Gyro bias cancellation by rotation

Estimating a gyro bias requires observability, time, and a filter that is willing to trust its measurements. There is an alternative that needs none of those: rotate the instrument. If the sensor turns relative to the navigation frame, a bias that is constant in the sensor becomes a sinusoid in the navigation frame, and a sinusoid integrates to zero. This is the principle behind carouselling, indexing and dithering, and it converts an unbounded error into a bounded one by geometry rather than by estimation.

Symbols used on this page are defined in the Nomenclature → inertial navigation.

The one-line idea. Integrated bias error grows as $b\,t$ forever. Integrated modulated bias error oscillates with amplitude $b/\Omega$ and never grows. You have traded a secular drift for a bounded ripple whose size you control by choosing the rotation rate $\Omega$.

1 · Why you would do this instead of estimating the bias

Bias estimation works well when the vehicle manoeuvres, when aiding is available, and when there is time. Each of those can fail:

Situation	Why estimation struggles	Why rotation helps
Static or benign motion	bias is weakly observable — it aliases with tilt (see alignment)	modulation is created deliberately, independent of vehicle motion
No external aiding	nothing to form a residual against; filter is blind	needs no measurement at all — it is open-loop physics
Long unaided coast	even a well-estimated bias has residual error that grows as $t^3$ in position	error stays bounded for arbitrarily long durations
Bias instability / random walk	the quantity you estimated is no longer the quantity acting	modulation attenuates any slowly-varying bias, not just the constant part
High-integrity requirement	an estimate can be wrong and confident	a mechanical rotation is verifiable and does not depend on data quality

The two are complements, not competitors: rotation reduces the size of the error the filter has to find, and the filter cleans up what rotation leaves behind.

2 · The mathematics of bias modulation

2.1 Setup

Let the gyro measure angular rate in its own sensor frame $s$, with a constant bias vector $\mathbf b^s$. The navigation algorithm needs rate in the navigation frame $n$, so it applies the known rotation $C^n_s(t)$:

$$ \tilde{\boldsymbol\omega}^n = C^n_s(t)\big(\boldsymbol\omega^s+\mathbf b^s\big) = \boldsymbol\omega^n + \underbrace{C^n_s(t)\,\mathbf b^s}_{\text{error term}} $$

The error injected into the attitude solution is the integral of that last term:

$$ \delta\boldsymbol\theta(T)=\int_0^T C^n_s(t)\,\mathbf b^s\,dt $$

Everything follows from this integral. If $C^n_s$ is constant — a strapdown instrument bolted to the vehicle — then

$$ \delta\boldsymbol\theta(T)=C^n_s\,\mathbf b^s\,T \quad\Longrightarrow\quad \text{grows linearly, forever} $$

which is the familiar $b_g t$ attitude drift. The entire trick is to make $C^n_s(t)$ time-varying so that the integral does not accumulate.

2.2 Continuous rotation (carouselling)

Rotate the sensor about the navigation-frame $z$ axis at constant rate $\Omega$, so $\psi(t)=\Omega t$:

$$ C^n_s(t)=\begin{bmatrix}\cos\Omega t & -\sin\Omega t & 0\\ \sin\Omega t & \cos\Omega t & 0\\ 0&0&1\end{bmatrix} $$

Integrating the horizontal components of the bias over one interval:

$$ \delta\theta_x(T)=\int_0^T\!\big(b_x\cos\Omega t-b_y\sin\Omega t\big)dt =\frac{b_x}{\Omega}\sin\Omega T+\frac{b_y}{\Omega}\big(\cos\Omega T-1\big) $$

and similarly for $\delta\theta_y$. The result is the key equation of the whole technique:

$$ \boxed{\;\|\delta\boldsymbol\theta_{horiz}\|\;\le\;\frac{2\,\|\mathbf b_{horiz}\|}{\Omega}\;} \qquad\text{bounded for all } T $$

Over a whole number of revolutions the error is exactly zero; in between it is a bounded ripple of amplitude $\sim b/\Omega$. Two immediate design consequences:

Faster rotation is better, and the benefit is $1/\Omega$. Spinning at $\Omega=1$ rad/s attenuates a horizontal bias by a factor of $\Omega T$ over a 1000 s flight — three orders of magnitude.
The rotation axis is not helped. $b_z$ is unaffected: $\delta\theta_z=b_zT$, still linear. This is the fundamental limitation of single-axis rotation and the reason two-axis or tumbling schemes exist.

Single-axis rotation cannot fix the axis it rotates about. A vector component along the rotation axis is invariant under that rotation, so it is never modulated. Any scheme must either rotate about more than one axis (sequentially or simultaneously), orient the rotation axis so that the critical error direction is perpendicular to it, or accept that one axis remains strapdown-grade. For a vertical rotation axis the unmodulated component is the one that drives heading drift — usually the error you cared about most.

2.3 Discrete indexing (two-position reversal)

Continuous rotation needs slip rings or a wrap-management scheme. A simpler mechanisation dwells at two orientations 180° apart for equal times $T/2$. In position A the horizontal bias contributes $+\mathbf b$; in position B the frame has been flipped, so the same physical bias now contributes $-\mathbf b$ in navigation axes:

$$ \delta\boldsymbol\theta=\underbrace{+\mathbf b\,\frac{T}{2}}_{\text{position A}} \;\underbrace{-\,\mathbf b\,\frac{T}{2}}_{\text{position B}}\;=\;\mathbf 0 $$

This is the same cancellation with a square wave instead of a sinusoid, and it is the classic "carousel"/"indexed IMU" arrangement. It is mechanically much simpler, at the cost of leaving a growing error within each dwell — the error reaches $bT/2$ at the moment of reversal before being walked back. The peak excursion, not the endpoint, is what the control system sees, so dwell time must be chosen against the transient requirement and not only the end-of-flight requirement.

Scheme	Error signature	Cancels	Mechanical cost
Strapdown (no rotation)	$bT$, unbounded	nothing	none
Continuous single-axis	bounded ripple $\le 2b/\Omega$	2 axes	continuous drive, slip rings
Two-position indexing	zero at period end; peak $bT/2$ mid-cycle	2 axes	indexer, two stops
Four-position / dual-axis	bounded on all axes	all 3	two gimbals
Continuous tumble	bounded on all axes	all 3	most complex; also averages scale factor

3 · The side effect that is arguably the main benefit

Rotation does something beyond cancellation: it makes errors observable. A strapdown bias and a tilt error are nearly indistinguishable because both are constant in the same axes. Once the sensor rotates, the bias signature becomes a sinusoid at the known rotation frequency $\Omega$ while a genuine attitude error does not, so a filter can separate them by frequency content.

$$ \text{measured error signal}\;\sim\;\underbrace{A\cos(\Omega t+\varphi)}_{\text{sensor-fixed bias}} \;+\;\underbrace{\text{slowly varying}}_{\text{navigation-frame errors}} $$

This is why rotating/indexed systems can self-calibrate: the modulation acts as a carrier, and demodulating at $\Omega$ recovers the bias amplitude and phase directly. It is the same mechanism that lets a multi-position tumble test measure bias on a bench (see IMU acceptance testing), applied continuously in flight. In observability terms, rotation breaks the symmetry that makes the observability matrix rank-deficient.

4 · What it costs — the errors rotation introduces

Nothing is free. A rotating instrument suffers error sources a strapdown one does not, and a real trade has to weigh these against the bias attenuation:

New error source	Mechanism	Why it can dominate
Angle pickoff error	$C^n_s(t)$ must be known; encoder error enters attitude directly	this is now a primary error path — an encoder error of $\epsilon$ is an attitude error of $\epsilon$, with no attenuation at all
Scale-factor & nonlinearity	the rotation itself is a large, persistent input rate $\Omega$	a scale-factor error $\kappa$ on the rotation axis produces a constant rate error $\kappa\Omega$ — a new bias, proportional to the very rate you added
Bearing wobble / axis wander	mechanical imperfection makes the true axis differ from the modelled one	appears as a rate-correlated bias, i.e. exactly what you were trying to remove
$g$-sensitivity and lever arm	the accelerometer triad now moves in a circle of radius $r$	centripetal $\Omega^2 r$ must be modelled exactly; it grows as $\Omega^2$ while the benefit grows only as $1/\Omega$
Vibration, wear, power	moving parts	reliability and lifetime become navigation concerns; also a jitter source for pointing

The trade closes at a finite $\Omega$, not at "as fast as possible." Bias attenuation improves as $1/\Omega$, but the scale-factor-induced error grows as $\kappa\Omega$ and the centripetal term as $\Omega^2r$. Total error therefore has a minimum: $\varepsilon(\Omega)\approx 2b/\Omega+\kappa\Omega$ is minimised at $\Omega^\star=\sqrt{2b/\kappa}$, and pushing past it makes things worse. This is the single most important quantitative result for sizing such a system — and it also explains why the rotation-axis scale factor is the parameter that must be calibrated most carefully.

5 · When this technique is the right answer

Long unaided missions — submarines, subsurface and GNSS-denied platforms, where $t^3$ position growth from gyro bias is the binding constraint and no aiding exists to estimate it away.
Lower-grade sensors, higher-grade system — rotation can buy an order of magnitude, which is sometimes cheaper than buying a better gyro, especially where the bias instability (not the bias itself) is the limit.
Where self-calibration matters — the demodulation benefit of §3 gives continuous in-situ estimates of sensor errors without external aiding.

And when it is not: short-duration, high-dynamics, aided flight. If the mission is minutes long and GNSS-aided, bias is observable from the dynamics anyway (the $[\mathbf f\times]$ coupling), the $t^3$ term never has time to matter, and adding a rotating mechanism buys little while adding failure modes, mass and a new set of calibration requirements.

6 · Further reading

Titterton & Weston, Strapdown Inertial Navigation Technology, 2nd ed. — chapters on gimballed vs strapdown mechanisations and rotation/indexing schemes.
Groves, Principles of GNSS, Inertial and Multisensor Integrated Navigation Systems — rotating INS, error modelling and calibration.
IEEE Std 952 — Specification Format Guide and Test Procedure for Single-Axis Interferometric Fiber Optic Gyros, for the error-term definitions used above.
Open literature on rotary/rotation-modulated INS (RINS) — a substantial body of published work analyses single- vs dual-axis schemes and the $\Omega$-optimisation trade.