Consider the Grassmanian $G(k,n)$ ($k\le \frac{n}{2}$) and take its Plucker embedding. Consider now the space of all normal rational curves of degree $k$, contained in the Plucker embedding of the Grassmanian. Finally, take the closure of this space. I would like to know if this closure has some nice description.

Example. The Grassmanian $G(2,4)$ is a quadric $Q$ in $\mathbb P^5$. A normal curve of degree two in this quadric is just the intersection of $Q$ with a $\mathbb P^2$. So the corresponding space in this case is $G(3,6)$.