We show that the Heisenberg double H(B^*) is a Yetter--Drinfeld module
algebra over the Drinfeld double D(B) for any Hopf algebra B with bijective
antipode. We use a braiding structure to generalize H(B^*) = B^{*cop} # B to
"Heisenberg n-tuples" and "chains" ... # B^{*cop} # B # B^{*cop} # B # ..., all
of which are Yetter--Drinfeld D(B)-modules. For B a particular Taft Hopf
algebra at a 2p-th root of unity, a certain truncation of these constructions
yields Yetter--Drinfeld module algebras and Yetter--Drinfeld modules over the
2p^3-dimensional quantum group U_q(sl_2).