Starting from the renormalisation theory for rough paths as introduced by Bruned, Chevyrev, Friz and Preiss '19, we will reformulate and extend this approach on any Hopf algebra $H$ and over the class of "smooth rough paths", i.e. smooth paths with values in the character group of $H$. In this context, renormalization is equivalent to solving an ordinary differential equation containing the Mauer-Cartan form for G(H). Further applications to the renormalisation of differential equations driven by a smooth rough path are provided in the geometric and quasi-geometric setting. Joint work with Rosa Preiss, Sylvie Paycha, Peter Friz