physics - In a 2D platform game, how to ensure the player moves smoothly over sloping ground?

I'm developing a physics engine for a 2D platform game. I'm using the separating axis theorem for collision detection. The ground surface is constructed from oriented bounding boxes, with the player as an axis aligned bounding box. (Specifically, I'm using the algorithm from the book "Realtime Collision Detection" which performs swept collision detection for OBBs using SAT). I'm using a fairly small (close to zero) restitution coefficient in the collision response, to ensure that the dynamic objects don't penetrate the environment.

The engine mostly works fine, it's just that I'm concerned about some edge cases that could possibly occur. For example, in the diagram, A, B and C are the ground surface. The player is heading left along B towards A. It seems to me that due to inaccuracy, the player box could be slightly below the box B as it continues up and left. When it reaches A, therefore, the bottom left corner of the player might then collide with the right side of A, which would be undesirable (as the intention is for the player to move smoothly over the top of A). It seems like a similar problem could happen when the player is on top of box C, moving left towards B - the most extreme point of B could collide with the left side of the player, instead of the player's bottom left corner sliding up and left above B.

Box2D seems to handle this problem by storing connectivity information for its edge shapes, but I'm not really sure how it uses this information to solve the problem, and after looking at the code I don't really grasp what it's doing.

Any suggestions would be greatly appreciated.