 Correlated floating-pinion uncertainty set

A 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 $b=e+x_{\mathrm{eq}}$ andThe 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. Supposewith every allowed vector $u$ admissible. Since $\delta -|b|=\min (\delta -b,\delta +b)$, minimizing each affine expression over the box givesThe 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 $|u|\le 0.2$. The two minima are $4.4$ and $5.2$, so $M_{\mathrm{rob}}=4.4\mu \mathrm{m}$. Separate parameter ranges instead give ${\delta }_L=4.8$ and $b_U=0.6$, hence rectangular bound $4.2\mu \mathrm{m}$. A displacement envelope $A=4.3\mu \mathrm{m}$ leaves guaranteed compression $0.1\mu \mathrm{m}$ 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.^[ref] The value of a joint set is justified constraint information, not an assumption that all uncertainties cancel.