>> I am particularly interested in the possibility that modern algorithms (e.g. deep reinforcement learning) can compensate for not having completely repeatable behaviors from a robot by using sensory feedback to create behaviors that reliable despite their variability.
Deep Reinforcement Learning will definitely make a huge difference in robotics. Already, ~50 years after the first implementations of RL algorithms [1] robotic systems trained with (deep) RL have now become so proficient that a robotic hand has learned to manipulate a cube [2]. Always the same cube (in terms of size, weight, etc) but a cube, nonetheless! Full-blown general object manipulation cannot be more than a hundred years away, or two!
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[1] I'm counting from Donald Michie's MENACE, a tick-tack-toe playing machine originally implemented in matchboxes, because of a dearth of computing hardware of sufficient power- see Rodney Brooks' excellent write-up:
https://rodneybrooks.com/forai-machine-learning-explained/
[2] OpenAI trained a robot hand to manipulate a cube with RL:
https://blog.openai.com/learning-dexterity/
Note that it's always the same cube. If they could train it to manipulate anything else, you can bet they'd be showing it off (and probably publishing in Nature or something).