Let $E\to M$ be a smooth real vector bundle of rank $r$. Then $\operatorname{Fr}(E)$ admits a unique smooth structure for which the frame coordinates induced by vector bundle trivializations are smooth charts, and with this structure $q:\operatorname{Fr}(E)\to M$ is a smooth principal $GL_r(\mathbb R)$-bundle with the right action $\nu\cdot A=\nu\circ A$.