I've just rambo'd my way through the samurai castle and now I'm stuck on the room with a grate in the floor and a giant cage that can be pulled up from it.
I've tried just about everything, I even closed the metal grate again and let the cage fall, then pulled it from ground level and jumped on as it swung so i could get a little closer to the exit above the door, but it didn't work.
The cage should be resting close to the ground. Facing away from the blocked doors, go to your right and make your way up to the top tier of the room. Now standing on the far side of the cage facing the blocked doors, pull the cage toward you then jump on it.
As the cage swings back toward the locked doors, jump and catch the ledge then climb over.
Open the floor opening. Raise the cage. Close the floor opening. Lower the cage. Pull the cage with your rope to bang it against the gate, It should break to reveal a spot where you can grab on to, but more on that later. Raise the cage. On the side that has the lever to raise the cage there should be something that you can climb to get higher up. Go to the opposite side of the cage up on on the ledge where you just climbed up to. Pull the cage with your rope. Once you pull the cage towards you, jump on top of it, and then as it's swinging it will put you close enough to the gate to be able to jump to the spot that you revealed when you hit the gate with the cage earlier.
I may have screwed something up there, but that should give you the gist of it.