Required clearance margin ($d_{\mathrm{safe}}$)

A required clearance margin is the minimum separation requested by a geometric motion specification. In adaptive clearance sampling, it is the threshold that a continuous-interval lower bound must exceed. Positive separation from an obstacle is enough to exclude contact in an exact model, but can still be less than the requested margin.

Let $d(t)$ be the modeled nonnegative set distance between a rigid body and fixed obstacles. Using the strict convention of adaptive clearance sampling, an interval is accepted only when its guaranteed lower bound exceeds $d_{\mathrm{safe}}\ge 0$. A query with $0<d(t)\le d_{\mathrm{safe}}$ violates that requested separation without demonstrating contact. Choosing $d_{\mathrm{safe}}$ is part of the specification, not something determined uniquely by the collision detection algorithm.

Geometric uncertainty can be incorporated with explicit bounds. Suppose robot geometry, obstacle geometry and reference-point tracking error can each change the relevant sets by at most ${\epsilon }_B,{\epsilon }_O,{\epsilon }_T$ in Hausdorff distance. The triangle inequality then givesTo guarantee an actual separation greater than $d_{\mathrm{req}}$, a sufficient modeled threshold isThese are simultaneous deterministic bounds. No independence assumption is needed to add them conservatively. The configuration-space construction in the primary planning text similarly represents body size by enlarging forbidden regions.^[ref]

For an original error budget, require $d_{\mathrm{req}}=5\mathrm{mm}$ and take geometric bounds of $2$, $3$ and $4\mathrm{mm}$ respectively. The modeled threshold becomes $14\mathrm{mm}$. Suppose the smaller reported endpoint distance is $20\mathrm{mm}$, the distance-query error is bounded by $1\mathrm{mm}$, the clearance Lipschitz continuity constant is $L=0.2m/s$, and the interval duration is $h=20\mathrm{ms}$. The simple interval certificate givesSubtracting the $9\mathrm{mm}$ geometric budget leaves at least $8\mathrm{mm}$ actual separation, exceeding the requested $5\mathrm{mm}$ by $3\mathrm{mm}$ in this model.

The query error was already subtracted from the interval bound and must not be subtracted again as an unrecorded duplicate allowance. Conversely, the motion term $Lh/2$ addresses unsampled motion; it does not automatically include obstacle-location error. An endpoint certificate for continuous collision clearance should state which effects are included in each quantity, while unresolved intervals in adaptive collision checking retain intervals whose lower bound is insufficient.