free space vs obstacle space
Once you picture a robot's poses as a map (the configuration space), that map naturally splits into two regions. Free space is the collection of all poses in which the robot touches nothing it shouldn't — it sits cleanly in the world without overlapping a wall, a table, or its own body. Obstacle space is everything left over: the poses that are forbidden because, if the robot actually struck them, some part of it would be jammed into an obstacle or tangled with itself. So the same dividing line that separates "allowed" from "not allowed" in the real world reappears, drawn right onto the map of poses.
The point of slicing C-space this way is that planning a safe motion becomes a pure geometry puzzle: find a route for the dot that stays entirely inside free space and never crosses into obstacle space. A surprising and important fact is that an obstacle's shadow on the map looks nothing like the obstacle itself. A small round pillar in the room can carve out a large, oddly shaped forbidden blob in C-space, because the robot's whole body — not just one point — has to clear that pillar at every pose. This reshaping is why planners do their thinking in C-space: there, a forbidden zone is simply a region to be steered around, no matter how strange the real-world collision geometry was.
Imagine a wide cleaning robot in a hallway. Poses where it sits in the open middle are free space; poses pressed against a wall, where its rim would dig into the plaster, are obstacle space. The dividing border on the map sits a robot's-radius away from each wall, not right at the wall.
The forbidden region grows by the robot's own size — an obstacle's C-space shadow is bigger than the obstacle.
Free space is often written "C-free". A planner's whole job is to find a path that stays inside C-free from start to goal.