 Floating-pinion affine uncertainty bound

A floating-pinion affine uncertainty bound evaluates the minimum contact-compression margin when the approach and imbalance depend linearly plus a constant on bounded shared parameters. It gives an exact result for a box of uncertainty variables, while preserving the dependence those variables induce between physical quantities.

In the correlated floating-pinion uncertainty set, supposeAll uncertainty vectors in this box are assumed admissible. For the linear two-contact model, the floating-pinion robust contact margin is the infimum of $\delta -|b|$ over that box.

First resolve the absolute value:For any affine expression $q_0+{\sum }_j{q}_j{u}_j$, its minimum is $q_0-{\sum }_j{\rho }_j|q_j|$: choose each bounded variable with the sign opposing its coefficient. Applying this rule to the two branches givesThe branches can attain their minima at different parameter vectors. Taking the smaller of their independently attainable minima is valid; requiring both minima to occur at one vector would impose an unnecessary condition.

For an original example in micrometres, letThe two branch minima are $4.5-0.1-0.3=4.1$ and $5.5-0.3-0.1=5.1$, so $M_{\mathrm{rob}}=4.1\mu \mathrm{m}$. The first is attained at $(u_1,u_2)=(-1,-1)$. Treating the separate physical ranges independently instead gives ${\delta }_L=4.7$ and $\max |b|=0.8$, hence a looser $3.9\mu \mathrm{m}$ bound. A motion amplitude $4.0\mu \mathrm{m}$ leaves guaranteed compression $0.1\mu \mathrm{m}$ in the shared-variable model.

The result is exact for the stated box and affine model. Additional constraints can make it conservative; nonlinear parameter dependence or opening of a contact can invalidate the model itself. A floating-pinion ellipsoidal uncertainty bound addresses a different set shape, and a statistical covariance matrix alone does not certify a hard bound.

NASA's face-gear report supplies the physical motivation for examining mounting freedom and load sharing, not an experimental validation of these chosen uncertainty coefficients.^[ref] A valid guarantee requires defensible parameter bounds and the appropriate floating-pinion contact-loss check.