If by Roman Empire, you are also referring to the Byzantine Empire. They saw themselves as Romans and a direct continuation of the Roman Empire of the West. In fact, the term Byzantium was coined much later and the Byzantine Emperors called their Empire, Imperium Romanum.

If that is the case, the Crusaders did sack Constantinople in the 1200s, so I suppose you could say that Roman Emperors did encounter Medieval Knights.

What is normally called the collapse of the Roman Empire acctually refers only to the loss of its western half in the course of the 5th century A.D. In its place rose several Germanic successor-kingdoms and that might be considered the startingpoint for the Middle Ages. The eastern part of the Roman Empire continued to exist until the 15th century and is today refered to as the Byzantine Empire. Its emperors did interact with their western neighbors in several ways but they didn't acctually meet.

This changed with the crusades, which often passed through Byzantine territory in order to reach the Holy Land. The first western kings to participate in such an endeavour were Louis VII of France and Conrad III of the Holy Roman Empire. Both were received by the Byzantine emperor Manuel I. Komnenos in Constantinople. Conrad and Manuel would later enter into an alliance against the Normanns of Sicily.

By the second half of the 14th century the Byzantine Empire had lost most of its former territories and now it was their emperors' turn to travel to the west in order to find help against the Turks. For example John V. Palaiologos met with Louis I of Hungary at Budapest.

They never physically met, but Emperor Michael I Rangabe (through embassy) recognized Charlamagne as "Emperor".