The Prayers at the Foot of the Altar

I was thinking of the prayers at the foot of the altar, since many who are attached to the EF believe that they should be retained. Is there something equivalent in the Eastern liturgies? The prayers at the foot of the alter, as far as I know, are proper to celebrant and the servers, even though the laity have been encouraged to adopt the role of the server (instead of singing or praying the introit), since they are the private prayers of the priest. In the OF the priest "begins" the liturgy by invoking the Blessed Trinity. It seems that this is also the case in the Divine Liturgy of St. John Chrysostom. Though I think there is something equivalent to the introit that is sung before it as well. My question is: do the priest and the deacon have any prayers in the sanctuary before this point as well?

In the EF, there is no invocation of the Blessed Trinity to commence the liturgy for the laity, unless they take the prayers at the foot of the altar as their own. Otherwise from the Introit they go to the Kyrie Eleison. I do think the invocation of the Trinity is a fitting way to start the liturgy. Was this never a part of the Roman rite until the introduction of the missal of Pope Paul VI?

I'll have to read A Short History of the Roman Mass by Michael Davies, and then go through Jungmann one of these days...