Can the robot play duplicate developments in solo-play?
Yes it can.

1) The robot treats the tableau as an abstracted stack, and ignores the abilities of cards it places. From the rules (PDF here):


  The robot's tableau is a stack of cards, without any goods, where only the number of cards and their VPs matter.


What matters is the robot's economic development, which is independent of what the cards do. There are therefore no in-game consequences to placing a card twice.

2) The description of how the robot searches for developments provides no specific way to handle duplicates.


  Develop action: draw 1 card and then flip to find a development.
  
  ...
  
  Flip to find: means to flip over cards one at a time from the robot's draw stack, discarding all non-matching cards, until either the desired type of card is found or the robot's draw stack is exhausted.


Note that "non-matching" here refers to the card type (e.g. discard non-development cards), not duplicates. Race for the Galaxy's rules are famous for both their terseness, and their exhaustive completeness. If it's not defined, it doesn't happen.

3) Although not definitive, the question has also been asked several times at BoardGameGeek, with the same consensus (e.g. 1, 2).