One thing we must be careful of in talking about 'submission' is its contrast to the Islamic concept.
Muslims claim that Jesus was a Muslim because he 'submitted' to God, Also the OT prophets that they know about, while we 'submit' to God we do it in joy and love, admiration even; whereas they do it as 'slaves' to Allah.
So our 'submission' to our Lord is as those who love and follow him. We must be able to easily clarify this with Muslims whose concept is of slavery, whereas ours is of love (wives to their husbands is an example of this). For Christians 'submit' is part of a fellowship, not part of slavery.