Invariable generation of certain groups of piecewise linear homeomorphisms of the interval

Let $P$ be the group of all the orientation preserving piecewise linear homeomorphisms of the interval $[0,1]$. Given any $a>1$, let $P^a$ be the subgroup of $P$ consisting of all the elements with slopes in $a^\mathbb {Z}$, and let $P^\mathbb {Q}$ be the subgroup of $P$ consisting of all the elements with slopes and breaks in $\mathbb {Q}$. We show that the groups $P$, $P^a$, $P^\mathbb {Q}$, as well as Thompson group $F$, are invariably generated.