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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.

An orientation-dependent configuration-space obstacle is the forbidden set for an extended robot whose occupied geometry changes with its orientation. For a planar rigid shape $B$ referred to a body-fixed point, translation $t\in {\mathbb{R}}^2$ and rotation matrix $R_{\theta }$, the occupied set is $t+R_{\theta }B$. With obstacle set $O$, collision meansFor each fixed angle, the forbidden translations are thereforeIndeed, an intersection gives $t+R_{\theta }b=o$, hence $t=o-R_{\theta }b$, and the converse follows by substitution. This is a reflected Minkowski sum, not an unreflected sum of the two shapes. Hauser develops these angle-indexed slices explicitly for a translating and rotating polygon.^[ref]

The full configuration-space obstacle is $\{(t,\theta ):t\in C_{\theta }\}$. Its angular coordinate is periodic: orientations zero and $2\pi$ represent the same rotation. Translation and orientation cannot generally be replaced by one planar obstacle inflation. A circular robot centered at its reference point is a special case, because rotating that shape leaves every $C_{\theta }$ unchanged.

An original doorway example uses a rectangle with half-length $a=0.30\mathrm{m}$ and half-width $b=0.10\mathrm{m}$, centered in an infinitely long horizontal corridor of width $W=0.25\mathrm{m}$. Its vertical half-extent at angle $\theta$ isWith contact counted as collision, a centered placement is clear only if $h(\theta )<W/2$. At zero angle the total width is $0.20\mathrm{m}$, leaving $25\mathrm{mm}$ at each wall. At $10^{\circ }$, the total width is about $0.3012\mathrm{m}$, so the same center position collides. For small positive angles, the first touching angle solves $0.30\sin \theta +0.10\cos \theta =0.125$, giving approximately $4.85^{\circ }$. A collision-free center path alone would miss this rotational restriction.

Angular sampling needs a bound on the geometry between checked orientations. If every body point lies within radius $\rho$ of the reference point, changing orientation by $\Delta \theta$ moves each corresponding point by at mostThe first expression is the chord length; the inequality follows from $\sin u\le u$. An angular clearance bound for a rotating body uses this motion allowance to transfer a measured separation to nearby orientations. For the rectangle above, $\rho =\sqrt{a^2+b^2}\approx 0.3162\mathrm{m}$. A ten-degree grid has a five-degree worst-case distance to a sampled orientation, allowing about $27.6\mathrm{mm}$ of point motion. A sampled $20\mathrm{mm}$ gap would not certify every intermediate orientation under that bound.

A geometric validation can place a rectangular test piece at predicted near-touching angles in a measured corridor, accounting for reference-point offsets and wall tolerances. A planner must also check the continuous motions connecting clear configurations; clear grid vertices alone do not establish clear connecting rotations. A conservative enclosing disk may simplify that check, at the cost of excluding some feasible narrow passages.

Robot motion planning computes a robot's proposed motion from its modeled state, environment and task constraints. A geometric planning problem asks for a path through configuration space; a timed motion can additionally constrain velocity, acceleration, forces and actuator effort. Hauser's primary text explicitly distinguishes geometric path planning from the broader problem involving dynamics and execution.^[ref]

Let $Q_{\mathrm{free}}$ be the configurations allowed by joint limits and collision detection. A geometric solution is a continuous curveEach point represents a complete robot configuration. A configuration-space obstacle is a forbidden set of configurations, not necessarily the same shape as the physical obstacle in the workspace.

An original planar example uses a point robot starting at $(0,0)$ and aiming for $(1,0)$, in metres. A circular obstacle centered at $(0.5,0)$ with radius $0.10\mathrm{m}$ blocks the straight path. A two-segment detour through $(0.5,0.20)$ has lengthThe perpendicular distance from the obstacle center to either segment isso the detour clears the circular obstacle. Checking only the start and goal would instead miss the collision on the straight segment.

Now give the robot a circular footprint of radius $0.05\mathrm{m}$, with translation as its only degree of freedom. The forbidden center positions form a disk of radius $0.15\mathrm{m}$: the obstacle is expanded by the footprint through Minkowski addition. The same detour still passes, but its center-path margin above the expanded obstacle is only about $0.0357\mathrm{m}$. The geometric reduction depends on the stated footprint and permitted motion.

A path's corner does not supply a dynamically smooth trajectory. Traversing this sharp detour at nonzero constant speed would instantaneously change velocity at its midpoint. A physical controller must stop, smooth the corner or choose a different feasible trajectory, while continuing to respect the obstacle and actuator constraints. A larger path length can be an acceptable cost if it permits the required motion.

A proposed validation compares the continuous planned geometry, sampled collision tests and measured execution on a controlled obstacle course. Report the robot geometry, constraints, computational budget and observed clearance. Failure to find a path in a limited run does not alone prove none exists; success on a numerical path does not alone establish accurate execution. Hauser treats completeness and the distinction between paths and feedback policies separately.^[ref]

A robust swing-foot clearance margin is a guaranteed lower separation between a robot's swing foot and terrain over a specified crossing interval and set of allowed errors. It differs from clearance computed only for a nominal trajectory. The interval must exclude intended ground contacts when a strictly positive margin is required.

An original point-foot model lets the planned terrain be $H(x)$ and the planned clearance be $c_0=z-H(x)$. Assume actual vertical foot error $|{\delta }_z|\le {ϵ}_z$, horizontal error $|{\delta }_x|\le {ϵ}_x$, and terrain-model height error bounded by ${ϵ}_H$. If $H$ has Lipschitz constant $L_H$ over the relevant neighborhood, thenThe actual clearance consequently satisfiesThis follows from the triangle inequality and covers all errors in the stated bounds. Their effects add in a worst-case calculation; combining them as independent random standard uncertainties would answer a different question.

For a local sloped-terrain example, take ${ϵ}_z=3\mathrm{mm}$, ${ϵ}_H=4\mathrm{mm}$, ${ϵ}_x=2\mathrm{mm}$ and $L_H=0.5$. The total clearance allowance is $3+4+0.5(2)=8\mathrm{mm}$. Guaranteeing a $5\mathrm{mm}$ margin therefore requires nominal clearance at least $13\mathrm{mm}$ throughout that crossing. A nominal $5\mathrm{mm}$ gap gives only a $-3\mathrm{mm}$ lower bound and cannot certify the requested separation.

Finite path samples need another allowance. Suppose the nominal clearance as a function of path parameter $s$ is Lipschitz with constant $L_c$, and every point lies within $\Delta s/2$ of a checked sample. If the smallest sampled clearance is $c_{\mathrm{sample}}$, a clearance bound between trajectory samples givesWith $L_c=0.10\mathrm{m}$ per unit parameter and sample spacing $\Delta s=0.10$, between-sample uncertainty costs another $5\mathrm{mm}$. The preceding example would then require sampled clearance at least $18\mathrm{mm}$ for a certified $5\mathrm{mm}$ margin.

Bounds must match the geometry. A vertical obstacle edge is not covered by a finite height-slope bound across its discontinuity; use the full configuration-space obstacle instead. Shared coordinate-frame errors can also cancel in a relative clearance, so separate component bounds can be conservative when their joint dependence is known.

The primary robotics formulation requires constraints along the full path and distinguishes planned from executed motion.^[ref] A proposed test records terrain and foot geometry in one reference frame, validates the stated error bounds, and compares a continuous-path certificate with observed minimum separation. A passing sample grid without the intervening bound does not establish the same guarantee.