Unresolved intervals in adaptive collision checking

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.