Learn English – With what prepositions can I use the word “pain”

That's a "trick" question because it is not about what follows "best", but simply about "should you use 'in' or 'on'?".

The double-trick is that in some case, you could (with a slight variation of context) use 'for'!

I'll do my best for the TOEFL preparation so I'll get higher score on the exam.

Anyway, if we stick to the initial choice 'in' or 'on', the final trick is in some case, both could be used, like the two first examples ("on" maybe being more commonly used in the US)

I find (personal rule)

• "in" standing for "during" or "within" (or other similar meanings)
• "on" standing for "about" or "referenced by" (or other similar meanings)

Meaning I don't find your examples "wrong".

By doesn't have the right meaning for this situation. With could work, but it sometimes connotes along with, as in with a companion, which is not the case. Using is the best bet, since that connotes using a tool, which is what we want. Also, do exploration is clunky, and sounds wrong. Explore means the same thing, but sounds much more natural:

Explore using Mobile Robot.