When a robot is using a tool that may normally be repaired after it breaks (such as those from Tinkers' Construct), the robot will delete the tool once it has broken.

The robot should be smarter about these types of items, and put them into a chest so that they may be repaired.

Is there a way to check this in a mod-independent manner?

not sure, i also think it might be a good idea to call the item update function when in the robot's inventory though for self-reparing tools