Combined pivot and axis uncertainty in rotational clearance

Combined pivot and axis uncertainty in rotational clearance bounds the occupied-point error of a rotating robot when its modeled pivot and axis direction are both uncertain. Adding separate bounds is valid only after specifying the same physical points, coordinate frame and endpoint quantity. A rigid rotation is represented by a rotation matrix, obtainable from a unit axis and angle through Rodrigues rotation formula.^[ref]

Let the nominal initial point and pivot be $\hat{p},\hat{c}$, with $r=\hat{p}-\hat{c}$ and $\Vert r\Vert \le \hat{R}$. Let actual values be $p=\hat{p}+\delta p$ and $c=\hat{c}+e$, where $\Vert \delta p\Vert \le {ϵ}_p$ and $\Vert e\Vert \le {ϵ}_c$. The actual and nominal rotations $Q,\hat{Q}$ have the same known principal angle $\theta \in [0,\pi ]$, while their unit axis directions differ by at most ${\alpha }_{\max }$. Expanding the actual endpoint and subtracting the nominal one gives the exact identityThe three terms isolate initial-point error, pivot error and direction-induced rotation error without assigning one error twice.

An orthogonal matrix preserves length, and $\Vert I-Q{\Vert }_2=2\sin (\theta /2)$. The axis-direction result in rotation-axis direction uncertainty in clearance bounds givesThereforeThe radius in this expression is the nominal $\hat{R}$, because the initial-point perturbation has already been accounted for by $Q\delta p$. Replacing it by an inflated radius is safe but may count some uncertainty conservatively twice. The pivot term uses the actual $Q$, whose rotation angle is shared and known in this model.

For an original tool-motion example, take $\hat{R}=0.50\mathrm{m}$, $\theta =90^{\circ }$, ${\alpha }_{\max }=1^{\circ }$, ${ϵ}_c=1\mathrm{mm}$ and ${ϵ}_p=0.2\mathrm{mm}$. The respective allowances are approximately $0.200$, $1.414$ and $12.341\mathrm{mm}$, totaling $E\approx 13.955\mathrm{mm}$. If every nominal occupied point is at least $20\mathrm{mm}$ from a fixed, exactly known obstacle, the actual clearance is at least $20-E\approx 6.045\mathrm{mm}$, provided point correspondence and the enclosure hold for the entire body at that orientation.

That last inference follows from the triangle inequality for set distance. An additional obstacle-position allowance must be subtracted separately when the obstacle is uncertain. A final-orientation calculation alone does not establish a bound along the intervening trajectory; robot motion planning constrains the whole continuous path.^[ref]

Shared-frame cancellation in rotational clearance uncertainty explains why a joint model can be tighter: translating the robot, pivot and obstacle coordinate descriptions together changes none of their relative clearances. A physical pivot displacement with a fixed obstacle is different. A proposed rotary-stage experiment fits pivot and direction in one frame, withholds fiducial observations at intermediate angles, and checks measured point errors against the bound. Uncertain rotation angle or changing structural deformation requires a further model; neither is included silently in ${\alpha }_{\max }$.