nah robert is his old time friend, and he never hated him, only what he became when he was king, but theres no joining for a common cause for this as it all happens during AGOT so robert is a no. and tywin, like the rest of the lannisters joined the war at the very end, without giving anyone a real choice, not fighting any of the battles with ned, so he never really joined them, he opposed them from the start, same with jaime etc. SPOILER FOR R+L=J THEORY: My idea was maybe rhaegar as at that time ned would have hated him for kidnapping lyannna as even if the theory was true, untill ned gets to lyanne he doesnt know otherwise. and when he made the 'promise' he was forced to join the common cause of raising the boy with the man he despised. Of course, this may not stand up as by this time rhaegar is dead so it may not work as well but it was just an idea.