 Uncertainty in a rigid-body bounding radius

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 $\hat{B}$ and modeled pivot $\hat{c}$ satisfy $\Vert \hat{p}-\hat{c}\Vert \le \hat{R}$ for every $\hat{p}\in \hat{B}$. Suppose every point $p$ of the actual body lies within ${ϵ}_B$ of some modeled point, and the actual pivot obeys $\Vert c-\hat{c}\Vert \le {ϵ}_c$. The triangle inequality givesThus a certified radius isThe 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 $\hat{R}=0.500\mathrm{m}$, shape coverage error ${ϵ}_B=3\mathrm{mm}$ and pivot-location error ${ϵ}_c=2\mathrm{mm}$. Then $R_{\mathrm{cert}}=0.505\mathrm{m}$. A ten-degree rotation about the actual fixed pivot moves any actual point by at mostUsing 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.^[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 $v_{\max }+R_{\mathrm{cert}}{\omega }_{\max }$ 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 $s\le 1+{ϵ}_s$ about the modeled pivot, a sufficient bound is $(1+{ϵ}_s)\hat{R}+{ϵ}_B+{ϵ}_c$, under the explicitly separated scale and residual-shape assumptions. With ${ϵ}_s=0.01$ 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.