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 $d(t)\ge0$ 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 $L\ge0$, and let the exact endpoint distances be $d_0,d_1$. The two endpoint inequalities imply
$$
d(t)\ge\max\{d_0-Lt,\ d_1-L(h-t),\ 0\}.
$$
This 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.<Robotic Systems by Kris Hauser>[Chapter 8, Stating the motion planning problem; Chapter 9, rotational configuration-space obstacles]{ref}

For $L>0$ and consistent endpoint data $|d_1-d_0|\le Lh$, the two sloping bounds intersect at
$$
t_* =\frac{d_0-d_1+Lh}{2L}\in[0,h].
$$
Their maximum is minimized there, giving the sharp general lower bound
$$
d(t)\ge\max\left\{0,\frac{d_0+d_1-Lh}{2}\right\}.
$$
It 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 $d_0=12\,\mathrm{mm}$, $d_1=20\,\mathrm{mm}$, $L=100\,\mathrm{mm/s}$ 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 $t_*=0.01\,\mathrm s$. Thus the sharper certificate proves separation greater than a $10\,\mathrm{mm}$ <required clearance margin> where the simpler one does not.

\Image[/endpoint-clearance-cones.png]
{title=Paired Lipschitz endpoint bounds certify at least 11 mm clearance over 100 ms, while the simpler constant bound gives 7 mm. Original constructed certificate; the blue envelope is not a measured trajectory.}

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.


 Endpoint certificate for continuous collision clearance (source code)