= Local contact pressure from additional spatial measurements
<Local contact pressure from additional spatial measurements> requires more information than total <normal force> and two first moments. One explicit reconstruction model partitions the contact into cells of known areas $A_i$ and centroids $(x_i,y_i)$, assuming uniform <pressure> $p_i$ within each cell. Writing $F_i=A_ip_i$ gives
$$
N=\sum_iF_i,\qquad Nx_P=\sum_i x_iF_i,\qquad Ny_P=\sum_i y_iF_i.
$$
These are three linear measurements of the cell loads. They need not determine every $F_i$, as the nonuniqueness argument in <pressure distributions with identical resultant force and center of pressure> shows. The primary centroid definition specifies precisely which <force>/moment information is retained.<Underactuated Robotics by Russ Tedrake>[Highly-articulated Legged Robots, Center of pressure definition]{ref}
For an original four-cell example, use equal $0.01\,\mathrm{m^2}$ cells centered at successive positions $(-a,-a),(a,-a),(a,a),(-a,a)$ with $a=0.05\,\mathrm m$. Loads $(10,20,30,40)\,\mathrm N$ give $N=100\,\mathrm N$ and centroid $(0,0.020)\,\mathrm m$. Adding $s(1,-1,1,-1)$ to these loads changes neither the sum nor either first moment. Nonnegative loads permit $-10\le s\le20\,\mathrm N$. The alternative $(15,15,35,35)\,\mathrm N$ therefore gives the same global measurements.
An independently calibrated measurement of the first cell load resolves this one missing degree of freedom under the four-cell model. If it reads $10\,\mathrm N$, the global moment equations recover the original other three loads. Their cell <pressures> are $(1,2,3,4)\,\mathrm{kPa}$. Measuring all four cells instead provides redundant data for checking total load and the assumed geometry.
The local sensor output must actually represent a known cell-integrated <force>, or have a validated response model; a point sample cannot automatically be treated as an area average. Calibration errors, cell cross-coupling and imperfect coverage become additional measurement inputs. <Spatial pressure-map resolution below a sensor cell> remains limited even when every cell average is known: unmeasured within-cell peaks are not recovered by simply drawing a finer image. Constitutive or smoothness assumptions can add information, but must be identified as assumptions rather than extra measurements.
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