= Floating-pinion ellipsoidal uncertainty bound
A \b[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 functions
$$
\delta=\delta_0+a^{\mathsf T}u,\qquad b=b_0+c^{\mathsf T}u,
$$
and describe the set by $u=u_0+Lz$ with $\lVert z\rVert_2\le1$. 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 formula
$$
M_{\rm rob}=\min\left\{
\bar\delta-\bar b-\lVert L^{\mathsf T}(a-c)\rVert_2,\quad
\bar\delta+\bar b-\lVert L^{\mathsf T}(a+c)\rVert_2
\right\}.
$$
To prove it, write $\delta-|b|=\min(\delta-b,\delta+b)$. For either linear coefficient vector $q$, the <Cauchy-Schwarz inequality> gives
$$
\inf_{\lVert z\rVert_2\le1}q^{\mathsf T}Lz=-\lVert L^{\mathsf T}q\rVert_2.
$$
If $L^{\mathsf T}q\ne0$, equality is attained at $z=-L^{\mathsf T}q/\lVert L^{\mathsf T}q\rVert_2$. 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 and
$$
\delta=5+0.2u_1+0.1u_2,\qquad b=0.5+0.1u_1-0.2u_2.
$$
Over $u_1^2+u_2^2\le1$, the two branch minima are $4.5-\sqrt{0.1}$ and $5.5-\sqrt{0.1}$, so the margin is about $4.1838\,\mathrm{\mu m}$. The first branch attains it at $u=(-1,-3)/\sqrt{10}$. The encompassing box $|u_1|,|u_2|\le1$ instead gives $4.1\,\mathrm{\mu m}$ 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.<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 two-contact force law must also remain valid; contact opening requires the <floating-pinion contact-loss check>.
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