 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]