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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.