  We study asymptotic error distributions of rough differential equations
driven by the fractional Brownian motion whose Hurst parameter $H$ satisfies
$\frac{1}{3}<H<\frac{1}{2}$ for several approximation schemes. To this end, we
introduce an interpolation process between the solution and the approximate
solution. Also we study limit theorems of weighted sum processes of Wiener
chaos of order 2. Our proof is based on the fourth moment theorem for Wiener
chaos of finite order, estimates of multidimensional Young integrals and
integration by parts formula in the Malliavin calculus.
