  This article presents methods to efficiently compute the Coriolis matrix and
underlying Christoffel symbols (of the first kind) for tree-structure
rigid-body systems. The algorithms can be executed purely numerically, without
requiring partial derivatives as in unscalable symbolic techniques. The
computations share a recursive structure in common with classical methods such
as the Composite-Rigid-Body Algorithm and are of the lowest possible order:
$O(Nd)$ for the Coriolis matrix and $O(Nd^2)$ for the Christoffel symbols,
where $N$ is the number of bodies and $d$ is the depth of the kinematic tree.
Implementation in C/C++ shows computation times on the order of 10-20 $\mu$s
for the Coriolis matrix and 40-120 $\mu$s for the Christoffel symbols on
systems with 20 degrees of freedom. The results demonstrate feasibility for the
adoption of these algorithms within high-rate ($>$1kHz) loops for model-based
control applications.
