Tipping of a rigid body

Tipping of a rigid body begins when a supported configuration can no longer balance its applied forces and torques with admissible contact forces, and rotation about a support edge becomes possible. For gravity alone on a horizontal surface, the center of mass projection must lie in the support polygon. A boundary projection marks incipient unloading of contacts; it is not a general guarantee about subsequent dynamics. The unilateral-contact and force/torque conditions are explicit in the primary robotics treatment.^[ref]

A planar horizontal pull gives a useful distinction between tipping and sliding. Let the gravitational weight be $mg$, the inward distance of the center of mass projection from the candidate tipping edge be $d>0$, and the pull's application height be $h>0$. Taking torque about that edge gives the limiting balanceThe contact forces elsewhere on the base then unload. With coefficient of static friction $\mu$, the horizontal sliding limit in this same model is $F_{\mathrm{slide}}=\mu mg$. Tipping can occur first only if $d/h<\mu$; the geometric support polygon by itself does not decide this competition.

For an original example, choose $m=20\mathrm{kg}$, $d=0.15\mathrm{m}$, $h=0.60\mathrm{m}$, and $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$. The tipping threshold is $49.05\mathrm{N}$. With $\mu =0.40$, sliding would require $78.48\mathrm{N}$, so tipping is the first predicted loss of this planar static configuration. With $\mu =0.15$, sliding starts at $29.43\mathrm{N}$ instead. Equal thresholds require a coupled contact analysis rather than a strict choice between the two labels.

A bench comparison can use a rigid box, a horizontal force gauge at a known height, and a slow force increase while observing the first loss of full-base contact or translation. Include the box's actual combined center of mass, not just its visible geometric centre. The static tipping margin measures the gravity-only geometric reserve; a pull, anchoring, changing contact set, or nonhorizontal ground changes the governing balance problem. The full friction constraints must then be retained.^[ref]