 Configuration-space obstacle

A configuration-space obstacle is a set of robot configurations whose occupied geometry intersects an obstacle. It translates a collision question about an extended robot into an excluded region of configuration space. The excluded coordinates can include rotation and joint angles as well as position. Hauser's primary planning formulation describes feasibility using the robot's occupied workspace set at each configuration.^[ref]

For an original fixed-orientation translation model, let $R$ be the robot's shape relative to its reference point, $O$ the physical obstacle and $q$ the reference-point translation. Collision meansThis is equivalent to $q+r=o$ for some $r\in R$, $o\in O$, and thereforeThe Minkowski sum must use the reflected robot shape. When $R$ is symmetric about its reference point, reflection leaves it unchanged; for an asymmetric reference choice, omitting the minus sign changes the collision region.

A one-dimensional example makes the reflection visible. Let a robot occupy $R=[0,0.30]$ metres relative to its left endpoint, and let the obstacle be the point $O=\{1\}$ metre. The forbidden reference positions are $[0.70,1.00]$ metres. At $q=0.80$, the robot spans $[0.80,1.10]$ and hits the obstacle. Using $O\oplus R=[1.00,1.30]$ would wrongly miss this collision and exclude some clear positions.

For a centered circular footprint of radius $r$ and a circular obstacle of radius $a$, the forbidden center positions form a disk of radius $a+r$. For axis-aligned rectangular shapes, each obstacle side expands by the robot's corresponding half-width. These are exact reductions for the stated geometry and fixed orientation.

An orientation-dependent configuration-space obstacle instead has a different translation-space cross-section for each orientation. A thin centered rod of length $0.40\mathrm{m}$ collides with a radius-$0.03\mathrm{m}$ obstacle centered at $(0.18,0)$ when the rod lies along the horizontal axis. At the same center position, a vertical rod is clear. The shared center point cannot encode the difference without the orientation coordinate.

For closed occupied sets, touching belongs to the forbidden set under this definition; a required clearance margin can enlarge it further. A proposed geometric check places the modeled robot at selected boundary configurations, compares the predicted intersections with measured geometry, and then verifies the path between them. Testing a few clear endpoint configurations alone does not establish a collision-free continuous motion.