Are there terms to distinguish the auxiliary verbs in a multi-auxiliary verb phrase?

In phrase constructions that include multiple auxiliary verbs, as in "he has been sleeping all day", is there terminology commonly used to distinguish them? For example, might you call the first one "the primary auxiliary" and the next "a secondary auxiliary"? I understand the distinction for "modal" auxiliaries, but here I am talking about non-modal forms that co-occur.

If there are no such formal terms, are there understood roles that each play in the construction of a VP?