Robot stories have never really gotten past Asimov--his Three Laws of Robotics have pretty much defined how robot stories get told. It's like Hobbes in political science--everyone has to respond to Asimov. But Rachel Swirsky's Eros, Philia, Agape is a good move away--the robot is a familiar humanoid, but his thinking is very much not in the logical vein of Asimov's artifical humans.
We have two very equal protagonists in this one. Adriana, a rich woman with a difficult past and much too little to do, purchases a robot as a companion. Lucien is a fine robot, learning about Adriana and assembling his personality as her perfect companion. She views him as human, and assists groups that work for robot personhood. The story starts with his taking advantage of this, leaving to find his own programming. The story got a Hugo nomination for Best Novelette in 2010, and it's quite a good read, if rather relaxed and sad. I enjoyed reading it, and I think you will too. Three stars from me.