7. A participial phrase at the beginning of a sentence must refer to the grammatical subject.
Walking slowly down the road, he saw a woman accompanied by two children.
The word walking refers to the subject of the sentence, not to the woman. If the writer wishes to make it refer to the woman, he must recast the sentence:
He saw a woman, accompanied by two children, walking slowly down the road.
Participial phrases preceded by a conjunction or by a preposition, nouns in apposition, adjectives, and adjective phrases come under the same rule if they begin the sentence.
On arriving in Chicago, his friends met him at the station.
When he arrived (or, On his arrival) in Chicago, his friends met him at the station.
A soldier of proved valor, they entrusted him with the defence of the city.
A soldier of proved valor, he was entrusted with the defence of the city.
Young and inexperienced, the task seemed easy to me.
Young and inexperienced, I thought the task easy.
Without a friend to counsel him, the temptation proved irresistible.
Without a friend to counsel him, he found the temptation irresistible.
Sentences violating this rule are often ludicrous.
Being in a dilapidated condition, I was able to buy the house very cheap.