  In this paper, we firstly provide the correspondence between the FRT
formalism and the Drindeld-Jimbo presentation for $U_{r,
s}(\mathfrak{so}_{2n+1})$, using the theory of finite dimensional weight
modules. In the affine case $U_{r, s}(\widehat{\mathfrak{so}_{2n+1}}),$ we give
its RLL realization via the Gauss decomposition. Thus we establish the
correspondence between this realization and the Drinfeld realization.
