= Correlated floating-pinion uncertainty set
A \b[correlated floating-pinion uncertainty set] describes combinations of contact-model parameters that can occur together, rather than treating every separate parameter bound as independently attainable. Here correlated means linked by the parameter model; it need not mean a particular statistical <correlation coefficient>.
For the <floating-pinion robust contact margin>, write $b=e+x_{\rm eq}$ and
$$
M_{\rm rob}=\inf_{(\delta,b)\in\mathcal U}(\delta-|b|).
$$
The set $\mathcal U$ can represent shared mounting motion, a common geometric error or another justified constraint. A rectangle based only on separate ranges includes the true set, but can include combinations that its physical cause cannot produce. Minimizing over this larger set gives a conservative lower bound.
An original <floating-pinion affine uncertainty bound> gives an exact result for bounded shared variables. Suppose
$$
\delta=\delta_0+\sum_j a_j u_j,\qquad b=b_0+\sum_j c_j u_j,\qquad |u_j|\le\rho_j,
$$
with every allowed vector $u$ admissible. Since $\delta-|b|=\min(\delta-b,\delta+b)$, minimizing each affine expression over the box gives
$$
M_{\rm rob}=\min\left\{
\delta_0-b_0-\sum_j\rho_j|a_j-c_j|,\quad
\delta_0+b_0-\sum_j\rho_j|a_j+c_j|
\right\}.
$$
The uncertainty variables are independently bounded here, but the resulting physical parameters can remain linked. Each absolute-value term follows by choosing the sign of $u_j$ that minimizes its coefficient times $u_j$.
For example, in micrometres take $\delta=5+u$, $b=0.5+0.5u$ and $|u|\le0.2$. The two minima are $4.4$ and $5.2$, so $M_{\rm rob}=4.4\,\mathrm{\mu m}$. Separate parameter ranges instead give $\delta_L=4.8$ and $b_U=0.6$, hence rectangular bound $4.2\,\mathrm{\mu m}$. A displacement envelope $A=4.3\,\mathrm{\mu m}$ leaves guaranteed compression $0.1\,\mathrm{\mu m}$ under the joint model, although the rectangular test cannot certify it.
A statistical <covariance matrix> is not itself a hard admissible set: a high-probability region and an absolute bound answer different questions. The physical model also needs to remain valid throughout the selected set. If a contact opens, the linear two-contact force law must be replaced in the <floating-pinion contact-loss check>.
NASA's face-gear work motivates examining mounting freedom and measured load division, while its slow-roll observations do not establish the particular uncertainty set used in this derivation.<Development of Face Gear Technology for Industrial and Aerospace Power Transmission by Gregory F. Heath and others>[Introduction, printed pp. 1–2; Conclusions, printed p. 84]{ref} The value of a joint set is justified constraint information, not an assumption that all uncertainties cancel.
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