We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the p-adic de Rham period map j_n^dr on elliptic and hyperelliptic curves over number fields via their universal unipotent connections U. Several algorithms forming part of the computation of finite level versions j_n^dr of the unipotent Albanese maps are presented. The computation of the logarithmic extension of U in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a result due to Vologodsky and Hadian on the computation of the Hodge filtration on U over affine elliptic and odd hyperelliptic curves. We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated p-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well. Finally, we present the output of implementations of the aforementioned algorithms in Magma for small level on both elliptic curves and hyperelliptic curves.