  This paper introduces a data-dependent approximation of the forward
kinematics map for certain types of animal motion models. It is assumed that
motions are supported on a low-dimensional, unknown configuration manifold $Q$
that is regularly embedded in high dimensional Euclidean space
$X:=\mathbb{R}^d$. This paper introduces a method to estimate forward
kinematics from the unknown configuration submanifold $Q$ to an $n$-dimensional
Euclidean space $Y:=\mathbb{R}^n$ of observations. A known reproducing kernel
Hilbert space (RKHS) is defined over the ambient space $X$ in terms of a known
kernel function, and computations are performed using the known kernel defined
on the ambient space $X$. Estimates are constructed using a certain
data-dependent approximation of the Koopman operator defined in terms of the
known kernel on $X$. However, the rate of convergence of approximations is
studied in the space of restrictions to the unknown manifold $Q$. Strong rates
of convergence are derived in terms of the fill distance of samples in the
unknown configuration manifold, provided that a novel regularity result holds
for the Koopman operator. Additionally, we show that the derived rates of
convergence can be applied in some cases to estimates generated by the extended
dynamic mode decomposition (EDMD) method. We illustrate characteristics of the
estimates for simulated data as well as samples collected during motion capture
experiments.
