Angular clearance bound for a rotating body

A rigid body can hit an obstacle while its reference point stays fixed. An angular clearance bound for a rotating body controls the motion of every body point, allowing a collision detection calculation to reason about orientations between sampled poses.

Let a compact set $B$ lie within a bounding radius for rigid-body clearance $R$ of a fixed pivot. For a planar rotation matrix $Q_{\theta }$, a point $p\in B$ moves by a chord of a circle:The same correspondence bounds the Hausdorff distance between the two body sets. For a fixed nonempty closed set of obstacles $O$, let $d(\theta )$ be the nonnegative set distance between $Q_{\theta }B$ and $O$. The triangle inequality, applied to each point pair and then to their infima, givesThis is an explicit Lipschitz continuity bound. It concerns a fixed obstacle and the specified rigid geometry, rather than a universal constant for every robot.

To cover every intermediate orientation with excursion at most $\alpha \ge 0$ from a sampled pose, use $2R\sin (\min \{\alpha ,\pi \}/2)$, or the simpler $R\alpha$. The cap matters because the sine is not increasing beyond $\pi /2$. With translation displacement bounded by $\Delta$, the triangle inequality gives the sufficient clearance reduction $\Delta +R\alpha$.

For an original example, a body with $R=0.40\mathrm{m}$ rotating by at most $5{}^{\circ }$ moves any point by at most $2R\sin (2.5{}^{\circ })=34.90\mathrm{mm}$. An exact sampled clearance of $50\mathrm{mm}$ leaves at least $15.10\mathrm{mm}$ throughout that angular excursion when the pivot and obstacles stay fixed.

The whole excursion must be bounded, not just the difference between endpoint orientations modulo $2\pi$. A full revolution has identical endpoint orientations but can sweep through an obstacle. Likewise, a unit half-length rod rotating from $-30{}^{\circ }$ to $+30{}^{\circ }$ about its center clears an obstacle point at $(1,0)$ by $0.5$ at both endpoints but touches it midway. Adaptive clearance sampling therefore uses bounds on the actual interpolated motion. Configuration-space orientation and clearance are central to the continuous-path planning problem.^[ref]

A bounding radius for rigid-body clearance bounds the distance of every point of a rigid body from the reference point used to describe its motion. It converts angular motion into a bound on point displacement, so collision detection can account for the whole body between sampled poses rather than only its reference point.

Let $B$ be a nonempty compact set in body coordinates and let $c$ be the chosen pivot in those coordinates. A valid radius satisfiesThe maximum exists by compactness and continuity of the Euclidean norm. A bound larger than $R_{\ast }$ is conservative; a smaller one does not cover the specified geometry. For a convex polytope with vertices $v_i$, it is sufficient and exact to use $R_{\ast }={\max }_i\Vert v_i-c\Vert$: expressing a point as a convex combination and applying the triangle inequality proves that no interior point is farther away than the farthest vertex.

For an original rectangular-body example, take side lengths $0.6\mathrm{m}$ and $0.8\mathrm{m}$. A pivot at the centre gives $R_{\ast }=\sqrt{0.3^2+0.4^2}=0.5\mathrm{m}$. A pivot at a corner instead gives $R_{\ast }=\sqrt{0.6^2+0.8^2}=1.0\mathrm{m}$. For a $5{}^{\circ }$ rotation about the respective fixed pivot, the angular clearance bound for a rotating body isThe smaller radius cannot be used for the corner-pivot motion unless the changed motion of the reference point is also included. A convenient geometric centre does not automatically remain stationary.

For a moving reference point with speed at most $v_{\max }$ and angular speed at most ${\omega }_{\max }$, a point-speed bound is $v_{\max }+R{\omega }_{\max }$. This supplies a Lipschitz continuity constant for clearance against fixed obstacles. The role of body geometry and orientation in configuration space is developed in the primary planning text.^[ref]

The modeled body must include attached tools and protrusions relevant to the query. Uncertainty in a rigid-body bounding radius can be accommodated with a conservative geometric enlargement. This radius controls motion around the declared reference point; it does not replace the separate required clearance margin for uncertain obstacle locations or tracking errors.

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.

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.

Rotation-axis direction uncertainty in clearance bounds accounts for a robot rotating about a direction different from the modeled direction. Even an exactly located pivot does not determine the final occupied geometry if its axis direction is uncertain. Rotation acts through a rotation matrix; Rodrigues rotation formula computes that matrix from a unit axis and rotation angle.^[ref]

Assume the pivot $c$ and principal rotation angle $0\le \theta \le \pi$ are known and shared by both models. Let the unit axes be $u,\hat{u}$, separated by $\alpha$, and suppose every occupied point obeys $\Vert p-c\Vert \le R$. The endpoint discrepancy isIt measures error against a nominal endpoint, not total motion from the initial position.

One exact operator norm calculation uses the unit quaternions $(\cos (\theta /2),u\sin (\theta /2))$ and its modeled counterpart. Their scalar product isThe relative rotation has half-angle cosine $|d|$. A rotation through angle $\beta$ moves a perpendicular unit vector by $2\sin (\beta /2)$, givingConsequently, if the direction error is at most ${\alpha }_{\max }\le \pi$,The cap $2R$ follows because both endpoints lie on the same radius-$R$ sphere. The bound vanishes at zero rotation and, for small axis error, scales approximately as $2R\sin (\theta /2){\alpha }_{\max }$. Opposite oriented axes represent the same rotation at $\theta =\pi$; the conservative inequality need not capture that special cancellation.

For an original rotary-tool example, $R=0.50\mathrm{m}$, $\theta =90^{\circ }$ and ${\alpha }_{\max }=1^{\circ }$ give an endpoint-error allowance of $0.01234\mathrm{m}$, about $12.34\mathrm{mm}$. A direction tolerance that looks small numerically can consume a substantial clearance margin at a long tool tip.

A combined pivot and axis uncertainty in rotational clearance must add or jointly bound pivot-location error, angle error and direction error without double-counting a shared coordinate-frame error. The fixed-axis formula in pivot-location uncertainty in a rotational clearance bound addresses a different contribution. For a proposed test, observe multiple rigid fiducials over several angles, fit pivot and direction, and validate independent endpoint measurements against the claimed enclosure. Geometric robot motion planning needs the resulting occupied-body bound along the path, rather than only at its final orientation.^[ref]

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 ${\alpha }_{\max }$.