In Meditations on First Philosophy, Descartes uses doubt to determine what can be called into question and what can be made certain. He says that there are sensations he has while awake that are similar to sensations he has while dreaming. Then he says there is no way to truly distinguish dreaming states from waking states. Therefore, it is "possible that I am dreaming now and that all my perceptions are false."
Early empiricism dates back at least to Aristotle but it really became a movement in the 17th and 18th centuries with modern science, the scientific method, and technological advances. Empiricism is the theory that knowledge should be based on sense experience. Descartes (1596-1650) favored Rationalism, a theory that states that knowledge should be based on reason (in the mind), based on intellectual inquiry (deduction). So, as empiricism was on the rise (or soon to be) in Descartes' time, he began to doubt sense experience in favor of what he could be certain of: his own existence as a thinking thing. He famously concludes that he can only be certain that he thinks and therefore is a thing that thinks. He can not be certain of being asleep, awake, or that his perceptions may or may not be the deception of some demon controlling him.
If you are asking how "robot doubt" would undermine rationalism, then we might formulate the notion this way. If a robot is programmed to be able to doubt, that robot is programmed with empirical things: wires, chips, metal, etc. So, this also seems to undermine rationalism's (Descartes') doubt of sense experience and/or the external world since the robot's mechanisms that allow him to doubt are actually built by empirical means and empirical objects of the external world.
If "robot doubt" refers to the ability to tell whether a being is a human or a robot, the proof would again come empirically, by taking each apart, thus showing that rationalism would not be enough to determine whether X was a man or a robot. In strict terms of cognitive analysis, this argument becomes very tricky. For example, if we had the technology to design a robot that walks, talks, thinks, and reasons like a human, it would be very difficult (without disassembling, an empirical practice) to determine who was the human and who was the robot.