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Adaptive clearance sampling spends more collision detection queries on portions of a trajectory that are difficult to certify, instead of accepting a path merely because a fixed set of samples is clear. Its guarantee comes from a bound on motion between samples. A configuration space path must satisfy the geometric constraints continuously.^[ref]

Represent the body by a nonempty compact set and the fixed obstacles by a nonempty closed set. Let $d(t)$ be their nonnegative set distance, and suppose a known Lipschitz continuity bound gives $|d(t)-d(s)|\le L|t-s|$. On an interval of duration $h$ with endpoint clearances $d_0,d_1$, each time is at most $h/2$ from one endpoint. Hence an endpoint certificate for continuous collision clearance isIf this lower bound is strictly positive, the entire interval is clear. For a required clearance margin $d_{\mathrm{safe}}\ge 0$, require the bound to exceed $d_{\mathrm{safe}}$ instead. Otherwise query the midpoint and subdivide. A sharper certificate uses both endpoint cones; when $L>0$ and the distances satisfy the assumed bound, its minimum is $(d_0+d_1-Lh)/2$. The simpler bound is sufficient for the example below.

For a translating and rotating rigid body, a valid speed estimate is $L=v_{\max }+R{\omega }_{\max }$, where $R$ is a bounding radius for rigid-body clearance, provided these are bounds over the whole interval and the obstacles stay fixed. The angular clearance bound for a rotating body proves the rotational contribution. A distance-estimation error bounded by $\epsilon$ can be accommodated by replacing each reported endpoint distance by its lower bound ${\hat{d}}_i-\epsilon$.

In an original point-motion example, $q(t)=(vt,0.01\mathrm{m})$ with $v=1m/s$ and $0\le t\le 1\mathrm{s}$ passes the obstacle point $(0.5,0)\mathrm{m}$. Its exact clearance is $d(t)=\sqrt{(vt-0.5\mathrm{m}{)}^2+(0.01\mathrm{m}{)}^2}$ and $L=1m/s$ suffices. Require $d_{\mathrm{safe}}=0.005\mathrm{m}$. The initial endpoint certificate is positive but below this required margin, so more queries are needed. Recursive midpoint sampling certifies long intervals away from the obstacle and refines only near the closest approach. The stated midpoint rule uses 15 distance queries and certifies 14 intervals. The figure shows the exact function alongside the sampled certificate; the formula is known here only because the example is constructed.

Unresolved intervals in adaptive collision checking must remain unresolved if a chosen depth or time budget is exhausted: failure to certify positive clearance does not establish a collision. Conversely, a query with exact zero set distance establishes contact or intersection. A sampled positive distance at or below $d_{\mathrm{safe}}$ violates the requested margin but does not establish contact. With exact queries, a valid finite $L$ and a trajectory whose clearance has a positive lower bound, subdivision eventually certifies every interval for $d_{\mathrm{safe}}=0$; a positive requested margin requires a positive surplus above that margin. Without positive clearance, finite termination is not promised.

A clearance certificate with bounded distance-query error proves separation over a continuous motion interval when sampled distances are known only within stated error bounds. A positive returned value from collision detection is not by itself a lower bound on the true geometric distance. Approximation, floating-point error and model error must be included if they affect the intended claim.

Let true set distance $d(t)\ge 0$ satisfy $|d(t)-d(s)|\le L|t-s|$ on $[0,h]$. At endpoints, assume certified query errors $|d_i-{\hat{d}}_i|\le {ϵ}_i$. DefineProvided the query bounds are consistent with nonnegative distances, ${\ell }_i\le d_i\le u_i$. The endpoint lower bounds propagate intoThis extends the endpoint certificate for continuous collision clearance. The primary planning model requires feasibility along the whole continuous path; checking endpoint poses alone is insufficient.^[ref]

The minimum of this envelope must be computed on the actual interval. Its general closed form, including unequal lower bounds, isFor $L>0$ and $|{\ell }_1-{\ell }_0|\le Lh$, the two sloping bounds meet inside the interval and the average expression dominates the other nonzero candidates. If ${\ell }_1-{\ell }_0>Lh$, their intersection lies before the interval; the minimum is instead $\max (0,{\ell }_1-Lh)$. The opposite inequality exchanges the endpoints. For $L=0$, the formula reduces to $\max ({\ell }_0,{\ell }_1)$ because true clearance is constant.

The lower bounds themselves need not satisfy the consistency condition for exact endpoint distances. For example, ${\ell }_0=2\mathrm{mm}$, ${\ell }_1=20\mathrm{mm}$ and $Lh=10\mathrm{mm}$ permit a true clearance rising from $10$ to $20\mathrm{mm}$. Rejecting these lower bounds because their difference exceeds $Lh$ would wrongly treat them as exact observations. The general certificate gives $10\mathrm{mm}$; using only the intersecting-cone expression would give the weaker $6\mathrm{mm}$.

A second original example takes reported distances $12$ and $20\mathrm{mm}$, errors $1$ and $3\mathrm{mm}$, and $Lh=10\mathrm{mm}$. Then ${\ell }_0=11$ and ${\ell }_1=17\mathrm{mm}$, so $b=9\mathrm{mm}$. The error-free formula would have claimed $11\mathrm{mm}$. For a required strict margin above $10\mathrm{mm}$, the uncertain queries do not certify the interval. Tightening both errors to $0.5\mathrm{mm}$ raises the lower bound to $10.5\mathrm{mm}$ and supplies the requested certificate.

Compatibility of bounded endpoint distance queries also uses the upper bounds. The intervals must obey ${\ell }_0\le u_0$, ${\ell }_1\le u_1$, ${\ell }_0\le u_1+Lh$ and ${\ell }_1\le u_0+Lh$. A violated condition signals inconsistent assumptions. An upper distance bound at or below a required margin can establish a sampled margin violation; failure of a lower-bound certificate alone does not establish a collision.

A proposed checker records the query’s enclosure guarantee, units and geometric model, propagates the motion bound with the actual interpolation, and leaves uncertified intervals unresolved for further subdivision or stronger queries. A standard deviation is not a deterministic query enclosure. A certificate against a nominal obstacle model also needs a separate justified relation to the actual obstacle geometry before it can be interpreted physically.

An endpoint certificate for continuous collision clearance uses distances at two sampled configurations and a bound on intervening motion to prove separation for an entire time interval. Its guarantee concerns every intermediate configuration, rather than only the sampled poses.

Let $d(t)\ge 0$ be the set distance between a moving nonempty compact set and fixed nonempty closed set of obstacles, with $0\le t\le h$. Suppose $|d(t)-d(s)|\le L|t-s|$, with $L\ge 0$, and let the exact endpoint distances be $d_0,d_1$. The two endpoint inequalities implyThis uses a valid clearance Lipschitz continuity constant, such as the rigid-body motion bound derived in angular clearance bound for a rotating body. The continuous-path requirement itself is fundamental to the primary planning model.^[ref]

For $L>0$ and consistent endpoint data $|d_1-d_0|\le Lh$, the two sloping bounds intersect atTheir maximum is minimized there, giving the sharp general lower boundIt is sharp among nonnegative functions with that Lipschitz continuity bound and those endpoints: the displayed envelope itself satisfies the assumptions. This does not say that the unknown actual trajectory attains the lower bound.

For an original example, take $d_0=12\mathrm{mm}$, $d_1=20\mathrm{mm}$, $L=100mm/s$ and $h=0.1\mathrm{s}$. Since $Lh=10\mathrm{mm}$, the simple nearest-endpoint certificate gives $\min (d_0,d_1)-Lh/2=7\mathrm{mm}$. The paired bounds give $(12+20-10)/2=11\mathrm{mm}$, with their intersection at $t_{\ast }=0.01\mathrm{s}$. Thus the sharper certificate proves separation greater than a $10\mathrm{mm}$ required clearance margin where the simpler one does not.

For $L=0$, consistency requires $d_0=d_1$, and the clearance is constant. An endpoint difference larger than $Lh$ contradicts the exact-data assumptions; the motion bound, query errors or model must be checked. A clearance certificate with bounded distance-query error replaces exact observations with justified lower bounds and must preserve the assumptions used in minimizing their envelope.

Under the strict convention of adaptive clearance sampling, certification requires the guaranteed lower bound to exceed the requested margin. If it does not, unresolved intervals in adaptive collision checking remain undecided until additional queries or other information resolve them. Failure of this sufficient certificate alone does not establish a collision.

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.