He was forced to ask for their help 'on bended knee'. [=he was forced to beg for their help; he was forced to ask for their help in a very humble way]
If you want to clean the floor properly, you have to 'get down on your (hands and) knees' [=you have to kneel down on the floor] and start scrubbing.
She 'dropped/fell/sank to her knees' [=she knelt down] and begged for forgiveness. = She 'got down on her knees' and begged for forgiveness.
(..) to synchronise the different kicks. EAMES: He hasn't got any surgery scheduled, no dental, nothing. COBB: Wasn't he supposed to have a knee operation? Nothing. Nothing that they'll put him under for, anyway. And we need... We need at least a good hours. Sydney to Los Angeles. (..)2010 Inception