
