Easiest Way: In the Beginning
Hitting their turbo boosts, the cops caught right up with him.         Jason

Shaking uncontrollably, I could not get the arrow to stay on my rest as I turned to face him.       Ben

Shivering in the darkness, the girl felt great relief from fear.          Lake

Not expecting it, Roxanne let the ball come right back over.       Melissa

Not knowing what the other guy was going to do, he decided to bring the girl down to the station.   Allison

Thinking that it was one of his partners calling from the office about the casehe ran to the phone.   Lake

Another Way: At the End
He thrust the weapon once he was close enough, moving like a serpentTyler

This was an average day for the two, fighting, yelling, and just being friendsBrynn 

"Then it will get washed this evening," said the large woman starting up the street, dragging the frightened boy behind her.                      "Thank You, Ma'am"   Langston Hughes

Hunger had always been more or less at my elbow when I played, but now I began to wake up at night to find hunger standing at my bedside, staring at me gauntly.          Black Boy  Richard Wright

But the sobs went on, unstifled, finally quieting until I could hear my mother's voice, deep and rich, humming softly as she used to hum to a frightened child"Marigolds" Eugenia W. Collier

A dirty harbor tug pushed his submarine's bow around to the north, facing down the channel. 
Tom Clancey, The Hunt for Red October

Contemplating her day at school, she walked slowly, not knowing the dangers that would comeRoxanne

Shifting the weight of the line to his left shoulder and kneeling carefully, he washed his hand in the ocean and held it there, submerged, for more than a minute, watching the blood trail away and the steady movement of the water against his hand as the boat moved.   Ernest Hemmingway, The Old Man & the Sea

These sentences DON'T have participle phrases:
Sometimes the supply drop doesn't come or is running late.  (This is a compound verb.)

After working all day, Emily went to the big house for dinner.  (This is a prepositional phrase.)

It was running, and it screamed loudly.  (This is a compound sentence.)