  In this paper, we study two-sided tilting complexes of preprojective algebras
of Dynkin type. We construct the most fundamental class of two-sided tilting
complexes, which has a group structure by derived tensor products and induces a
group of auto-equivalences of the derived category. We show that the group
structure of the two-sided tilting complexes is isomorphic to the braid group
of the corresponding folded graph. Moreover we show that these two-sided
tilting complexes induce tilting mutation and any tilting complex is given as
the derived tensor products of them. Using these results, we determine the
derived Picard group of preprojective algebras for type $A$ and $D$.
