= Uncertainty in a rigid-body bounding radius
{title2=$R+\varepsilon_R$}
<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 $\widehat B$ and modeled pivot $\widehat c$ satisfy $\|\widehat p-\widehat c\|\le\widehat R$ for every $\widehat p\in\widehat B$. Suppose every point $p$ of the actual body lies within $\epsilon_B$ of some modeled point, and the actual pivot obeys $\|c-\widehat c\|\le\epsilon_c$. The triangle inequality gives
$$
\|p-c\|\le\|p-\widehat p\|+\|\widehat p-\widehat c\|+\|\widehat c-c\|
\le\epsilon_B+\widehat R+\epsilon_c.
$$
Thus a certified radius is
$$
\boxed{R_{\rm cert}=\widehat R+\epsilon_B+\epsilon_c.}
$$
The 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 $\widehat R=0.500\,\mathrm m$, shape coverage error $\epsilon_B=3\,\mathrm{mm}$ and pivot-location error $\epsilon_c=2\,\mathrm{mm}$. Then $R_{\rm cert}=0.505\,\mathrm m$. A ten-degree rotation about the actual fixed pivot moves any actual point by at most
$$
2R_{\rm cert}\sin5^\circ\approx88.03\,\mathrm{mm}.
$$
Using 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.<Robotic Systems by Kris Hauser>[Chapter 9, C-obstacles for rotational motion and Motion planning in SE(2)]{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 $v_{\max}+R_{\rm cert}\omega_{\max}$ 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 $s\le1+\epsilon_s$ about the modeled pivot, a sufficient bound is $(1+\epsilon_s)\widehat R+\epsilon_B+\epsilon_c$, under the explicitly separated scale and residual-shape assumptions. With $\epsilon_s=0.01$ 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.
Back to article page
