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Let 
$X$ be a topological space. We consider certain generalized configuration spaces of points on 
$X$, obtained from the cartesian product 
$X^{n}$ by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call ‘twisted commutative dg algebra models’ for the cochains on 
$X$. Suppose that 
$X$ is a ‘nice’ topological space, 
$R$ is any commutative ring, 
$H_{c}^{\bullet }(X,R)\rightarrow H^{\bullet }(X,R)$ is the zero map, and that 
$H_{c}^{\bullet }(X,R)$ is a projective 
$R$-module. We prove that the compact support cohomology of any generalized configuration space of points on 
$X$ depends only on the graded 
$R$-module 
$H_{c}^{\bullet }(X,R)$. This generalizes a theorem of Arabia.
$F(G\times \mathbb{R}^{n},2)$
The Lusternik–Schnirelmann category 
$G\times \mathbb{R}^{n}$ and apply the results to the planar and spatial motion of two rigid bodies in 
$\mathbb{R}^{2}$ and 
$\mathbb{R}^{3}$ respectively.
