  It is a classical result that configuration spaces of labelled particles in
$\mathbb{R}^d$ are free $E_d$-algebras and that their $d$-fold bar construction
is equivalent to the $d$-fold suspension of the labelling space. In this paper,
we study a variation of these spaces, namely configuration spaces of labelled
clusters of particles. These configuration spaces are again $E_d$-algebras, and
we give geometric models for their iterated bar construction in two different
ways: one establishes a description of these configuration spaces of clusters
as cellular $E_1$-algebras, and the other one uses an additional verticality
constraint. In the last section, we apply these results in order to calculate
the stable homology of certain vertical configuration spaces.
