Multi-sample consistency of distance-query intervals

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.

A multi-sample clearance envelope bounds every possible instantaneous clearance consistent with several uncertain distance queries and a known motion-rate limit. It sharpens isolated endpoint estimates by requiring one Lipschitz continuous clearance history to explain all samples. Such a bound addresses the between-sample gap in continuous robot motion planning, where valid sampled configurations alone do not guarantee a valid path.^[ref]

At times $t_i$, let $0\le {\ell }_i\le d(t_i)\le u_i$, and suppose the nonnegative clearance $d$ has Lipschitz constant $L$. Assuming multi-sample consistency of distance-query intervals, defineEvery admissible history obeys $d_{-}\le d\le d_{+}$. Each lower cone follows from $d(t)\ge d(t_i)-L|t-t_i|$; the upper cones follow by reversing the inequality. Under consistency, both envelope functions themselves satisfy the sample intervals and the same Lipschitz bound. Thus the enclosure is pointwise sharp among the scalar histories allowed by these assumptions.

A positive minimum of $d_{-}$ over a time interval certifies separation throughout it. If that minimum is zero, the assumptions permit a scalar history touching zero; this is lack of a strictly positive certificate, not an observation of collision. Some admissible scalar histories may not correspond to a realizable rigid-body path, so sharpness here concerns the stated scalar model.

The exact interval minimum is inexpensive to compute. On $[t_k,t_{k+1}]$, all sample times lie on one side or the other. PutThen $d_{-}(t)=\max (0,A_k-Lt,B_k+Lt)$. For $L>0$, its minimum occurs at the crossing $t_{\ast }=(A_k-B_k)/(2L)$ if that crossing lies in the interval; otherwise it occurs at the nearer endpoint. Prefix and suffix maxima therefore allow all segment minima to be computed in linear time. For $L=0$, every admissible clearance is constant and the lower envelope is ${\max }_i{\ell }_i$.

For an original example, queries at $0,1,2\mathrm{s}$ give intervals $[5,7]$, $[6,8]$, $[5,7]\mathrm{mm}$, and $L=4mm/s$. On the first segment, $A_0=5$ and $B_0=2$ in the corresponding millimetre/second coordinates. The crossing is $t_{\ast }=0.375\mathrm{s}$ and the lower bound there is $3.5\mathrm{mm}$. Symmetry gives a second minimum at $1.625\mathrm{s}$, so the full interval is certified with margin $3.5\mathrm{mm}$. Using only the two outer queries would give a minimum of $1\mathrm{mm}$.

A proposed checker first tests consistency, then computes segment minima with conservative arithmetic and reports the limiting time and active queries. Keep query errors, timestamps and the bound $L$ traceable to the same geometry and motion. Clearance envelope with segment-dependent speed bounds can use more motion information, but replacing $L|t-t_i|$ by an unsupported smaller number would invalidate the certificate.