 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 $\varphi$ leaves residual amplitude $2A|\sin (\varphi /2)|$. For a budget $0\le E_{\max }<2A$, the permitted principal mismatch around zero isFor 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, $A=4m/{\mathrm{s}}^2$ and $E_{\max }=0.10m/{\mathrm{s}}^2$ give a bound of about0.02500rad. At $\Omega =20rad/s$ this is approximately1.25ms; at $200rad/s$ 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 $\hat{A}$ differs from $A$. Phasor subtraction givesEven at zero phase error, the residual is at least $|\hat{A}-A|$. 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 isWith $A=4$, $\hat{A}=3.96$ 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.^[ref] The numerical tolerances here are illustrative design calculations, not measured limits asserted for that apparatus.