  We establish the well-definedness of the barycenter (in the sense of Buser
and Karcher) for every integrable measure on the free nilpotent Lie group of
step $L$ (over $\mathbb{R}^d$). We provide two algorithms for computing it,
using methods from Lie theory (namely, the Baker-Campbell-Hausdorff formula)
and from the theory of Gr\"obner bases of modules. Our main motivation stems
from measures induced by iterated-integrals signatures, and we calculate the
barycenter for the signature of the Brownian motion.
