 Bounding radius for rigid-body clearance

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.