We study the convergence rate of discretized Riemannian Hamiltonian Monte Carlo on sampling from distributions in the form of on a convex body $\mathcal {M}\subset\R^{n} $. We show that for distributions in the form of on a polytope with constraints, the convergence rate of a family of commonly-used integrators is independent of and the geometry of the polytope. In particular, the implicit midpoint method (IMM) and the generalized Leapfrog method (LM) have a mixing time of to achieve total variation distance to the target distribution. These guarantees are based on a general bound on the convergence rate for densities of the form in terms of parameters of the manifold and the integrator. Our theoretical guarantee complements the empirical results of\cite {kook2022sampling}, which shows that RHMC with IMM can sample ill-conditioned, non-smooth and constrained distributions in very high dimension efficiently in practice.