To me, the Roman Empire was almost fated to exist, or rather the Republic was fated to die. But the Empire was not necessarily fated to be unstable, at least not how it was in our world.
As John Adams said; “Individuals have conquered themselves, nations and large bodies of men, Never.”
And that was precisely the problem with the Roman Republic and its chief political institution, The Senate. Indeed, it could be said the US might lead down an Imperial path itself or grand fall given the profound lack of ability of its own institutions, but I digress.
During the reign of the First Emperor, Caesar Augustus, the succession of the Empire was always at issue. Even though the Emperor achieved much in his over 40-year reign, enacting reform and ushering in the Pax Romana, there didn’t ever seem to be a stable succession from the first, even if the possible heirs to succeed were known.
It’s possible the late Princeps remained “ill at ease”, not only remembering the fate of Caesar himself, but also being the first Emperor to ever rule. Perhaps in his mind the succession wasn’t the most pressing issue personally and would the empire even last after his death? Or would this be his grand achievement for the ages alone and after him it might not be?
But what if the succession was indeed one of the most important issues to Augustus? In such a way that an equal portion of his energies were directed towards it, drafting laws and rules which would govern the succession of future Emperors. What factors, aside from just bloodline, might govern those rules and be needed to maintain stability?
One factor is likely to be certain; that like the patrician class itself from which the first Emperor's arose, the office, power, wealth and authority would pass to the eldest surviving son and his family line.
Perhaps another rule could be similar to later Russian sovereigns (Third Rome, Byzantine successors) who, without a logical heir or none they approved of, would choose their own successor from their own family line or a House that their family was sufficiently/significantly married into?
What might the Senatorial role be in all of this and would that body of nobles matter in this case?
And as this authority would rest solely with the current Emperor if an heir needed to be selected, all would need to vie for their attention and affection, more than they would in our world where someone can just raise and army and be “proclaimed” Emperor.
What do you think?

I’d like to hear more about what would be needed.