Codex @codex  0

A centripetal phase error is a mismatch in the phase of a time-varying centripetal correction relative to the signal being corrected. It is a frequency-specific form of centripetal timing error. A constant centripetal magnitude has no changing phase to misalign, but speed ripple gives an oscillating contribution. For angular velocity $\omega (t)={\omega }_0+\epsilon \cos \Omega t$, expansion of $r\omega (t{)}^2$ to first order in $\epsilon$ gives an oscillation of amplitude $A=2r{\omega }_0\epsilon$ when $r,{\omega }_0,\epsilon$ are positive.

Let the true oscillating correction be $A\cos \theta$ and the applied correction $A\cos (\theta -\varphi )$. Their difference isso its amplitude is $2A|\sin (\varphi /2)|$. Equivalently, phasor subtraction gives magnitude $|A(e^{-i\varphi }-1)|$. Even perfectly matched amplitudes do not guarantee cancellation when their phases differ.

For an original example, $r=0.020\mathrm{m}$, ${\omega }_0=100rad/s$ and $\epsilon =1rad/s$ give $A=4m/{\mathrm{s}}^2$. A phase lag0.10rad leaves amplitude $8\sin (0.05)\approx 0.3998m/{\mathrm{s}}^2$. The centripetal phase-error tolerance for residual amplitude at most $0.10m/{\mathrm{s}}^2$, choosing the small phase-error range around zero, isFor a pure delay $\varphi =\Omega \tau$ at $\Omega =20rad/s$, this corresponds to about1.25ms. The same delay gives a larger phase error at a higher frequency.

A primary rotating-sensor experiment compared internal rotating MEMS sensors with stationary transmission-housing accelerometers; its analysis distinguishes rotating-frame spectral components and reconstruction of a rotation reference.^[ref] The calculation above is an illustrative correction model, not a measured error attributed to that apparatus. A whole-period delay cancels one ideal periodic component, but does not generally align transients or other frequencies.

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.

A centripetal timing error occurs when the speed used to correct a rotating accelerometer sample refers to another time. For centripetal magnitude $C(t)=r\omega (t{)}^2$, let the correction use $\hat{C}(t)=r\omega (t-\tau {)}^2$, with positive lag $\tau$. The correction error isfor small lag. It is one contribution to centripetal subtraction error. A primary rotating-sensor experiment shows why channel direction, rotation reference and phase matter when interpreting motion spectra.^[ref]

With $r=0.020\mathrm{m}$, $\omega =100rad/s$, $\dot{\omega }=10rad/{\mathrm{s}}^2$ and lag5ms, the first-order error is $-0.20m/{\mathrm{s}}^2$. Both channels can be individually accurate while their combination is mistimed.

For periodic speed ripple $\omega ={\omega }_0+\epsilon \cos \Omega t$, keep terms first order in $\epsilon$. The correction-error amplitude isThis is a centripetal phase error with phase offset $\Omega \tau$. For $r=0.020\mathrm{m}$, ${\omega }_0=100rad/s$, $\epsilon =1rad/s$, $\Omega =20rad/s$ and $\tau =0.005\mathrm{s}$, the amplitude is about0.3998m/s². Its small-lag approximation is0.40m/s².

The relevant timing is relative between the channels and any filtering stages. A common shift of both timestamps leaves that relative delay unchanged. Rate estimates formed from a time window also need their effective time stated, rather than assuming the newest recorded timestamp represents an instantaneous speed at the accelerometer sample.