= Pressure distributions with identical resultant force and center of pressure
{title2=$\int_A\delta p\,dA=\int_A x\delta p\,dA=\int_A y\delta p\,dA=0$}
<Pressure distributions with identical resultant force and center of pressure> demonstrate that a <pressure> centroid is not a reconstruction of a <pressure> field. A normal-load measurement and its two first spatial moments constrain only
$$
\int_Ap\,dA,\qquad \int_Axp\,dA,\qquad\int_Ayp\,dA.
$$
A perturbation $\delta p$ preserving all three integrals leaves the resultant and <center of pressure> unchanged. It is physically admissible only when the perturbed field remains nonnegative under the nonadhesive <pressure> model. The polynomial profiles in the parent article give one smooth example; the following construction shows that even a peak-pressure bound is missing.
Take a rectangular region of length $L$ and width $B$, centered about its midpoint, and fix a normal load $N>0$. A uniform field has <pressure> $N/(LB)$ and centroid $(L/2,B/2)$. Instead place the entire load uniformly on a smaller centered square of side $a\le\min(L,B)$, with zero <pressure> outside that square. Its <pressure> is
$$
p_a(x,y)=\begin{cases}N/a^2,&(x,y)\text{ in the centered square},\\0,&\text{elsewhere}.
\end{cases}
$$
Symmetry fixes the same centroid, and integration still gives total <force> $N$. Yet its peak is $N/a^2$, which has no upper bound as $a$ shrinks. Smooth symmetric concentrated profiles can provide the same conclusion if a continuous <pressure> field is required.
For an original numerical example, let $L=0.40\,\mathrm m$, $B=0.20\,\mathrm m$, and $N=800\,\mathrm N$. Uniform <pressure> is $10\,\mathrm{kPa}$. Concentrating the same load on a centered $0.05\,\mathrm m$ square gives $320\,\mathrm{kPa}$ while preserving the measured point $(0.20,0.10)\,\mathrm m$. Both satisfy the stated nonnegative-field moment constraints.
Consequently a <force>/moment sensor cannot infer local peaks from the centroid alone. <Local contact pressure from additional spatial measurements> needs more data or justified material/contact assumptions, such as a constitutive model and known geometry. The source's normal-force centroid definition explains exactly which resultant information was retained.<Underactuated Robotics by Russ Tedrake>[Highly-articulated Legged Robots, Center of pressure definition]{ref} The examples are inverse-problem constructions; they are not claims that arbitrary <pressure> maps can be realized by the same elastic surfaces.
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