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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.

Compatibility of bounded endpoint distance queries asks whether two uncertain distance readings could arise from any nonnegative clearance function obeying a stated motion bound. It tests consistency of the observations and assumptions; it does not itself prove that the entire motion is collision-free.

Suppose endpoint queries yield intervals $I_0=[{\ell }_0,u_0]$ and $I_1=[{\ell }_1,u_1]$ for the true set distances, with $0\le {\ell }_i\le u_i$. Let the interval duration be $h>0$ and clearance have Lipschitz continuity constant $L\ge 0$. Write $c=Lh$. Compatibility means there exist $a\in I_0$ and $b\in I_1$ satisfying $|a-b|\le c$. The exact condition isalong with the two intervals being nonempty and nonnegative. Equivalently, the gap between the two intervals, $\max (0,{\ell }_0-u_1,{\ell }_1-u_0)$, must not exceed $c$.

Necessity follows because $a\ge {\ell }_0$, $b\le u_1$ and $a-b\le c$, with the reversed argument for the other inequality. For sufficiency, overlapping intervals allow $a=b$ at any common point. If they are disjoint, choose their nearest endpoints; the boxed condition bounds that separation by $c$. Linear interpolation between the selected nonnegative endpoints has slope at most $L$, proving that at least one admissible continuous clearance function exists.

The feasible endpoint intervals can be tightened toThese are the exact projections of the feasible endpoint-pair set onto each coordinate. A large difference between the lower bounds alone is not a contradiction: lower bounds are not exact distances.

For an original example, take $I_0=[4,8]\mathrm{mm}$, $I_1=[15,20]\mathrm{mm}$ and $Lh=10\mathrm{mm}$. Their nearest endpoints differ by $7\mathrm{mm}$, so the data are compatible. Tightening gives $I_0^{\prime }=[5,8]\mathrm{mm}$ and $I_1^{\prime }=[15,18]\mathrm{mm}$. By contrast, $[1,3]$ and $[15,17]\mathrm{mm}$ have a $12\mathrm{mm}$ minimum gap and are incompatible with the same motion bound.

Incompatibility means that some query enclosure, motion bound, timing or model assumption is wrong. It is not evidence that a collision occurred. Compatibility also need not certify separation: exact endpoint distances of $5\mathrm{mm}$ over one second with $L=10mm/s$ permit both constant clearance and a V-shaped clearance touching zero halfway through. The clearance certificate with bounded distance-query error evaluates a lower envelope over the full interval to answer that additional question.

For $L=0$, compatibility reduces to overlap of the intervals because clearance is constant. With more query times, multi-sample consistency of distance-query intervals must preserve one common clearance model rather than checking unrelated pairs with incompatible bounds. The primary motion-planning formulation requires constraints along a continuous path, motivating these between-sample checks.^[ref]

A proposed checker retains query lower and upper bounds, the clock interval, geometry version and motion allowance. It should report inconsistent inputs separately from undecided clearance intervals and certified margin violations. Tight numerical arithmetic alone cannot turn an unsupported standard uncertainty into a deterministic distance enclosure.

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.

Multi-sample consistency of distance-query intervals asks whether every bounded distance reading can belong to one nonnegative clearance function with the claimed Lipschitz constant. The same motion history must explain all samples. Checking each adjacent pair in isolation can select a different intermediate distance for each check and miss a contradiction.

Let strictly increasing times $t_0,\dots ,t_n$ have nonempty intervals $[{\ell }_i,u_i]$, with $0\le {\ell }_i\le u_i$, and let $L\ge 0$ bound clearance change per unit time. Necessary and sufficient conditions areNecessity follows by comparing any feasible distances $d(t_i),d(t_j)$ using Lipschitz continuity. For sufficiency, constructEach envelope has Lipschitz constant at most $L$. The pairwise conditions and the triangle inequality imply $f_{-}(t)\le f_{+}(t)$ everywhere. At each sample time, ${\ell }_i\le f_{-}(t_i)\le u_i$, so $f_{-}$ itself is an admissible nonnegative clearance history. This proves existence, not that this constructed history is the robot's actual history.

The intervals $[f_{-}(t_i),f_{+}(t_i)]$ are exact feasible projections at sample times. Both bounding functions are feasible and convex combinations of them remain feasible, so every intermediate value at one fixed time is attainable. The associated multi-sample clearance envelope also gives between-sample lower bounds for a clearance certificate with bounded distance-query error.

For an original counterexample to adjacent-only testing, let successive times differ by one second, take $L=4mm/s$, and useThe first adjacent pair can use middle distance $0$, and the second can use middle distance $10$; each pair is compatible. Yet the endpoints require a $10\mathrm{mm}$ change in two seconds, exceeding the allowed $8\mathrm{mm}$. No common history exists.

An equivalent linear-time feasibility test propagates a reachable interval. Start $J_0=I_0$ and recursively setStop if an interval becomes empty. In the example, $J_1=[0,4]$ and the next propagated upper bound is $8$, so $J_2$ is empty. Retaining the tightened interval is the key difference from independent adjacent-pair checks. If all intervals remain nonempty, select a final value and work backward through the reachable intervals; piecewise linear interpolation yields a feasible history.

Incompatibility diagnoses inconsistent readings or assumptions, not a demonstrated collision. Compatibility proves only the existence of a scalar clearance history; it does not prove that some rigid robot trajectory realizes that history or that the true clearance stays positive. Continuous collision avoidance remains a constraint along the physical path, as in the primary motion-planning formulation.^[ref] A proposed implementation tests synthetic histories with known bounds, deliberately inconsistent triples and timestamp errors, keeping model inconsistency separate from insufficient clearance.

Uncertainty in a rigid-body bounding radius concerns whether the radius used in a motion bound actually encloses the robot about its declared pivot. A nominal mesh radius can miss manufacturing tolerances, an attached tool or an uncertain reference-point location. The bounding radius for rigid-body clearance must bound the relevant actual occupied geometry, not only a convenient displayed shape.

Let modeled body $\hat{B}$ and modeled pivot $\hat{c}$ satisfy $\Vert \hat{p}-\hat{c}\Vert \le \hat{R}$ for every $\hat{p}\in \hat{B}$. Suppose every point $p$ of the actual body lies within ${ϵ}_B$ of some modeled point, and the actual pivot obeys $\Vert c-\hat{c}\Vert \le {ϵ}_c$. The triangle inequality givesThus a certified radius isThe one-sided geometric coverage assumption is sufficient; it does not require every modeled point to lie close to actual material. A conservative overfilled model is allowed.

For an original example, take $\hat{R}=0.500\mathrm{m}$, shape coverage error ${ϵ}_B=3\mathrm{mm}$ and pivot-location error ${ϵ}_c=2\mathrm{mm}$. Then $R_{\mathrm{cert}}=0.505\mathrm{m}$. A ten-degree rotation about the actual fixed pivot moves any actual point by at mostUsing the nominal radius gives only $87.16\mathrm{mm}$, missing about $0.872\mathrm{mm}$ of possible motion. The angular clearance bound for a rotating body translates that displacement allowance into a clearance bound. Geometry and rotation must be considered together in the continuous-path planning model.^[ref]

Pivot-location uncertainty in a rotational clearance bound is distinct from an uncertain body translation. If the chosen reference point also moves, its translation contributes separately to the occupied-point motion. A speed certificate becomes $v_{\max }+R_{\mathrm{cert}}{\omega }_{\max }$ when the stated translational and angular speed bounds are valid. Substituting a smaller radius about a different center without accounting for that center’s motion changes the problem.

If the geometry has an additional bounded scale factor $s\le 1+{ϵ}_s$ about the modeled pivot, a sufficient bound is $(1+{ϵ}_s)\hat{R}+{ϵ}_B+{ϵ}_c$, under the explicitly separated scale and residual-shape assumptions. With ${ϵ}_s=0.01$ in the example it becomes $0.510\mathrm{m}$. Unknown errors and repeated descriptions of the same geometric tolerance must not be combined as if they were independent extra effects.

A radius enlargement addresses the motion-rate bound. It does not by itself correct an inaccurate sampled distance between the nominal robot surface and an obstacle. That query still needs a suitable geometric allowance or a clearance certificate with bounded distance-query error. Likewise, hard enclosure tolerances are different from standard uncertainties; a root-sum-square of standard deviations is not automatically a guaranteed radius.

A proposed geometric check measures extreme points of the assembled robot, including attachments, relative to the same pivot used by the motion representation. Account for measurement error and verify coverage between sampled surface points using a geometric model or validated tolerance. Recompute the bound when the tool, pivot or assembly changes, and record the radius’s provenance alongside the trajectory certificate.

Unresolved intervals in adaptive collision checking are trajectory intervals for which the available queries and motion bounds prove neither the requested continuous clearance nor a violation at a known configuration. They are a computational result of incomplete information, rather than evidence that a collision occurred.

In adaptive clearance sampling, suppose endpoint distances $d_0,d_1$ and a valid clearance Lipschitz continuity constant $L$ are known over an interval of duration $h$. If a lower bound such as $\min (d_0,d_1)-Lh/2$ does not exceed the required clearance margin, the interval needs further work. Its endpoints can both be clear while its interior remains undecided. The continuous-path requirement motivating this distinction is stated in the primary planning text.^[ref]

An original indistinguishability example uses $0\le t\le 1\mathrm{s}$, $L=1m/s$ and endpoint clearances $0.4\mathrm{m}$. Both of the following functions satisfy the same endpoint data and motion bound:They can be realized by the height of a point moving above a fixed horizontal wall. The first path stays clear; the second reaches the wall. For a requested $0.05\mathrm{m}$ separation, the initial endpoint certificate is $0.4-0.5=-0.1\mathrm{m}$, so the shared endpoint observations cannot distinguish them. A negative lower estimate does not establish a negative physical distance.

Querying the midpoint resolves this constructed example. Model A gives another $0.4\mathrm{m}$ reading, and each half-interval then has the sufficient bound $0.4-0.25=0.15\mathrm{m}>0.05\mathrm{m}$. Model B gives zero, demonstrating contact. More generally, a strictly positive sampled distance at or below the requested margin establishes a margin violation but not a collision. Exact zero set distance establishes contact or intersection when the modeled body is a nonempty compact set and the obstacles form a nonempty closed set.

If the chosen query, time or subdivision budget expires first, retain the undecided intervals explicitly. Accepting the whole trajectory would discard the unverified portions; labeling every undecided portion a collision would also claim information not obtained.

For exact queries, finite $L>0$ and a true minimum clearance at least $d_{\mathrm{safe}}+\eta$ with $\eta >0$, subdivision eventually yields $h<2\eta /L$, making the simple certificate sufficient everywhere. A trajectory merely touching the required threshold has no such surplus. A clearance certificate with bounded distance-query error must also account for how conservative query bounds affect that remaining surplus.