 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 $F_i=A_i{p}_i$ givesThese 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.^[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 $-10\le s\le 20\mathrm{N}$. 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.