The theorem of Rips about free actions on R-trees relies
on a careful analysis of finite systems of partial isometries of R.
In this paper we associate a free action on an R-tree to any finite
system of isometries without reflection. Any free action may be approximated
(strongly in the sense of Gillet-Shalen) by actions arising in this way.
Proofs use in an essential way separation properties of systems of isometries.
We also interpret these finite systems of isometries as generating sets
of pseudogroups of partial isometries between closed intervals of R.