A participial preposition is a participle (an -ed or –ing verb) that functions as a preposition. Some of the most common examples are assuming, barring, considering, during, given, notwithstanding, provided, regarding, and respected. Unlike other participles, participial prepositions don’t necessarily create dangling modifiers when they don’t correspond to a subject. So, to take a counterexample, consider the following sentence:
Sitting on the porch, it started to get cold.
Here the phrase sitting on the porch is a dangling modifier because its grammatical position indicates that it corresponds to the subject of the sentence, it. It is obviously not sitting on the porch.
Participial prepositions are a subset of participles that, due to widespread, long-time usage habits, have become acceptable as prepositions. Here are a few examples of conventional participial prepositions used well:
Considering that the whole idea behind poetry is to convey meaning through sound, “Jabberwocky” is actually a brilliant work of art. [Shmoop]
Given that the profit margin is high on derivatives trading, Bernstein’s estimates seem to be somewhat on the mark. [NY Times Dealbook]
Assuming the talks start, they may still go nowhere. [Financial Times]
The opening phrases in these sentences look like dangling participles and may seem illogical to a strict grammarian (“Jabberwocky” is not considering, Bernstein’s estimates are not given, they are not assuming), but few English speakers would have trouble with them.