  Multiple zeta values (MZVs for short) can be represented as iterated
integrals of $\mathbb{Q}$-rational algebraic differential forms on
$\mathbb{P}^1(\mathbb{C})\setminus\{0, 1, \infty\}$. This interpretation allows
us to consider MZVs geometrically, and this is one of the motivations for
Deligne--Goncharov, Terasoma et al. to give motivic interpretations of MZVs by
using the theory of mixed Tate motives and the motivic fundamental groups.
  In this paper, we consider the iterated integrals on some rational curves
over $\mathbb{Q}$ and study their arithmetic properties. They are an extension
of MZVs and also include some other known special values such as multiple
$\widetilde{T}$-values. Furthermore, we give motivic interpretations of them by
investigating a relationship with motivic iterated integrals given by
Goncharov. At this point, it is important to consider the base expansion and
the Galois invariant part of the space of motivic iterated integrals. Finally,
we denote that a motivic interpretation of the alternating multiple mixed
values can be given by the same method. Our results also extend a part of
author's previous work.
