 Pivot-location uncertainty in a rotational clearance bound

Pivot-location uncertainty in a rotational clearance bound describes how an error in the point about which a rigid rotation is modeled changes predicted occupied-point motion. A radius measured about one pivot is not automatically valid about another. The underlying planning problem must include both translation and orientation of the robot’s actual geometry.^[ref]

Let $Q$ be a fixed rotation matrix, $\hat{c}$ the modeled pivot, and $c=\hat{c}+e$ the actual pivot. First suppose the initial physical point $p$ is known exactly and both descriptions use the same rotation $Q$. The modeled and actual endpoints areso their difference is exactlyThis vanishes for zero rotation even if the pivot location is uncertain. Treating pivot error as an arbitrary full translation at every angle can therefore be unnecessarily conservative.

For a rotation through principal angle $0\le \theta \le \pi$ about a known axis direction, the component of $e$ along that direction is unchanged by $Q$. Its perpendicular component giveswhen $\Vert e\Vert \le {ϵ}_c$. The chord-length factor is the same geometry used in the angular clearance bound for a rotating body. For a $30^{\circ }$ rotation and ${ϵ}_c=2\mathrm{mm}$, the maximum endpoint error from pivot location alone is about $1.035\mathrm{mm}$. An equally large pivot offset parallel to the axis produces no endpoint change in this model.

If the initial point also has error $p-\hat{p}=\delta p$ with $\Vert \delta p\Vert \le {ϵ}_p$, thenFor the difference of displacement vectors, rather than final positions, the shape term is $(Q-I)\delta p$, giving the different bound $2({ϵ}_p+{ϵ}_c)\sin (\theta /2)$. The requested geometric quantity determines which allowance applies.

A complementary radius argument bounds total actual motion. If modeled points satisfy $\Vert \hat{p}-\hat{c}\Vert \le \hat{R}$, then $\Vert p-c\Vert \le \hat{R}+{ϵ}_p+{ϵ}_c$, and total point displacement is at most $2(\hat{R}+{ϵ}_p+{ϵ}_c)\sin (\theta /2)$. This is the enclosure approach in uncertainty in a rigid-body bounding radius; it need not equal an endpoint-error calculation against a nominal path.

Correlations can reduce the bound. If the same coordinate-frame translation shifts both the modeled point and pivot, relative geometry may be unchanged; treating those two descriptions as independent physical errors loses that cancellation. Conversely, rotation-axis direction uncertainty in clearance bounds changes $Q$ itself and is not covered by the fixed-$Q$ pivot formula.

A proposed rotary-stage check observes several rigid fiducials, estimates the pivot and axis in one coordinate frame, and compares independent endpoint measurements at several angles with the predicted chord scaling. Include attached tools and measurement uncertainty. When the robot moves against an uncertain obstacle, combine this occupied-point model with a separately justified distance-query enclosure; a correct pivot bound does not establish the obstacle model’s accuracy.