Wal Mart has product that you put on the floor and then lay the rug over it. You wouldn't sew it.
https://www.walmart.com/ip/Ultra-Sto...d5aa594ee93c83
I had read something about someone using either Elmer's glue or some type of silicon drizzled on the back of a slippery item to stop the slipping. You have to let it dry afterwards. I don't know if it would be washable. Here are some possible solutions.
https://www.reference.com/home-garde...b6c42d0870e82a