  The $f(R)$ gravity can be cast into the form of a scalar-tensor theory, and
scalar degree of freedom can be suppressed in high-density regions by the
chameleon mechanism. In this article, for the general $f(R)$ gravity, using a
scalar-tensor representation with the chameleon mechanism, we calculate the
parameterized post-Newtonian parameters $\gamma$ and $\beta$, the effective
gravitational constant $G_{\rm eff}$, and the effective cosmological constant
$\Lambda_{\rm eff}$. In addition, for the general $f(R)$ gravity, we also
calculate the rate of orbital period decay of the binary system due to
gravitational radiation. Then we apply these results to specific $f(R)$ models
(Hu-Sawicki model, Tsujikawa model and Starobinsky model) and derive the
constraints on the model parameters by combining the observations in solar
system, cosmological scales and the binary systems.
