  We use Very Long Baseline Interferometry to measure the proper motions of
three black hole X-ray binaries (BHXBs). Using these results together with data
from the literature and Gaia-DR2 to collate the best available constraints on
proper motion, parallax, distance and systemic radial velocity of 16 BHXBs, we
determined their three dimensional Galactocentric orbits. We extended this
analysis to estimate the probability distribution for the potential kick
velocity (PKV) a BHXB system could have received on formation. Constraining the
kicks imparted to BHXBs provides insight into the birth mechanism of black
holes (BHs). Kicks also have a significant effect on BH-BH merger rates, merger
sites, and binary evolution, and can be responsible for spin-orbit misalignment
in BH binary systems. $75\%$ of our systems have potential kicks
$>70\,\rm{km~s^{-1}}$. This suggests that strong kicks and hence spin-orbit
misalignment might be common among BHXBs, in agreement with the observed
quasi-periodic X-ray variability in their power density spectra. We used a
Bayesian hierarchical methodology to analyse the PKV distribution of the BHXB
population, and suggest that a unimodal Gaussian model with a mean of
$107\pm16\,\rm{km~s^{-1}}$ is a statistically favourable fit. Such relatively
high PKVs would also reduce the number of BHs likely to be retained in globular
clusters. We found no significant correlation between the BH mass and PKV,
suggesting a lack of correlation between BH mass and the BH birth mechanism.
Our Python code allows the estimation of the PKV for any system with sufficient
observational constraints.
