Directional tipping margin

A directional tipping margin is the distance an interior projection of the center of mass can move along one specified direction before reaching a boundary of its support polygon. It differs from the nearest-edge static tipping margin, which protects against movement in every horizontal direction. For a bounded polygon $P=\{x:a_i\cdot x\le b_i\}$, interior point $c$, and unit direction $u$, the ray $c+su$ stays inside exactly while all edge inequalities hold. HenceEdges with nonpositive denominator do not limit positive motion along that ray. The underlying gravity-only feasibility criterion is the nonnegative-reaction convex-hull condition in support polygon.^[ref]

For an original triangular footprint, take $x\ge 0$, $y\ge 0$, $x+y\le 1$ in metres and $c=(0.20,0.30)\mathrm{m}$. Moving right gives $s_{\max }=0.50\mathrm{m}$; moving downward gives $0.30\mathrm{m}$; moving along $u=(1,1)/\sqrt{2}$ gives $0.50/\sqrt{2}\approx 0.3536\mathrm{m}$. The all-direction margin is only $0.20\mathrm{m}$, set by the left edge. A displacement allowance restricted to the right can therefore be larger than the isotropic allowance without contradiction.

The same geometry can enter a planar load calculation. A horizontal force $Fu$ applied at height $h$ shifts the effective support-resultant point by $(Fh/mg)u$ under the corresponding static torque balance. The directional threshold is $F_{\mathrm{tip}}=mg{s}_{\max }/h$, subject also to friction and the other contact assumptions. With $m=10\mathrm{kg}$ and $h=0.50\mathrm{m}$, the rightward example gives $98.1\mathrm{N}$, whereas the nearest-edge worst-direction value is $39.24\mathrm{N}$.

A measured pull direction and contact layout are therefore part of the specification. Changes in contact set, applied couples, or slipping require the fuller analysis in tipping of a rigid body; this geometric directional reserve alone is not a dynamical stability certificate.