= Required clearance margin
{title2=$d_{\rm safe}$}
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 $d_{\rm safe}\ge0$. A query with $0<d(t)\le d_{\rm safe}$ violates that requested separation without demonstrating contact. Choosing $d_{\rm safe}$ 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 $\varepsilon_B,\varepsilon_O,\varepsilon_T$ in <Hausdorff distance>. The <triangle inequality> then gives
$$
d_{\rm actual}(t)\ge d(t)-\varepsilon_B-\varepsilon_O-\varepsilon_T.
$$
To guarantee an actual separation greater than $d_{\rm req}$, a sufficient modeled threshold is
$$
d_{\rm safe}=d_{\rm req}+\varepsilon_B+\varepsilon_O+\varepsilon_T.
$$
These 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.<Robotic Systems by Kris Hauser>[Chapter 9, Translating robots and Minkowski sums and differences]{ref}
For an original error budget, require $d_{\rm req}=5\,\mathrm{mm}$ 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 $L=0.2\,\mathrm{m/s}$, and the interval duration is $h=20\,\mathrm{ms}$. The simple interval certificate gives
$$
d(t)\ge20-1-\frac{200\times0.020}{2}=17\,\mathrm{mm}.
$$
Subtracting 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.
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