= 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.<Robotic Systems by Kris Hauser>[Chapter 12, Contact modeling and Equilibrium conditions]{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 balance
$$
F_{\rm tip}h=mgd,\qquad F_{\rm tip}=\frac{mgd}{h}.
$$
The contact forces elsewhere on the base then unload. With coefficient of <static friction> $\mu$, the horizontal sliding limit in this same model is $F_{\rm slide}=\mu mg$. 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 $g=9.81\,\mathrm{m\,s^{-2}}$. 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.<Robotic Systems by Kris Hauser>[Chapter 12, Contact modeling; Multi-modal motion planning, Example 1]{ref}
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