  In this note we show that the configuration spaces of the kinematic system
constructed in [4] and [12] gives rise to a natural tower of sphere bundles.
Moreover, we prove that, each tower of projective bundles associated to special
multi- flags (cf [1], [13], [2], [3]), we can associate such a tower of sphere
bundles which is a two-fold covering of the previous one. In particular we give
a positive answer of some conjecture proposed in [3]
