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Free space
Definition
Free space is the set of configurations in which a robot or moving object satisfies the modeled collision and configuration constraints. In motion planning, a valid geometric path must remain within this space from its start configuration to its goal configuration.
What it means
An empty-looking patch of a room is not automatically free space for every robot. A large robot may collide there even though its center lies outside an obstacle. A robot arm may place its end effector in an open location while an elbow hits a wall.
Configuration space represents the robot's complete arrangement, such as position and orientation or a collection of joint angles. Free space is the allowed subset of that representation.
This entry uses the robotics and motion-planning sense. It is different from unused disk capacity, available memory or the physical concept of electromagnetic propagation through free space.
How it works
Start with a configuration space C and a model of the robot's geometry and environment. For each configuration q, determine the space occupied by the robot. Configurations that cause forbidden intersections with obstacles belong to the configuration-space obstacle region.
The collision-free configurations form its complement within the permitted configuration domain. Depending on the formulation, joint limits, self-collision and other configuration restrictions are included in that domain or checked alongside obstacle avoidance.
A planner seeks a continuous path through the allowed configurations. Collision-free endpoints do not guarantee a collision-free connecting motion. Moreover, geometric freedom does not by itself establish that a path respects steering, velocity or acceleration constraints. Those require additional motion or dynamics modeling.
Three practical examples
- Mobile robots: A planner searches for positions and orientations that keep the whole vehicle clear of walls and furniture.
- Robot arms: A manipulation planner checks joint configurations for environmental and self-collision.
- Virtual assembly: A simulation checks whether a part can move through a modeled opening while maintaining the required clearance.
Think of it this way
Think of carrying a wide tray through a doorway. Your hands may fit through the opening while the tray's corners do not. Free space concerns the complete carried shape and its orientation, not just a reference point. The analogy does not capture all the dimensions of a multi-joint robot, but it explains why object size matters.
Worked example
Consider a circular robot with radius 0.3 meters moving along the x-axis. A circular obstacle has radius 0.5 meters and is centered at x = 2.0 meters on the same axis.
The circles overlap or touch whenever the distance between their centers is at most 0.3 + 0.5 = 0.8 meters. If touching is treated as collision, the robot center cannot lie in the closed interval [1.2, 2.8]. Along this one-dimensional route, the free positions are x < 1.2 or x > 2.8.
A start at x = 0 and a goal at x = 4 are each collision-free, but no continuous path restricted to that line connects them without crossing the forbidden interval. The free set has two disconnected components.
If motion is allowed in the plane, the center must stay more than 0.8 meters from the obstacle center, and a route around the obstacle may exist if no other restrictions block it. Adding 0.1 meters of required clearance expands the forbidden center-distance threshold to 0.9 meters.
The calculation uses exact circles and a static obstacle. Localization error, uncertain geometry or moving obstacles would require additional modeling rather than being silently covered by this geometric result.
Illustrative case study
Fictional case: Birch Robotics tests a small mobile platform in a simulated storeroom. A point-based path planner finds routes through narrow gaps, but the simulated vehicle clips the shelving because the planner ignores its footprint.
The team revises collision checking to use the full vehicle geometry and an explicit clearance margin. It tests the whole interpolated motion between waypoints instead of checking only the endpoints.
When the vehicle carries a wider tray, the team updates the geometry and replans. A route that was valid for the unloaded platform is no longer assumed valid. The case illustrates how free space depends on the modeled robot and constraints, not only on a map of obstacle locations.
Common misconceptions and mistakes
- An open workspace point is free for every robot. Collision depends on the robot's full geometry and arrangement.
- Two free endpoints imply a valid connecting path. An obstacle or disconnected free-space component can separate them.
- A collision-free path is automatically executable. Steering and dynamic constraints may prevent following it.
- Boundary contact has one universal convention. The model must state whether touching is allowed; many formulations treat it as collision.
Business use
Free-space modeling is fundamental to robotic navigation, manipulation and motion simulation. Errors in the representation can invalidate otherwise sophisticated planning algorithms.
Specify the robot footprint, payload, joint limits and contact convention. Check whether the environment is static or time-dependent and whether clearance accounts for relevant uncertainty. Validate complete motions rather than isolated waypoints. When comparing planners, use the same collision model so that apparent performance improvements are not caused by allowing different motions.
People also ask
Is free space always two-dimensional?
No. Its dimension follows the configuration variables, so a robot arm's free space can have many dimensions.
Can a collision-free configuration be unreachable?
Yes. It may lie in a disconnected region or be unreachable under the robot's motion constraints.
Does free space stay fixed?
Only under the modeled conditions. Moving obstacles, changed payloads or altered joint restrictions can change the allowed configurations.
Check your understanding
A robot's center is outside an obstacle, but its body overlaps the obstacle. Is that configuration free?
Answer: No, under a collision-avoidance model. The whole robot geometry must satisfy the constraint, not just its reference point.
Related terms
- Configuration space
- Motion planning
- collision detection
- obstacle region
- robot footprint
- clearance
- reachability
- kinodynamic planning
Learn next
Increase the robot radius in the worked example to 0.6 meters and recompute the forbidden interval. Then explain why adding orientation as a variable matters for a rectangular robot even though it does not change a perfect circle's footprint.
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Sources
LaValle's Planning Algorithms develops configuration-space obstacles and free space: https://msl.cs.uiuc.edu/planning/node156.html
Hauser's Robotic Systems defines free configurations and distinguishes geometric path planning from other motion constraints: https://motion.cs.illinois.edu/RoboticSystems/WhatIsMotionPlanning.html
The circle calculation and Birch Robotics case are original illustrations. Contact and clearance conventions are stated explicitly rather than treated as universal physical guarantees.
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