How did the federalists convince the anti-federalists to ratify the united states constitution?

The Federalists promised a bill of rights if the anti-federalists would vote for the ratification, as well as they promised that the Federal Government wouldn't be over powerful, like the British and that they couldnt screw the states with their power (which they limited in the constitution but the Federalists left it open to very broad arguments such ad the Elastic clause and the commerce clause, to give them power of the states in the end.