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Occupied space
Definition
Occupied space is the region physically taken up by a robot, obstacle, or other object. In robotics, the occupied region depends on the object's shape and, for a moving robot, its configuration. Collision checking asks whether these regions overlap. This entry uses the geometric motion-planning meaning and distinguishes physical workspace from configuration space.
What it means
A robot is not just the point used to mark its location on a map. Its body, wheels, arm, load, and other relevant parts occupy space around that reference point. Two robots at the same reference position can occupy different regions if their orientations or joint settings differ.
Obstacles also occupy regions. A planner must determine whether the robot's occupied region intersects an obstacle at each proposed configuration. The resulting set of forbidden configurations is related to occupied physical space, but it is a different mathematical object with potentially different dimensions.
How it works
First define the workspace, the coordinate system in which physical objects are represented. Let O denote the obstacle region and A(q) denote the space occupied by the robot at configuration q. Under a convention that counts touching as collision, q is forbidden whenever A(q) intersects O.
The collection of all such forbidden configurations forms configuration-space obstacles. The remaining configurations form free configuration space. A motion planner seeks a continuous path through that free space, subject to any additional movement restrictions.
For a circular robot translating on a floor, obstacle expansion can sometimes reduce checking the full footprint to checking a reference point. For an articulated arm, the relationship is more complex because joint angles change the occupied geometry. Real systems may also enlarge modeled obstacles or robot shapes to allow for uncertainty and required clearance. These margins are planning choices, not extra physical material.
Three practical examples
- Delivery robot: check that its entire footprint, including a carried tray, fits through a doorway.
- Robot arm: test whether any modeled link intersects a shelf during a proposed movement.
- Warehouse planning: update the obstacle region when a cart is parked in an aisle, then reconsider routes that pass through it.
Think of it this way
Think of moving a sofa through a room. Tracking only the center of the sofa would miss collisions involving its corners. Turning it changes which parts of the room it occupies.
The analogy has limits: a mathematical robot model can include many joints, uncertain measurements, and constraints that a person moving furniture handles informally. A clear floor plan alone does not establish that every intermediate movement is feasible.
Worked example
Consider a one-dimensional teaching model. An obstacle occupies the closed interval [2,3] meters. A robot with center q extends 0.4 meters to either side, so its occupied interval is [q−0.4,q+0.4]. Count boundary contact as collision.
The intervals intersect when q+0.4≥2 and q−0.4≤3. Therefore the forbidden robot-center positions are [1.6,3.4]. This is the configuration-space obstacle for this simple translating robot.
At q=1.7, the center is outside the physical obstacle, but the robot occupies [1.3,2.1] and overlaps it. At q=1.5, the robot occupies [1.1,1.9], leaving a 0.1-meter gap. Checking the center against [2,3] would incorrectly approve the first position.
If the planning rule requires strictly more than 0.1 meters of clearance, expanding the forbidden center interval to [1.5,3.5] implements that conservative boundary convention. The physical obstacle remains [2,3]. This hand calculation assumes exact geometry and translation only; it does not establish safe operation with uncertain sensing or moving obstacles.
Illustrative case study
Fictional case: a small mobile robot navigates successfully without a load, but its route planner continues using the same footprint after a wide box is attached. The robot's center follows an apparently valid path while the box clips a doorway in simulation.
The team adds a load-specific geometry model and checks the swept movement rather than just a few center positions. It also distinguishes measured geometry from clearance margins in its records. The case illustrates a modeling failure and its investigation, not a reported accident or a measured commercial result.
Common misconceptions and mistakes
Occupied physical space is not the same as forbidden configuration space. The latter answers which robot settings cause collision and may include joint-angle dimensions rather than ordinary spatial coordinates.
A point marked free on a map does not prove that a robot centered there fits. Conversely, an expanded obstacle used for conservative planning should not be described as the obstacle's measured physical size.
An occupancy-grid estimate also differs from exact geometric occupancy. A cell may be uncertain or unobserved, and a thresholded map can hide that distinction. Finally, checking only a path's endpoints can miss collisions between them.
Business use
Occupied-space models support layout reviews, robot deployment, and evaluation of changes in payload or equipment. They help teams translate questions such as “will it fit?” into explicit geometry and movement checks.
Useful review records include geometry versions, reference frames, load assumptions, treatment of boundary contact, uncertainty margins, and how intermediate motion is checked. For an operational system, these checks belong within a broader engineering validation process that accounts for sensing and control limitations.
People also ask
Does occupied space include a safety margin?
Physical occupied space describes geometry; a planner may use an enlarged approximation that includes a separately stated margin.
Can the same location be both safe and unsafe for one robot?
Yes, different orientations or joint configurations at that location can occupy different regions.
Is an unknown map cell occupied?
Not necessarily. Unknown describes insufficient information; the planner must specify how it handles that uncertainty.
Check your understanding
Why is center position 1.7 invalid in the interval example?
Answer: The robot reaches 2.1 meters and overlaps the obstacle beginning at 2 meters, even though its center lies outside the obstacle.
Related terms
- Configuration space
- Free space
- Occupancy grid
- Motion planning
- Workspace
- collision detection
- robot footprint
- obstacle inflation
Learn next
Draw a rectangular robot at several orientations beside a doorway. Compare its physical outline with the set of allowable center positions. Then study configuration-space obstacles and continuous collision checking to understand why both geometry and intermediate movement matter.
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Sources
Lydia Kavraki and Steven LaValle, Motion Planning: https://lavalle.pl/papers/KavLav08.pdf — robot-occupied workspace regions and the collision-based definition of configuration-space obstacles.
Alberto Elfes, Occupancy Grids: A Stochastic Spatial Representation for Active Robot Perception: https://arxiv.org/pdf/1304.1098 — probabilistic occupancy representations. Scope is geometric robotics; examples are original hand calculations.
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