Is it possible to make floor constraint base on surface detection? Let's say I created a plane as the floor
And then I created a cube.
Instead of detecting the center of the cube against the floor, can blender use the surface of the cube.So whenever the surface of both object met, the cube can't cross.
Added Info:
Well actually I'm searching for the solution of my rig problem. Often time when I make an animation, the tip of the foot often times went through the ground. So I was thinking, maybe if there's a way to prevent this by using mesh collision detection. Rigid body seems for simulation only
Try changing the origin of the cube to any vertex on the bottom of the cube:

Select the vertex in edit mode.

Press ⇧ Shift + S to get the cursor menu and then choose Cursor to Selected.

Then in object mode get the object menu and set the origin to the cursor.