Why did Abimelech ask one of his men to kill him with his sword?

So that no one would say he was killed by a woman.

And a certain woman cast a piece of a millstone upon Abimelech's head, and all to brake his skull.

Then he called hastily unto the young man his armourbearer, and said unto him, Draw thy sword, and slay me, that men say not of me, A woman slew him. And his young man thrust him through, and he died. Judges 9:53-54