 Static tipping margin

A static tipping margin is the distance of an interior center of mass projection $c$ from the boundary of a horizontal support polygon $P$. It quantifies the horizontal displacement reserve in the gravity-only rigid-contact model. For a convex polygon written as $P=\{x:a_i\cdot x\le b_i\}$, with nonzero edge normals $a_i$, the margin isThis is the radius of the largest Euclidean ball about $c$ contained in $P$: each normalized edge inequality bounds how far the projection can move toward that edge. The admissible contact-reaction argument is proved in support polygon and in the primary equilibrium model.^[ref]

For an original rectangular footprint, let $P=[-0.30,0.30]\times [-0.20,0.20]$ in metres and $c=(0.10,0.05)\mathrm{m}$. Its four edge distances are $0.20$, $0.40$, $0.15$, and $0.25\mathrm{m}$, so $d=0.15\mathrm{m}$. Any horizontal projection perturbation of norm at most $0.05\mathrm{m}$ leaves a margin of at least $0.10\mathrm{m}$. This follows because each normalized edge gap changes by at most the perturbation norm.

Unsigned distance alone must not classify an exterior point as stable. At $c=(0.40,0.05)\mathrm{m}$, distance to the polygon boundary is still positive, namely $0.10\mathrm{m}$, but gravity-only balance is impossible. The minimum signed edge gap is then negative. For an exterior point, that minimum is a useful feasibility indicator; its magnitude is not generally the Euclidean distance to the polygon, especially beyond a corner.

A directional tipping margin can exceed the nearest-edge margin when the allowed displacement direction is restricted. The scalar $d$ protects against every horizontal direction, so it is the appropriate geometric reserve only for an isotropic displacement bound. It omits center of mass height, static friction, applied moments, and contact compliance. Under a specified horizontal load, tipping of a rigid body instead compares the load's overturning torque with the restoring gravitational torque.