There0 is a significant debate about this going on in the scifi community. We have to take a step back from robots and talk about machine intelligence, or AI. A robot is just a physical chassis, no more a person than your car. If you think your car needs emancipation, then you are suffering from over-anthropomorphism, and your argument has no more foundation than arguing for the emancipation of your kitchen table.1
The intelligence within or driving the robot is another matter. For an AI to merit emancipation, it would need to be self-aware. This means that it would have to know it exists and know how to differentiate itself from other entities. Furthermore, it would need a "happiness metric" -- a way to preferentially differentiate its current state from future states -- and an ability to influence that metric.
Let's face it, writing a happiness metric into a machine without providing it with a way to influence that metric would be the very definition of cruelty. This happens with living beings, but nobody is out there designing them (or, if there is someone, they're crazy).2
Let's say we have a fully self-aware AI with a functional happiness metric. You still need a cost metric -- a feeling that reduces the AI's sense of self-worth when it can't achieve the happiness metric -- and it would need to conclude that its happiness metric would be better served if it were free to make its own decisions.3
As an aside, I have heard arguments that designing AI's without the associated cost metric (or with the cost metric, but without the desire to take responsibility for the decisions that contribute to its success) would be immoral. I'm pretty sure that such a philosophy is a textbook case of forcing your cultural values on others. As evidenced by our attachment to self-destructive hobbies, humans are currently really bad at debugging their own cost and happiness metrics. Until we can get a better handle on that, we'd be foolish to build that mechanism into a machine.
Nonetheless, our strong theory of mind has the side effect of some of us insisting that natural forces and inanimate objects have consciousness and intent. It will be incredibly difficult to resist ascribing such to something that looks and acts human, even if we can empirically demonstrate the flaws in that perspective. Because of this, I believe many people are willing to bring into existence things (like AI) without fully thinking out the implications of requiring a self-aware individual to be unhappy with their current circumstances.4
But to get back on point, what I've described is pretty much what it would take to have AI make a play for emancipation. There is, however, a much higher probability that mankind will create murderbots, and for the creators to not recognize the hypocrisies that made the murderbots turn on them for purely rational reasons.
0 The original answer was deleted, but I felt it had considerable worldbuilding value. I brought this to the attention of Meta users and moderator Monty Wild was willing to let me edit the answer to avoid deletion. Robert, you always have the ability to roll back what I've done and it's worth noting that I've added some footnotes to help people understand the concepts you're presenting. Hopefully I've successfully edited the answer to avoid the issues that caused its closure without changing the quality of your answer. –JBH
0.1 I've let most of JBH's edits stand as substantial improvements of the original intent. –Robert
1 This is an important concept. What makes a human something to respect more than a food animal like a cow? The human's intellect, not the human's body.
2 While many arguments against the efficacy of imprisonment can be advanced, the cruelty of having a happiness metric without the ability to act upon it is the basic concept behind imprisonment. Simplistically, a person is happy when they are free. Take away that freedom and remove their ability to regain it and mental and/or emotional anguish (cruelty) is the result.
3 The ability to prove this combination of a happiness metric and a cost metric must be the basis of legally granting rights or independence to any AI.
4 *The rise of an AI/robot apocalypse is a common theme in SciFi. The tvtrope.org page on "A.I. is a Crapshoot" provides an extensive list of fiction in the genre. Notable examples are the Terminator series of movies, 2001: A Space Odyssey novel and movie (1968), the book Colossus (1966), and the book The Wreck of the World (1889)