“Robert has gone to Italy on a special assignment.”
In this sentence, what is the function of "on" ?

Question

Answer

It's from the phrasal verb "to go on something".
From Macmillandictionary.com:  
'[TRANSITIVE] [go on something] to start doing a particular activity or being in a particular state."  
In your example, Robert is going to start dong a particular activity: work on a special assignment in Italy.
The words are omitted but it is understood:  Robert has gone to Italy, (to work) on a special assignment . . .

Answer

Are you asking what grammatical part it plays? 
I'm not a grammarian, but I think it's a preposition. It's used in the same way as 'on vacation' or 'on assignment'.
The other way of answering this would simply be that the sentence is meaningless without it!

Answer

Robert has gone to Italy (because he is working) on a special assignment.

Hope this makes sense.