Contact mechanics

Contact mechanics studies how touching bodies deform, transmit forces and separate. Even a ball that appears to touch a flat surface at one point develops a finite contact patch: local elastic deformation spreads the load into a nonuniform pressure distribution. Hertzian contact is a useful ideal model when the surfaces are smooth, the contact patch is small and adhesion and friction can be neglected.

The contact patch is part of the solution, rather than a prescribed loaded area. In nonadhesive contact the nonpenetration constraint prevents overlap, and the normal contact force can push but cannot pull; unilateral contact complementarity joins those requirements. Pressing a spherical probe into a specimen while recording force and indentation tests the contact law and can determine a reduced elastic modulus. Surface roughness, plastic deformation and adhesion require additional modeling, especially when interpreting a measured pull-off force.

The center of pressure summarizes where a distributed normal pressure acts. On a planar contact region $A$ in $z=0$, take a nonnegative pressure field $p(x,y)$ and positive total normal force $N$. Its center of pressure is the pressure-weighted positionIt is a continuous weighted mean, rather than necessarily the point of largest pressure. For discrete normal forces $N_i\ge 0$ at $(x_i,y_i)$, replace the integrals by sums. The resulting convex combination places the center of pressure inside the contacts' convex hull, or support polygon.^[ref]

This definition preserves the torque as well as the resultant normal force. About the origin in the contact plane,A single vertical force $(0,0,N)$ applied at $(x_P,y_P,0)$ gives the same torque. Thus its line of action passes through the center of pressure. Conversely, force-and-torque measurements reconstruct the point throughThe origin and sign convention are part of these equations; moving the moment reference above the contact plane requires the corresponding torque translation.

For an original discrete example, four vertical reactions of $100$, $200$, $300$ and $200\mathrm{N}$ act at the successive corners $(0,0)$, $(0.40,0)$, $(0.40,0.30)$ and $(0,0.30)$ in metres. Their total normal force is $800\mathrm{N}$, with $M_x=150\mathrm{N}\mathrm{m}$ and $M_y=-200\mathrm{N}\mathrm{m}$. Hence the center of pressure is $(0.250,0.1875)\mathrm{m}$. A rigid plate on four load cells provides a simple force platform implementation: calibrate the force channels and positions, then test known vertical loads at known locations. Subtracting the unloaded plate's own force and moment removes its contribution when reconstructing the added load. This is a proposed measurement test, not experimental data.

A continuous example shows the inverse problem hidden by this useful compression. On a rectangle of length $L$ and width $B$, put $u=x/L$ and choose profiles independent of $y$:Both profiles are positive on $0\le u\le 1$. The added polynomial has zero integral and zero first weighted integral:Thus both profiles give $N=p_{0}LB$ and the same center of pressure $(0.60L,B/2)$, while their peak pressures are $1.6p_0$ and $1.9p_0$. These are pressure distributions with identical resultant force and center of pressure. For $L=0.40\mathrm{m}$, $B=0.30\mathrm{m}$ and $N=800\mathrm{N}$, both give $(0.240,0.150)\mathrm{m}$ despite different local loads. A force platform alone therefore does not identify every local pressure.

Two limits matter. If $N=0$, the normal-pressure center of pressure is undefined. At fixed $N$, an absolute moment error $\delta M_y$ changes the reconstructed $x_P$ by $-\delta M_y/N$, so fixed absolute measurement uncertainty is amplified as a contact unloads. Also, tangential contact forces can leave a residual contact torque at the center of pressure: with full resultant $\mathbf{F}$,Selecting the normal-pressure centroid does not force this yaw torque to vanish.

Under the gravity-only, resting assumptions of support polygon, the center of pressure coincides with the horizontal center of mass projection. During motion it need not. For a planar model driven only by gravity and ground contact forces, at fixed center of mass height $h$ and with constant angular momentum about that centre, force and torque balance give $x_P=x_{\mathrm{CM}}-(h/g){\ddot{x}}_{\mathrm{CM}}$. The acceleration term explains why the center of pressure is a contact-force measurement rather than a general substitute for center of mass position.^[ref]

In terrestrial robot locomotion, the center of pressure is the ground point summarizing the normal load shared by the active contacts. With contact positions $(x_i,y_i)$ and nonnegative normal reactions $N_i$, its coordinates are the weighted averages $x_P={\sum }_i{N}_i{x}_i/{\sum }_i{N}_i$ and $y_P={\sum }_i{N}_i{y}_i/{\sum }_i{N}_i$, provided the total normal force is positive. It lies in the current support polygon, but is a contact-force quantity rather than the robot's center of mass position.^[ref]

Force-sensor coordinates matter when reconstructing this point. Suppose a sensor origin lies height $h$ above a horizontal contact plane, with parallel coordinate axes. If it reports force $(F_x,F_y,F_z)$ and moment $(M_x,M_y,M_z)$, the plane point is located at $(x_P,y_P,-h)$ relative to that origin. Moment translation givesThe ground-plane transformation of a measured contact wrench must therefore precede interpreting moment/force ratios as a contact location.

For an original sensor example, take $h=0.05\mathrm{m}$, force $(30,-20,600)\mathrm{N}$, and measured roll/pitch moments $(17,-49.5)\mathrm{N}\mathrm{m}$. The reconstructed point is $(0.080,0.030)\mathrm{m}$. Ignoring the origin height would instead give $(0.0825,0.02833)\mathrm{m}$. Those millimetre differences are coordinate effects, not a change in the physical pressure distribution.

During motion, the contact point need not equal the horizontal projection of the center of mass. The source's constant-height, negligible-angular-momentum-rate model relates their difference to horizontal acceleration; those assumptions must be established before using that simplified relation.^[ref] A robot-foot experiment can compare known applied loads with the sensor's translated force/moment measurements. When $F_z$ approaches zero, reconstruction becomes sensitive to moment error; at zero normal load the normal-pressure point is undefined. A residual yaw couple may remain, as explained in residual contact torque at the center of pressure.

A ground-plane transformation of a measured contact wrench combines a change of coordinate axes with a change of the point about which moments are reported. A six-component wrench is incomplete without both pieces of metadata. Drake's primary SpatialForce reference explicitly tracks the application point and expressed-in frame separately from the stored force and torque values.^[ref]

Let $R$ rotate sensor-frame components into ground-frame components, and let $p$ be the ground-frame vector from ground origin $O$ to sensor origin $S$. For measured force $f_S$ and moment $M_S$ about $S$, the equivalent result about $O$ isThe plus sign uses the stated vector from the new origin to the old one. A shift expressed with the opposite vector uses the opposite sign. The offset and force must be expressed in the same axes before taking the cross product.

For a planar normal-pressure reconstruction with $f_{G,z}>0$, computeThe special parallel-axis height correction in center of pressure is recovered with $p=(0,0,h)$ and $R=I$.

For an original rotated-sensor example, use a 90-degree rotation about $z$, sensor readings $f_S=(10,0,100)\mathrm{N}$ and $M_S=(1,2,3)\mathrm{N}\mathrm{m}$, and $p=(0.05,-0.02,0.10)\mathrm{m}$. Rotation gives $f_G=(0,10,100)\mathrm{N}$ and $R{M}_S=(-2,1,3)\mathrm{N}\mathrm{m}$. The translated moment is $M_O=(-5,-4,3.5)\mathrm{N}\mathrm{m}$, yielding $(x_P,y_P)=(0.040,-0.050)\mathrm{m}$. The yaw moment remaining at that point is $3.1\mathrm{N}\mathrm{m}$, so translating to the centroid does not erase the entire wrench.

A bench check applies known loads at known ground locations and repeats after changing sensor orientation. Geometry, tare loads, units and axis signs must all be included. A correct rotation with the wrong moment origin can still give an incorrect pressure location; a small normal load also amplifies reconstruction uncertainty.

The center of pressure locates the resultant of a distributed fluid pressure load relative to its moment. On a planar surface carrying parallel pressure forces, the first-moment ratio in center of pressure applies. On an airfoil, pressure variation and its resultant can change with the flow state, so the point is not a permanently fixed geometric centre. NASA's primary account explicitly distinguishes a moving center of pressure from retaining a force and pitching moment about an aerodynamic center.^[ref]

For an original hydrostatic plate example, take a vertical rectangle of width $w$, extending from the free surface to depth $H$ in a constant-density fluid. If the same atmospheric pressure acts on its other face, the net gauge pressure at depth $y$ is $p(y)=\rho gy$. This hydrostatic relation follows from vertical force balance.^[ref] The resultant and its depth areFor $\rho =1000\mathrm{kg}{\mathrm{m}}^{-3}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, $w=1\mathrm{m}$ and $H=2\mathrm{m}$, the net force is $19.62\mathrm{kN}$ at depth $1.333\mathrm{m}$. The area centroid is only one metre below the surface; the increasing pressure shifts the load's centroid downward.

A tank demonstration can infer the resultant line from the force and hinge moment on a small plate while changing the water depth. The plate's own gravitational moment and the pressure on both faces must be included. This tests a specified pressure-load model, not a general claim that the point always lies below every geometric centre.

For aerodynamic net loads, differences of pressures on opposite faces can have both signs. The convex-hull property proved for nonnegative contact pressure then need not apply, and a small resultant with a finite moment makes its location ill-conditioned. A curved surface or a general wrench may also require a residual couple rather than a single force that removes every component of torque. The resultant convention and moment reference are therefore part of the definition.

A residual contact torque at the center of pressure is the moment left after translating a planar contact's resultant force to its normal-pressure centroid. Let the resultant wrench at an origin in $z=0$ be force $F=(F_x,F_y,N)$ and moment $M=(M_x,M_y,M_z)$, with $N>0$. Put $r_P=(x_P,y_P,0)$, where $x_P=-M_y/N$ and $y_P=M_x/N$. Translation givesThe two horizontal moment components vanish, but the yaw component need not. The zero-moment-point construction for flat robot contact removes moments perpendicular to the surface normal; it does not assert that every component is zero.^[ref]

A useful invariant makes the limitation exact. Since $F\cdot (r_P\times F)=0$,If this scalar is nonzero, no change of the moment origin can eliminate the entire moment vector. Replacing the whole contact by only one force at the center of pressure would discard that information.

For an original two-contact example, put equal normal loads of $100\mathrm{N}$ at $(+0.10,0)$ and $(-0.10,0)$ metres. Add tangential forces $(0,+15,0)\mathrm{N}$ at the positive position and $(0,-15,0)\mathrm{N}$ at the negative position. The total force is $(0,0,200)\mathrm{N}$, the pressure centroid is the origin, and the opposing tangential forces give a pure yaw couple of $3\mathrm{N}\mathrm{m}$. A friction coefficient of at least $0.15$ permits the stipulated tangential magnitudes at these loads.

A normal-load-only reconstruction would identify the correct centroid while missing the couple. A sensor measuring all force and moment components can distinguish the two situations. The origin, surface orientation and axis signs must be retained when comparing measurements. Near zero normal load, the centroid reconstruction itself becomes singular, independently of whether a tangential moment remains.

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 onlyA 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 isSymmetry 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.^[ref] The examples are inverse-problem constructions; they are not claims that arbitrary pressure maps can be realized by the same elastic surfaces.

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.

Spatial pressure-map resolution below a sensor cell concerns which variations of a contact pressure field can actually be distinguished by spatial measurements. A smaller display pixel does not imply a smaller independently measured contact region. The response of a pressure sensor, its sampling positions and any imposed reconstruction assumptions must be specified.

For an original ideal array model, cell $A_j$ of known area reports an area-average pressure,It determines that cell’s load $F_j=|A_j|y_j$, but not every value inside it. For a more general linear response, $y_j=\int K_j(x,y)p(x,y)dA$ with known sensitivity $K_j$. The null space of a spatial pressure measurement consists of perturbations $\delta p$ whose weighted integrals vanish for all measured channels. Nonnegative fields that differ by such a perturbation remain indistinguishable in that model.

Consider square cells of side $a$ aligned with $x=ja$, and compare fields independent of $y$:Every cell contains one full cosine period, so both fields give every channel the identical value $p_0$. Yet $p_2$ varies between $0.5p_0$ and $1.5p_0$. With $a=10\mathrm{mm}$ and $p_0=100\mathrm{kPa}$, a perfectly uniform field and a field spanning $50$–$150\mathrm{kPa}$ produce the same ideal map. On an aligned finite rectangular array, the added cosine also has zero first moment within each cell, so even the global resultant and center of pressure agree. This is an inverse-problem construction, not a claim that both profiles arise from the same elastic contact geometry.

Cell averaging has a calculable spatial response. For a one-dimensional centered interval of width $a$, averaging a complex wave $e^{2\pi i\nu x}$ multiplies its value at the interval center byThe first zero at $\nu =1/a$ explains the erased cosine above. Sampling introduces further ambiguity; even translating this ideal array cannot recover a component erased by its aperture at every position.

A regularity assumption can supply a conditional bound. If $p$ has a validated spatial Lipschitz constant $L_p$ over a square cell, its center value differs from its area average by at mostEvery point in the cell is at most half the diagonal from its center, so integrating the Lipschitz inequality proves this bound. For $a=10\mathrm{mm}$ and $L_p=2kPa/mm$, it gives about $14.14\mathrm{kPa}$. The array readings alone do not establish that slope bound.

Interpolation chooses a finer representation from the measured values. SciPy explicitly describes its array zoom operation as spline interpolation; it does not describe collecting new pressure data.^[ref] A proposed apparatus test scans a known small indenter across a sensor array at controlled total load, comparing independent load measurements with channel responses. The indenter’s actual contact area, load spreading, sensor cross-coupling and repeatability must be measured before inferring a spatial response. Additional smaller-aperture measurements or justified mechanical constraints can improve identifiability; a smoother image alone supplies neither.

The null space of a spatial pressure measurement consists of changes to a contact pressure field that leave every measured channel unchanged. For a linear response model on a contact region $A$, writeThen $\ker \mathcal{M}=\{\delta p:\mathcal{M}\delta p=0\}$. Two fields have identical outputs precisely when their difference lies in this kernel of a linear map. The sensitivity kernels $K_j$, their support and the admissible class of pressure fields are part of the model.

For an ideal area-average sensor, $K_j=1/|A_j|$ inside cell $A_j$ and zero elsewhere. A positive and negative perturbation within one cell can cancel in its integral while remaining invisible to the other cells. This is the mechanism behind spatial pressure-map resolution below a sensor cell. It is a nonuniqueness of the observations, even if their numerical values are exact.

An original finite model divides each of two equal-area sensor cells into two equal halves with constant pressures $p_1,\dots ,p_4$. The observed averages areThis matrix has rank two and a two-dimensional null space spanned by $(1,-1,0,0{)}^{\mathsf{T}}$ and $(0,0,1,-1{)}^{\mathsf{T}}$. Thus the fieldsboth produce $y=(100,100{)}^{\mathsf{T}}\mathrm{kPa}$. Their peak pressures differ by $40\mathrm{kPa}$. Adding an independent measurement of $p_1$ determines the first pair, but still leaves the second pair’s difference undetermined. An extra channel helps only if it adds an independent constraint on the missing directions.

The linear null space is not itself a set of physical pressure fields: its perturbations can have either sign. For a nonadhesive contact, require $p+\delta p\ge 0$ as well. With the uniform $100\mathrm{kPa}$ field above, perturbations $t(1,-1,0,0)$ are admissible for $|t|\le 100\mathrm{kPa}$. Positivity restricts the ambiguity without removing it. In contrast, a zero area-average reading together with a nonnegative pressure assumption forces pressure to vanish almost everywhere in that cell under the ideal model.

For a chosen finite basis of pressure fields, a singular value decomposition identifies the matrix null space. SciPy’s pinned implementation constructs its basis from right singular vectors and uses a relative singular-value cutoff to determine effective numerical rank.^[ref] A numerical cutoff and an exact zero singular value are different notions. Near-null directions of a pressure sensor array have small nonzero responses and can amplify measurement error: a dimensionless singular value of $0.01$ turns a $1\mathrm{kPa}$ channel-error component into a $100\mathrm{kPa}$ coefficient error when inverted without regularization.

A proposed apparatus check applies controlled local loads to estimate the sensitivity kernels, then compares fields with similar predicted channel outputs using independent spatial measurements. Contact area, load spreading and cross-coupling must enter that comparison. A reconstruction algorithm may select one smooth or nonnegative solution, but the selected structure is additional modeling information; it is not automatically an independently observed subcell pressure map.

Near-null directions of a pressure sensor array are changes of a modeled pressure field that produce small, rather than exactly zero, changes in the sensor outputs. They distinguish an ill-conditioned reconstruction from the exact ambiguity in the null space of a spatial pressure measurement. A formally invertible response can still contain very little usable information about some pressure patterns.

Choose a finite basis for the pressure field and a linear response $y=Hp+\eta$, where $p$ contains pressure coefficients, $y$ the readings, and $\eta$ measurement error. With common pressure units for the coefficients and readings, letbe a singular value decomposition. A change $\Delta p=a{v}_j$ changes the noiseless readings by $a{s}_j{u}_j$. Inverting that component divides the measurement error by $s_j$. If the errors have covariance ${\sigma }^{2}I$, the standard deviation of the reconstructed coefficient along $v_j$ is $\sigma /s_j$. An exact zero gives no inverse; a small nonzero value gives noise amplification.

An original two-region example has uniform pressures $p_1,p_2$ and sensor matrixBoth sensors mostly measure the same average. Write $\bar{p}=(p_1+p_2)/2$ and $d=(p_1-p_2)/2$. Then $y_1=\bar{p}$ and $y_2=\bar{p}+0.01d$. Uniform pressures $(100,100)\mathrm{kPa}$ give readings $(100,100)\mathrm{kPa}$; the markedly different pair $(140,60)\mathrm{kPa}$ gives only $(100,100.4)\mathrm{kPa}$. The matrix is invertible, but these outputs differ by just $0.4\mathrm{kPa}$ in one channel.

If each channel has independent $1\mathrm{kPa}$ standard uncertainty, the direct contrast estimate is $\hat{d}=(y_2-y_1)/0.01$, with standard deviation about $141.4\mathrm{kPa}$. The matrix’s smallest singular value is approximately $0.005$, consistent with roughly $200\mathrm{kPa}$ uncertainty in the unit-length contrast direction. Nonnegative pressure constraints can restrict the admissible answers, but do not turn these noisy readings into precise contrast information.

Numerical rank depends on a chosen tolerance. SciPy’s pinned null-space implementation compares singular values with a relative cutoff and returns right singular vectors beyond the retained rank.^[ref] A floating-point cutoff detects numerical scale; a physical decision about usable information must also consider sensor noise, response uncertainty and the chosen pressure basis. Rescaling a coefficient can change a raw singular value without changing the apparatus.

A noise-whitened pressure sensing matrix accounts for unequal or correlated measurement errors before comparing response strength. For positive-definite error covariance $R$, analyze $R^{-1/2}H$ together with the same coefficient units and normalization. A duplicate noisy channel does not provide the same information as an independent measurement of the weak contrast.

Tikhonov regularization can limit amplification by using factors $s_j/(s_j^2+{\alpha }^2)$ instead of $1/s_j$. Relative to the true noiseless coefficient, this introduces attenuation $s_j^2/(s_j^2+{\alpha }^2)$. For $s_j=0.005$ and $\alpha =0.05$, that factor is about $0.0099$: the weak component is nearly suppressed. A smoother map can therefore have less noise while omitting real contrast. A proposed validation applies independently known pressure patterns that excite both average and contrast directions, repeats the measurements, and reports the error covariance and response-model uncertainty alongside reconstruction results.

A noise-whitened pressure sensing matrix measures how a contact pressure pattern changes sensor outputs relative to their joint noise, rather than their raw numerical size. For a linear measurement modelassume the error covariance $R$ is positive definite. Choose an invertible matrix $W$ satisfying $WR{W}^{\mathsf{T}}=I$. ThenThe transformed errors have identity covariance. This is the whitening transformation documented in the primary SciPy covariance interface; its example obtains approximately identity sample covariance from simulated observations.^[ref]

A Cholesky decomposition $R=L{L}^{\mathsf{T}}$ supplies $W=L^{-1}$. Compute $Wy$ and $WH$ by solving triangular systems rather than forming an inverse unnecessarily. The least-squares residual becomesThus an output change is significant according to its noise covariance, not merely the Euclidean length of the raw readings. The original pressure basis and its units must still be stated; whitening rows does not remove arbitrary column rescalings.

For an original two-channel example, use numerical pressure values in kilopascals andThe separate standard deviations are $2$ and $1\mathrm{kPa}$ with correlation $0.9$. A residual $(2,1)\mathrm{kPa}$ whitens to $(1,0.1/\sqrt{0.19})$, giving squared length about $1.0526$. The residual $(2,-1)\mathrm{kPa}$ instead whitens to $(1,-1.9/\sqrt{0.19})$, giving squared length $20$. Treating the channels as independent would assign both residuals squared standardized length two. The positively correlated noise makes opposite-sign discrepancies much less plausible than a shared shift.

A simple repeated-measurement model illustrates the information consequence. If two sensors measure the same scalar pressure with equal standard deviation $\sigma$ and correlation $\rho$, their mean has variance ${\sigma }^2(1+\rho )/2$. At $\rho =0.9$ the standard deviation is about $0.9747\sigma$, rather than the independent-channel value $0.7071\sigma$. Adding an almost duplicate channel therefore supplies little improvement for that common pressure.

The pressure-map information matrix is $A^{\mathsf{T}}A=H^{\mathsf{T}}{R}^{-1}H$. In a full-column-rank linear Gaussian model with known $R$, the unconstrained weighted least-squares estimate has covariance $(A^{\mathsf{T}}A{)}^{-1}$. Small singular values of $A$ identify near-null directions of a pressure sensor array after accounting for noise. Alternative whiteners differ by an orthogonal row transformation and give the same information matrix; an individual whitened row need not represent one physical sensing cell.

A proposed array characterization applies repeatable known pressure patterns, estimates the channel covariance from repeated residuals, and checks the covariance on independent repeats after whitening. Drift and response-model uncertainty must be included if they belong to the intended error model. If $R$ is singular, a simple inverse whitening is unavailable: exact zero-noise constraints and redundant stochastic directions need separate treatment. A visually smooth pressure reconstruction does not establish that its weak directions were measured precisely.

A pressure-map information matrix quantifies which coefficients of a modeled contact pressure field are constrained by a sensor array, relative to its measurement noise. For $y=Hp+\eta$ with fixed positive-definite error covariance $R$, defineHere $A$ is the noise-whitened pressure sensing matrix. Whitening removes the output-noise covariance from the residual metric; the primary SciPy covariance interface documents this transformation and demonstrates near-identity covariance on simulated observations.^[ref]

The matrix is positive semidefinite because $v^{\mathsf{T}}\mathcal{I}v=\Vert Av{\Vert }^2\ge 0$. Its null space is exactly the null space of $H$: an unobserved pressure change remains unobserved after invertible row whitening. With independent Gaussian errors of known covariance, the log likelihood is a constant minus $(y-Hp{)}^{\mathsf{T}}{R}^{-1}(y-Hp)/2$. Differentiating twice with respect to $p$ gives negative Hessian $\mathcal{I}$, identifying it as the Fisher information for this stated model.

If $H$ has full column rank, weighted least squares givesThis covariance follows by substituting $y=Hp+\eta$ and multiplying the error matrices. It concerns the unconstrained linear estimator with known response and covariance, not every reconstruction algorithm. Nonnegativity constraints, regularization and uncertain sensitivity kernels require additional treatment.

An original two-cell example begins with one average-pressure measurement, $H=(1/2,1/2)$, with $1\mathrm{kPa}$ standard error. In numerical kilopascal coordinates,Its eigenvalues are $1/2$ and zero: the mean is observed, but the pressure difference is not. A second independent copy doubles the information matrix while leaving the difference unobserved. More repeats of the same mixture improve precision only in the already measured direction.

Instead add a direct measurement of the first cell with the same independent error. NowThe inferred first-cell pressure has standard deviation $1\mathrm{kPa}$ and the second $\sqrt{5}\approx 2.236\mathrm{kPa}$. Their errors are negatively correlated because the second pressure is reconstructed by subtracting the first from twice the measured mean.

A rank-one update of pressure-map information follows when an independent new channel has sensitivity row $a^{\mathsf{T}}$ and error variance ${\sigma }_a^2$: its added information is $a{a}^{\mathsf{T}}/{\sigma }_a^2$. A channel aligned with a weak pressure direction can therefore be more useful than another precise copy of a strong direction. Correlated new errors cannot be added by that independent-channel formula.

A proposed characterization uses repeatable known loading patterns to estimate $H$, independent repeats to estimate $R$, and withheld patterns to test predicted reconstruction covariance. State the pressure basis and units: rescaling a coefficient changes the numerical information entries. A sharp reconstructed image or a large matrix determinant alone does not establish that the underlying response model or noise assumptions are correct.

A rank-one update of pressure-map information describes the extra constraint supplied by one independent scalar pressure measurement. Suppose the existing pressure-map information matrix is $\mathcal{I}=H^{\mathsf{T}}{R}^{-1}H$, and a new sensor reports $z=a^{\mathsf{T}}p+{\eta }_a$ with known variance ${\sigma }_a^2>0$, independent of the existing errors. Adding its squared residual to the weighted least squares objective givesThe added matrix has rank one when $a\ne 0$. It is positive semidefinite because its quadratic form is $(a^{\mathsf{T}}v{)}^2/{\sigma }_a^2$. Thus the new observation cannot reduce information in any direction under this fixed linear map and noise model. The noise normalization is the same whitening transformation used to express a correlated residual metric as an ordinary squared norm.^[ref]

The update can remove a missing direction only if the new channel actually sees it. More precisely,This follows because a sum of two nonnegative quadratic forms is zero exactly when both are zero. Repeating an existing mixture improves its precision but cannot identify a pressure contrast that all mixtures annihilate.

When $\mathcal{I}$ is positive definite, write $C={\mathcal{I}}^{-1}$. The Sherman–Morrison formula yields the updated estimator covariance matrixMultiplication by ${\mathcal{I}}_{+}$ verifies the identity directly. For a particular quantity $v^{\mathsf{T}}p$, its variance falls by $(v^{\mathsf{T}}Ca{)}^2/({\sigma }_a^2+a^{\mathsf{T}}Ca)$. A small noise level is valuable only together with a useful sensitivity direction.

For an original calculation in kilopascal coordinates, let $C=\mathrm{diag}(4,1)$, $a=(1,1{)}^{\mathsf{T}}$ and ${\sigma }_a^2=1$. ThenThe variance of the sum drops from $5$ to $5/6\mathrm{kP}{\mathrm{a}}^2$. The negative off-diagonal entry means the two inferred coefficients now have negatively correlated errors; it does not mean either individual variance is negative.

A correlated-channel update of pressure-map information needs the new channel's residual after predicting it from old channel noise. Simply adding $a{a}^{\mathsf{T}}/{\sigma }_a^2$ double-counts information when a common amplifier or reference contributes correlated errors. A proposed array experiment uses repeated known loads to estimate sensitivity and noise separately, adds one channel, and compares withheld-load errors with $C_{+}$. Agreement tests the full assumed model, including independence and unchanged sensor response, rather than the matrix identity alone.

A correlated-channel update of pressure-map information measures what a new pressure channel adds after accounting for noise that it shares with the old channels. The rank-one update of pressure-map information for independent noise is a special case. An extra channel may be a precise-looking duplicate of an existing noisy signal and contribute almost no new information.

Write the old and new observations as $y=Hp+e$ and $z=a^{\mathsf{T}}p+\eta$. Suppose their known joint error covariance matrix is positive definite,Subtract the part predicted from old noise by definingDirect multiplication gives $\mathrm{Cov}(e,\nu )=0$ and ${\tau }^2=\mathrm{Var}\nu =s-r^{\mathsf{T}}{R}^{-1}r>0$, a Schur complement. Thus the invertible residual transformation separates the weighted residual objective into its old term and one scalar innovation term:For jointly normal random variables, uncorrelatedness here also gives independence, and the matrix is the parameter Fisher information. For more general zero-mean errors with this fixed covariance, the displayed algebra still describes the generalized least squares information matrix. This is a partial whitening transformation; the primary covariance interface documents whitening and its identity-covariance check.^[ref]

An original scalar example observes the same pressure twice, with unit error variances and error correlation $\rho \in (-1,1)$. Then $H=a=R=s=1$, $r=\rho$, soAt $\rho =0.8$, the true estimator variance is $0.9$, whereas an independent-noise calculation would claim $0.5$. At $\rho =-0.8$, the variance is $0.1$: opposite noise contributions can cancel. Correlation can reduce or increase the benefit relative to the independent case; it does not make the correctly computed added information negative.

The effective sensitivity is $b$, not merely the original $a$. If $b=0$ and ${\tau }^2>0$, the new residual carries no additional pressure information. A zero-innovation-variance pressure channel requires separate treatment: division by ${\tau }^2=0$ is invalid, and the limiting relation may be a redundant duplicate or an exact constraint, depending on $b$.

A proposed array characterization measures simultaneous repeated readings under fixed known loads, estimates the complete cross-channel covariance, and tests reconstruction on withheld repetitions. A common reference offset can survive indefinitely despite more channels. The derivation assumes known, unchanged sensitivities and covariance; estimating them from few samples adds uncertainty not included by this update.

Unilateral contact permits bodies to separate while preventing them from passing through each other. In an ideal rigid, nonadhesive model, the supporting normal contact force can push but cannot pull. It is therefore different from a permanently attached support: an upward pull may lift a body away instead of producing a downward support reaction.

Let $d$ be the signed normal gap, positive when the bodies are separated, and $N$ the outward compressive normal contact force. The nonpenetration constraint and absence of adhesion give the unilateral contact complementarity conditionsAn open gap requires $N=0$, while $N>0$ requires a closed gap. The pair $d=N=0$ is also allowed: touching does not by itself imply a loaded contact.^[ref] Tangential friction requires an additional relation such as Coulomb friction; it is not specified by the normal-gap conditions.

The equation of motion is still needed to determine the reaction. Consider a block of mass $m$ on a fixed horizontal floor, initially at rest at $d=0$, with an upward applied force $u$ through its center of mass. With upward positive, Newton's second law isIf the contact remains closed, $\ddot{d}=0$ would require $N=mg-u$. That is feasible for $u\le mg$. For $u>mg$, a negative normal reaction would violate unilateral contact, so the contact opens with $N=0$ and $\ddot{d}=(u-mg)/m>0$. At a closed contact with zero normal velocity, the appropriate acceleration-level contact complementarity iswhich yieldsThese acceleration conditions are being applied at a closed, initially stationary contact; they are not a rule forbidding downward acceleration during separated free flight.

For an original numerical example, let $m=2\mathrm{kg}$ and $g=9.81m/{\mathrm{s}}^2$. Pulling upward with $u=5\mathrm{N}$ leaves a normal reaction of $14.62\mathrm{N}$. At $u=25\mathrm{N}$, the reaction is zero and the initial upward acceleration is $2.69m/{\mathrm{s}}^2$. At $u=19.62\mathrm{N}$, both the reaction and the initial acceleration vanish, while touching remains possible.

A simple apparatus test places the block on a calibrated load cell and applies the upward pull through a spring balance, keeping the pull vertical through the center of mass to avoid an extra overturning torque. The support reading should decrease with the pull while contact persists, then vanish at lift-off. Recording the gap as well distinguishes an unloaded touch from a genuinely open contact. Real support elastic deformation and adhesion must be accounted for before treating that apparatus as the ideal rigid, nonadhesive model.

A penalty approximation of unilateral contact replaces perfect rigidity by a chosen stiff response, for exampleHere a small negative gap represents an allowed modeled overlap relative to undeformed reference surfaces. It is not an exact solution of $d\ge 0$. For the $14.62\mathrm{N}$ supported load above and chosen penalty stiffness $k=10^{5}N/m$, equilibrium requires $d=-0.1462\mathrm{mm}$. Increasing $k$ reduces that overlap at the same load. The penalty method and the ideal constraint are different approximations of the contact response.

Finally, impact requires a rule for the velocity change. The position-level inequalities alone do not decide whether an approaching body sticks or rebounds. A coefficient of restitution or another specified impact law determines that behavior, and the impulsive contact force is summarized by its impulse. Tedrake's particular time-stepping formulation identifies its block collisions as perfectly inelastic; that is a feature of the stated model, not a consequence of the three gap/reaction inequalities alone.^[ref]

A penalty approximation of unilateral contact allows a small modeled overlap and uses a stiff restoring force instead of enforcing an exact rigid nonpenetration constraint. With signed gap $d$, positive in separation, a simple nonadhesive law isAn open or just-touching contact has zero force in this spring law; a sustained compressive load requires negative $d$. Thus it approximates unilateral contact rather than satisfying its rigid gap condition exactly. Tedrake's primary compliant-contact treatment uses stiff springs and damping to resist penetration and explains the resulting numerical stiffness.^[ref]

For an original gravity-supported example, a $2\mathrm{kg}$ mass with no other vertical load has $N=mg=19.62\mathrm{N}$. With $k=2\times 10^5\mathrm{N}{\mathrm{m}}^{-1}$, its static overlap is $mg/k=98.1\mu \mathrm{m}$. Increasing $k$ by a factor of 100 reduces that overlap to $0.981\mu \mathrm{m}$, but also makes the contact dynamics faster.

While the contact stays compressed, linearization about equilibrium givesThe normal contact penalty time scale is therefore proportional to $\sqrt{m/k}$. For the first stiffness above, ${\omega }_c\approx 316.2\mathrm{rad}{\mathrm{s}}^{-1}$; after increasing stiffness it is ten times larger. At fixed numerical step size, a smaller overlap error need not mean a more accurate trajectory.

For example, the Forward Euler method applied to this undamped oscillator has amplification factors $1\pm i{\omega }_c\Delta t$, whose magnitudes exceed one for every positive step. Merely shrinking the step does not make that method asymptotically stable for indefinite undamped oscillation, although it can reduce error on a finite interval. A suitable integration method, damping model and convergence check must be chosen. The source notes specifically identify stiff differential equations as a challenge for compliant contact simulation.^[ref]

A penalty stiffness is a chosen model parameter, not automatically the actual material stiffness. Impact dissipation and tangential friction still need their own constitutive choices; a purely elastic normal spring does not imply a perfectly inelastic collision.

The normal contact penalty time scale describes the fast dynamics introduced by a stiff normal-contact approximation. For a compressed linear penalty spring with stiffness $k$ acting on a point mass $m$, small oscillations about equilibrium satisfy $m\ddot{x}+kx=0$. Their angular frequency, reference time and period areThe reference time $t_c$ is not the full oscillation period. Rescaling time by $s=t/t_c$ gives $d^{2}x/d{s}^2+x=0$, so a numerical step is controlled by its dimensionless size $h/t_c$, not just its duration in seconds. The primary compliant-contact treatment highlights the numerical stiffness caused by such springs.^[ref]

A modeling and numerical budget interact. Limiting equilibrium overlap under a specified maximum supported load $F_{\max }$ to $d_{\max }$ requires $k\ge F_{\max }/d_{\max }$ in this linear model. For an original choice $F_{\max }=10\mathrm{N}$ and $d_{\max }=10\mu \mathrm{m}$, choose $k=10^6\mathrm{N}{\mathrm{m}}^{-1}$. At $m=0.5\mathrm{kg}$,A chosen resolution target of twenty steps per contact oscillation would set $h\le 0.2221\mathrm{ms}$. This is a resolution target, not a universal accuracy or stability theorem for every integrator.

Increasing the stiffness by a factor of 100 reduces the same-load overlap by 100 but reduces these times by only ten. Keeping the old step then gives roughly two samples per new oscillation instead of twenty. The parent penalty approximation of unilateral contact also shows why the Forward Euler method's undamped-oscillator instability cannot be cured merely by calling the step small.

For a multibody contact, the effective mass in a normal-contact penalty mode replaces an arbitrary total body mass. Additional damping and nonlinear force laws can introduce other time scales. A convergence study should therefore check force histories and contact events as well as maximum overlap; smaller geometric penetration alone does not establish a more accurate simulation.

The effective mass in a normal-contact penalty mode is the inertia seen by one specified relative normal displacement, rather than automatically the total mass of either body. A contact away from a center of mass can excite rotation as well as translation, changing that inertia and the normal contact penalty time scale.

Use independent generalized velocities $v$, a positive-definite mass matrix $M$, and a normal gap $g$ whose local velocity is $\dot{g}=Jv$. A scalar normal force $\lambda$ produces generalized force $J^{\mathsf{T}}\lambda$. The primary multibody formulation gives this contact-Jacobian velocity/force pairing explicitly.^[ref] At a frozen configuration, the part of gap acceleration caused by that force isIf the scalar $a=J{M}^{-1}{J}^{\mathsf{T}}$ is positive, defineThe definition uses a physical normal gap measured in metres and a matching normal force measured in newtons. Arbitrarily rescaling the gap coordinate changes the associated inertia. Constraints must first be accounted for by independent coordinates or the appropriate constrained response.

A short normal impulse $P$ gives $\Delta v=M^{-1}{J}^{\mathsf{T}}P$, hence $\Delta \dot{g}=P/m_{\mathrm{eff}}$. This supplies a direct interpretation: less effective mass means a larger relative normal-velocity change for the same impulse. At a fixed local configuration, a compressed linear penalty spring of stiffness $k$ gives the small-motion estimateIt omits changing contact geometry, other driving forces and damping; the source separately explains why stiff compliant contact challenges numerical integration.^[ref]

For two freely translating particles, the relative gap sees $a=1/m_1+1/m_2$, so $m_{\mathrm{eff}}=m_1{m}_2/(m_1+m_2)$. This is the reduced mass. Equal $2\mathrm{kg}$ particles give $m_{\mathrm{eff}}=1\mathrm{kg}$; replacing one by a fixed wall gives $2\mathrm{kg}$ instead.

For a planar rigid body above a fixed plane, let its mass be $m$, rotational moment of inertia about its center be $I$, and the contact normal be vertical. A contact a horizontal distance $r$ from the center hasFor $m=2\mathrm{kg}$, $I=0.020\mathrm{kg}{\mathrm{m}}^2$ and $r=0.10\mathrm{m}$, $m_{\mathrm{eff}}=1\mathrm{kg}$. At the center it is $2\mathrm{kg}$. With $k=10^{5}N/m$, the off-center angular frequency is $316.2rad/s$, compared with $223.6rad/s$ at the center. Using the full mass in both cases misses this rotational response.

Several simultaneous contacts require more than separate scalar replacements. Their matrix $A=J{M}^{-1}{J}^{\mathsf{T}}$ generally has off-diagonal terms: a force at one contact changes another contact’s acceleration. With frozen geometry and positive diagonal stiffness matrix $K$, compressed gap perturbations satisfy $\ddot{\xi }+AK\xi =0$; its nonzero modal squared frequencies are those of the symmetric matrix $K^{1/2}A{K}^{1/2}$. Coupled normal-contact penalty modes can therefore be faster than estimates treating each contact independently. A proposed verification applies a small known impulse or displacement perturbation at a stated configuration and checks relative contact motion, followed by a timestep convergence study.

Coupled normal-contact penalty modes are small oscillations generated by several simultaneously compressed penalty contacts acting through shared body inertia. A displacement at one contact can accelerate another because both load the same translating or rotating bodies. Separate scalar effective mass in a normal-contact penalty mode estimates do not generally give the coupled frequencies.

Freeze a contact configuration, use independent generalized displacement perturbations $q$, positive-definite mass matrix $M$, and gap perturbations $\xi =Jq$. Let $K$ be a positive diagonal matrix of normal penalty stiffnesses. After subtracting equilibrium loads, the restoring generalized force is $-J^{\mathsf{T}}KJq$, soThe contact-Jacobian force/velocity pairing and stiff compliant-contact model come from the primary multibody formulation; the local modal reduction here follows by substitution.^[ref]

Although $AK$ need not be symmetric, it is similar to the symmetric positive-semidefinite matrix $K^{1/2}A{K}^{1/2}$. Set $z=K^{1/2}\xi$ to obtainIts positive eigenvalues are squared angular frequencies. A zero eigenvalue may describe redundant gap coordinates outside the realizable displacement subspace; it should not automatically be interpreted as an extra physical oscillation. Free generalized motions satisfying $Jq=0$ are a separate possibility.

For an original planar example, a rigid bar has vertical translation $y$, small rotation $\theta$, mass $m$, center-of-mass moment of inertia $I$, and two vertical contacts at horizontal offsets $-a,+a$. ThenWith equal contact stiffness $k$, the equal-gap mode is vertical translation and the opposite-gap mode is rocking. Their angular frequencies areTake $m=2\mathrm{kg}$, $I=0.005\mathrm{kg}{\mathrm{m}}^2$, $a=0.10\mathrm{m}$ and $k=10^{5}N/m$. Then $A$ has diagonal entries $2.5\mathrm{k}{\mathrm{g}}^{-1}$ and off-diagonal entries $-1.5\mathrm{k}{\mathrm{g}}^{-1}$. Its eigenvalues are $1$ and $4\mathrm{k}{\mathrm{g}}^{-1}$, giving $316.2$ and $632.5rad/s$. Treating the contacts independently would assign each $500rad/s$ and miss the faster rocking mode.