Null space of a spatial pressure measurement ($\ker \mathcal{M}=\{\delta p:\mathcal{M}\delta p=0\}$)

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.