Let $G$ be a smooth affine group scheme over a base $S$. $G$ acts on a scheme $X$ over $S$. Let $x$ be an $S$-point in $X$. Then we have an orbit map $G\to X$. I wonder when the image (set-theoretically) of this map is locally closed, and the induced scheme structure (the minimal one) on the orbit is smooth over $S$.