= Floating-pinion affine uncertainty bound
A \b[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>, suppose
$$
\delta=\delta_0+\sum_{j=1}^m a_ju_j,\qquad
b=b_0+\sum_{j=1}^m c_ju_j,\qquad |u_j|\le\rho_j.
$$
All 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:
$$
\delta-|b|=\min(\delta-b,\delta+b).
$$
For any affine expression $q_0+\sum_jq_ju_j$, 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 gives
$$
M_{\rm rob}=\min\left(
\delta_0-b_0-\sum_j\rho_j|a_j-c_j|,\;
\delta_0+b_0-\sum_j\rho_j|a_j+c_j|
\right).
$$
The 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, let
$$
\delta=5+0.2u_1+0.1u_2,\qquad
b=0.5+0.1u_1-0.2u_2,\qquad |u_1|,|u_2|\le1.
$$
The two branch minima are $4.5-0.1-0.3=4.1$ and $5.5-0.3-0.1=5.1$, so $M_{\rm rob}=4.1\,\mathrm{\mu m}$. 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 $3.9\,\mathrm{\mu m}$ bound. A motion amplitude $4.0\,\mathrm{\mu m}$ leaves guaranteed compression $0.1\,\mathrm{\mu m}$ 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.<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} A valid guarantee requires defensible parameter bounds and the appropriate <floating-pinion contact-loss check>.
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