  A Richardson variety $X_\ga^\gc$ in the Orthogonal Grassmannian is defined to
be the intersection of a Schubert variety $X^\gc$ in the Orthogonal
Grassmannian and a opposite Schubert variety $X_\ga$ therein. We give an
explicit description of the initial ideal (with respect to certain conveniently
chosen term order) for the ideal of the tangent cone at any $T$-fixed point of
$X_\ga^\gc$, thus generalizing a result of Raghavan-Upadhyay \cite{Ra-Up2}. Our
proof is based on a generalization of the Robinson-Schensted-Knuth (RSK)
correspondence, which we call the Orthogonal bounded RSK (OBRSK). The OBRSK
correspondence will give a degree-preserving bijection between a set of
monomials defined by the initial ideal of the ideal of the tangent cone (as
mentioned above) and a `standard monomial basis'. A similar work for Richardson
varieties in the ordinary Grassmannian was done by Kreiman in \cite{Kr-bkrs}.
