= Centripetal phase-error tolerance
A <centripetal phase-error tolerance> bounds the relative phase mismatch allowed when subtracting an oscillating centripetal correction. It translates an acceptable residual acceleration into a phase requirement and, for a known frequency, a timing requirement. With equal true and correction amplitudes $A>0$, a <centripetal phase error> $\phi$ leaves residual amplitude $2A|\sin(\phi/2)|$. For a budget $0\le E_{\max}<2A$, the permitted principal mismatch around zero is
$$
|\phi|\le2\arcsin\!\left(\frac{E_{\max}}{2A}\right).
$$
For a pure time lag at angular frequency $\Omega>0$, divide this bound by $\Omega$ to obtain the lag tolerance. An allowable phase at one frequency is not a universal allowable delay for every spectral component.
For an original example, $A=4\,\mathrm{m/s^2}$ and $E_{\max}=0.10\,\mathrm{m/s^2}$ give a bound of about0.02500rad. At $\Omega=20\,\mathrm{rad/s}$ this is approximately1.25ms; at $200\,\mathrm{rad/s}$ it is approximately0.125ms. The tenfold change follows from the frequency dependence of phase delay, without changing the acceleration budget.
A <centripetal amplitude-error budget> is needed if the applied correction amplitude $\widehat A$ differs from $A$. <Phasor> subtraction gives
$$
E^2=A^2+\widehat A^2-2A\widehat A\cos\phi
=(\widehat A-A)^2+4A\widehat A\sin^2(\phi/2).
$$
Even at zero phase error, the residual is at least $|\widehat A-A|$. If that exceeds $E_{\max}$, timing improvement alone cannot meet the requirement. Otherwise, for positive amplitudes and a budget smaller than their sum, the allowed principal phase is
$$
|\phi|\le2\arcsin\sqrt{\frac{E_{\max}^2-(\widehat A-A)^2}{4A\widehat A}}.
$$
With $A=4$, $\widehat A=3.96$ and budget0.10 in consistent acceleration units, the phase bound falls to about0.02303rad, or1.15ms at20rad/s.
The tolerance is a property of the correction model and the chosen operating range. A primary rotating MEMS-sensor study analyzes rotating-frame frequency components and reconstruction of a rotation reference; these are practical reasons to retain channel timing and phase information.<Evaluation of MEMS-Based Wireless Accelerometer Sensors in Detecting Gear Tooth Faults in Helicopter Transmissions by David G. Lewicki and others>[Printed pp.19–21, rotating-frame spectra; pp.24–25, timing reconstruction]{ref} The numerical tolerances here are illustrative design calculations, not measured limits asserted for that apparatus.
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