  Deterministically generating near-uniform point samplings of the motion
groups like SO(3), SE(3) and their n-wise products SO(3)^n, SE(3)^n is
fundamental to numerous applications in computational and data sciences. The
natural measure of sampling quality is discrepancy. In this work, our main goal
is construct low discrepancy deterministic samplings in product spaces of the
motion groups. To this end, we develop a novel strategy (using a two-step
discrepancy construction) that leads to an almost exponential improvement in
size (from the trivial direct product). To the best of our knowledge, this is
the first nontrivial construction for SO(3)^n, SE(3)^n and the hypertorus T^n.
  We also construct new low discrepancy samplings of S^2 and SO(3). The central
component in our construction for SO(3) is an explicit construction of N points
in S^2 with discrepancy \tilde{\O}(1/\sqrt{N}) with respect to convex sets,
matching the bound achieved for the special case of spherical caps in
\cite{ABD_12}. We also generalize the discrepancy of Cartesian product sets
\cite{Chazelle04thediscrepancy} to the discrepancy of local Cartesian product
sets.
  The tools we develop should be useful in generating low discrepancy samplings
of other complicated geometric spaces.
