  Exclusive production of $\rho^0$ mesons was studied at the COMPASS experiment
by scattering 160 GeV/$c$ muons off transversely polarised protons. Five
single-spin and three double-spin azimuthal asymmetries were measured as a
function of $Q^2$, $x_{Bj}$, or $p_{T}^{2}$. The $\sin \phi_S$ asymmetry is
found to be $-0.019 \pm 0.008(stat.) \pm 0.003(syst.)$. All other asymmetries
are also found to be of small magnitude and consistent with zero within
experimental uncertainties. Very recent calculations using a GPD-based model
agree well with the present results. The data is interpreted as evidence for
the existence of chiral-odd, transverse generalized parton distributions.
