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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.

A floating-pinion ellipsoidal uncertainty bound gives the smallest two-contact compression margin when the uncertain parameter vector lies in a specified ellipsoid. It retains shared parameter effects, as does the floating-pinion affine uncertainty bound, but uses a different admissible-set shape.

Let approach and imbalance be affine functionsand describe the set by $u=u_0+Lz$ with $\Vert z{\Vert }_2\le 1$. Put $\bar{\delta }={\delta }_0+a^{\mathsf{T}}{u}_0$ and $\bar{b}=b_0+c^{\mathsf{T}}{u}_0$. The floating-pinion robust contact margin then satisfies the exact formulaTo prove it, write $\delta -|b|=\min (\delta -b,\delta +b)$. For either linear coefficient vector $q$, the Cauchy-Schwarz inequality givesIf $L^{\mathsf{T}}q\ne 0$, equality is attained at $z=-L^{\mathsf{T}}q/\Vert L^{\mathsf{T}}q{\Vert }_2$. Thus the result supplies both a margin and an attaining direction. The image representation also handles rank-deficient sets without inverting a shape matrix.

For an original example in micrometres, take $u_0=0$, $L$ the identity andOver $u_1^2+u_2^2\le 1$, the two branch minima are $4.5-\sqrt{0.1}$ and $5.5-\sqrt{0.1}$, so the margin is about $4.1838\mu \mathrm{m}$. The first branch attains it at $u=(-1,-3)/\sqrt{10}$. The encompassing box $|u_1|,|u_2|\le 1$ instead gives $4.1\mu \mathrm{m}$ because it admits combinations excluded by the disk.

A covariance-derived ellipse does not automatically certify an absolute engineering bound. Its probability content requires distributional assumptions and an appropriate radius; a hard guarantee requires a defensible admissible set. NASA's face-gear mounting and slow-roll load-sharing observations motivate the physical contact question, not the particular coefficients or set used here.^[ref] The two-contact force law must also remain valid; contact opening requires the floating-pinion contact-loss check.