Talk to Iruki-Waraki at (K-9) in Aht Urghan Whitegate (on the top floor).
Talk to Ghatsad at (I-7) the Automaton Workshop (on the top floor).
Note that the area you need to travel to Arrapago Reef is also the same area for unlocking Corsair. So you may want to start that quest as well: Luck of the Draw.
Find the ??? at (H-10) on the first boat, hidden beneath the stairs, to receive a key item: Antique Automaton.
Return to Ghatsad (I-7) in Aht Urghan Whitegate. Talk to him to give him the Antique Automaton. Wait for the next game day and talk to him again.
Return to Iruki-Waraki (K-9) to receive your Animator and to complete the quest. You will also be given a chance to pick your Automaton's name from a list.