Sensor Error Sources

A complete spacecraft-GNC treatment of deterministic, stochastic, environmental, dynamic, digital, installation, timing, calibration, covariance, Monte Carlo, filtering, and fault-related sensor errors.

4.1.7 Sensor Error Sources

This chapter is intentionally cross-cutting. The individual sensor pages explain the physics and error mechanisms of gyroscopes, accelerometers, magnetometers, star trackers, Sun sensors, and GNSS. Here, those effects are placed into one common engineering framework so they can be used consistently in simulation, calibration, covariance design, Kalman filtering, Monte Carlo analysis, FDIR, and mission-level GNC verification.

The central engineering question is not simply, “What is the sensor accuracy?” It is: which errors are deterministic, which are stochastic, which vary with time or environment, which can be calibrated, which must be estimated online, which should appear in the process model, which should appear in the measurement covariance, and which must instead be handled by validity or fault logic?

A spacecraft sensor never provides the exact physical quantity that exists in reality. Every measurement is affected by imperfections associated with the sensing element, electronics, calibration, installation, spacecraft environment, timing system, signal processing, and stochastic processes.

The general idea is

\[ \boxed{ \text{True Physical Quantity} \rightarrow \text{Sensor Physics} \rightarrow \text{Deterministic Errors} \rightarrow \text{Stochastic Errors} \rightarrow \text{Sampling / Processing} \rightarrow \text{Measured Quantity} } \]

A useful generic sensor model is

\[ \boxed{ \mathbf z_m = h(\mathbf x) + \mathbf e_d + \mathbf e_s } \]

where

For a three-axis vector sensor, a more detailed model is

\[ \boxed{ \mathbf z_m = (\mathbf I+\mathbf S+\mathbf M) \mathbf z_{true} + \mathbf b + \mathbf n } \]

where

\[ \mathbf S= \begin{bmatrix} s_x&0&0\\ 0&s_y&0\\ 0&0&s_z \end{bmatrix} \]

represents scale-factor errors,

\[ \mathbf M \]

represents misalignment, non-orthogonality and cross-axis coupling,

\[ \mathbf b \]

is sensor bias, and

\[ \mathbf n \]

represents stochastic measurement noise.

For real spacecraft sensors, this can be extended further:

\[ \boxed{ \mathbf z_m(t_k) = \mathcal Q \left[ \mathcal S \left( h(\mathbf x(t_k-\tau)) + \mathbf e_{cal} + \mathbf e_{env} + \mathbf e_{dyn} + \mathbf n \right) \right] } \]

where \(\mathcal S\) represents sampling, \(\mathcal Q\) quantisation, and \(\tau\) measurement latency.

4.1.7.1 Why Sensor Errors Matter in GNC

Sensor errors propagate directly into the quantities used by guidance, navigation and control.

The complete chain is

\[ \boxed{ \text{Sensor Error} \rightarrow \text{Measurement Error} \rightarrow \text{State-Estimation Error} \rightarrow \text{Guidance Error} \rightarrow \text{Control Error} \rightarrow \text{Spacecraft Performance} } \]

For example, a gyroscope bias produces attitude drift:

\[ b_g \rightarrow \delta\omega \rightarrow \delta\theta. \]

An accelerometer bias produces velocity and position drift:

\[ b_a \rightarrow \delta v \rightarrow \delta r. \]

A magnetometer bias distorts the measured geomagnetic vector and therefore affects attitude estimation and magnetic control.

Star-tracker centroiding, alignment or thermal errors directly influence spacecraft attitude knowledge.

GNSS range, clock, geometry and antenna errors propagate into position, velocity and timing estimates.

Consequently, sensor modelling is not merely an instrumentation problem. It is fundamental to the entire GNC error budget.

4.1.7.2 Sensor Error Classification

Sensor errors can be classified in several ways.

A useful top-level classification is

\[ \boxed{ \text{Sensor Errors} = \text{Deterministic} + \text{Stochastic} + \text{Environmental} + \text{Dynamic} + \text{Digital} + \text{Installation} + \text{Operational} } \]

Deterministic errors

These can often be characterized and calibrated:

Stochastic errors

These cannot be predicted exactly for an individual measurement:

Environmental errors

Examples include:

Dynamic errors

Examples include:

Digital errors

Examples include:

Installation errors

Examples include:

4.1.7.3 Accuracy, Precision, Resolution and Repeatability

These terms should not be used interchangeably.

Accuracy

Accuracy describes closeness to the true value.

\[ \boxed{ e=z_m-z_{true} } \]

Small measurement error generally corresponds to high accuracy.

Precision

Precision describes how tightly repeated measurements cluster.

A sensor may be highly precise but biased.

Resolution

Resolution is the smallest input change that can be represented or detected.

For an \(N\)-bit sensor over range

\[ x_{min}\le x\le x_{max}, \]

an approximate digital resolution is

\[ \boxed{ \Delta x = \frac{x_{max}-x_{min}}{2^N} } \]

depending on encoding convention.

Repeatability

Repeatability describes the ability to reproduce measurements under nominally identical conditions.

Stability

Stability describes how sensor characteristics change with time.

These distinctions are essential when reading sensor datasheets.

4.1.7.4 Bias

Bias is an additive offset between the sensor output and the ideal measurement.

\[ \boxed{ z_m=z_{true}+b } \]

For a vector sensor,

\[ \boxed{ \mathbf z_m = \mathbf z_{true} + \mathbf b. } \]

Bias may arise from:

4.1.7.5 Constant Bias

The simplest model assumes

\[ \boxed{ b(t)=b_0. } \]

Thus

\[ z_m=z_{true}+b_0+n. \]

Constant bias can often be estimated during calibration.

However, even a small constant bias can become extremely important when the measurement is integrated.

For a gyro,

\[ \delta\theta(t)\approx b_g t. \]

For an accelerometer,

\[ \delta v(t)\approx b_a t \]

and

\[ \boxed{ \delta r(t)\approx\frac12b_a t^2. } \]

Therefore an apparently small inertial-sensor bias may produce large navigation errors.

4.1.7.6 Bias Repeatability

A sensor may not return to exactly the same bias after:

This is called bias repeatability or turn-on-to-turn-on bias variation.

A useful model is

\[ \boxed{ b_0\sim\mathcal N(\bar b,\sigma_{b0}^2). } \]

Each Monte Carlo realization can therefore begin with a different initial bias.

4.1.7.7 Bias Drift

Real bias usually changes slowly with time:

\[ \boxed{ b=b(t). } \]

The measurement becomes

\[ z_m=z_{true}+b(t)+n(t). \]

Bias drift may be caused by:

Bias drift is especially important for inertial sensors.

4.1.7.8 Bias Random Walk

A common stochastic model is

\[ \boxed{ \dot b=w_b } \]

where

\[ w_b(t)\sim\mathcal N(0,q_b). \]

In discrete time,

\[ \boxed{ b_{k+1}=b_k+w_{b,k}. } \]

More carefully, for a continuous white driving process with spectral density \(q_b\),

\[ \boxed{ b_{k+1} = b_k+\sqrt{q_b\Delta t}\,\xi_k } \]

where

\[ \xi_k\sim\mathcal N(0,1). \]

This model is widely used in Kalman filters.

4.1.7.9 First-Order Gauss-Markov Bias

Not all bias processes should be represented as unbounded random walks.

A first-order Gauss-Markov model is

\[ \boxed{ \dot b = -\frac{1}{\tau_b}b+w_b } \]

where \(\tau_b\) is the correlation time.

Its discrete form is

\[ \boxed{ b_{k+1} = e^{-\Delta t/\tau_b}b_k+w_k. } \]

This gives a correlated but bounded stochastic process.

4.1.7.10 Bias Instability

Bias instability describes slowly varying fluctuations that are not adequately represented by simple white noise.

It is commonly identified using Allan deviation.

Conceptually,

\[ \boxed{ \text{Bias Instability} \rightarrow \text{Slowly Varying Measurement Offset}. } \]

It is particularly important for gyroscopes and accelerometers.

Bias instability should not automatically be treated as identical to bias random walk. They represent different stochastic behaviours.

4.1.7.11 Scale-Factor Error

Scale-factor error causes the sensor response to differ from the ideal proportionality.

For a scalar measurement,

\[ \boxed{ z_m=(1+s)z_{true}. } \]

If

\[ s=0.01, \]

the scale-factor error is approximately \(1\%\).

For a three-axis sensor,

\[ \boxed{ \mathbf z_m = (\mathbf I+\mathbf S)\mathbf z_{true}. } \]

Scale-factor error becomes particularly important when the measured quantity is large.

4.1.7.12 Scale-Factor Asymmetry

Some sensors have different scale factors for positive and negative inputs:

\[ s_+\neq s_-. \]

Therefore,

\[ z_m= \begin{cases} (1+s_+)z, & z\ge0\\ (1+s_-)z, & z<0. \end{cases} \]

A single scale-factor coefficient may therefore be insufficient for high-fidelity modelling.

4.1.7.13 Scale-Factor Temperature Dependence

Scale factor can vary with temperature:

\[ \boxed{ s(T) = s_0+k_{s1}(T-T_0)+k_{s2}(T-T_0)^2+\cdots } \]

so that

\[ z_m=[1+s(T)]z_{true}. \]

This is one reason calibration tables are sometimes functions of temperature rather than fixed constants.

4.1.7.14 Axis Misalignment

A sensor's physical axes are not perfectly aligned with the assumed coordinate frame.

If

\[ C_B^S \]

maps a body-frame quantity into the sensor frame,

\[ \boxed{ \mathbf z^S=C_B^S\mathbf z^B. } \]

An error in \(C_B^S\) produces a mounting or alignment error.

For small angular misalignment,

\[ C\approx I-[\delta\boldsymbol\theta\times] \]

for one common convention.

Therefore,

\[ \boxed{ \delta\mathbf z \approx -[\delta\boldsymbol\theta\times]\mathbf z. } \]

The sign depends on the adopted frame/transformation convention.

4.1.7.15 Non-Orthogonality

Ideally, the three sensor axes are mutually perpendicular:

\[ \hat x\cdot\hat y = \hat y\cdot\hat z = \hat z\cdot\hat x = 0. \]

Manufacturing imperfections can cause small deviations.

A non-orthogonality matrix can be represented as

\[ \boxed{ M= \begin{bmatrix} 0&m_{xy}&m_{xz}\\ m_{yx}&0&m_{yz}\\ m_{zx}&m_{zy}&0 \end{bmatrix}. } \]

The sensor model becomes

\[ \mathbf z_m=(I+M)\mathbf z_{true}+\cdots. \]

4.1.7.16 Cross-Axis Sensitivity

An input along one physical axis may generate an output on another axis.

For example,

\[ z_{m,x} = z_x + c_{xy}z_y + c_{xz}z_z. \]

The complete model is

\[ \boxed{ \mathbf z_m=C_{cross}\mathbf z_{true}. } \]

Cross-axis sensitivity can result from sensing-element construction, electronics, mounting, or imperfect calibration.

4.1.7.17 Combined Calibration Matrix

Scale factor, misalignment and cross-axis effects can be grouped into one calibration matrix:

\[ \boxed{ \mathbf z_m = A\mathbf z_{true} + \mathbf b + \mathbf n. } \]

where

\[ A = \begin{bmatrix} a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33} \end{bmatrix}. \]

Calibration seeks estimates

\[ \hat A,\qquad\hat{\mathbf b}. \]

The corrected measurement becomes

\[ \boxed{ \hat{\mathbf z}_{true} = \hat A^{-1} (\mathbf z_m-\hat{\mathbf b}). } \]

This formulation is useful for accelerometers, gyroscopes and magnetometers.

4.1.7.18 Nonlinearity

A real sensor may not maintain a perfectly linear input-output relationship.

Instead of

\[ z_m=Kz, \]

the response may be

\[ \boxed{ z_m = K_1z + K_2z^2 + K_3z^3 +\cdots. } \]

Nonlinearity is often specified relative to full scale.

It becomes particularly important near the sensor's operating-range limits.

4.1.7.19 Hysteresis

Hysteresis means the output depends not only on the current input but also on its previous history.

Conceptually,

\[ \boxed{ z_m(t) = h[z(t),\text{history}]. } \]

The output for increasing input may differ from the output for decreasing input.

Hysteresis may appear in:

4.1.7.20 Dead Zone

A sensor may have a region in which small input changes produce little or no observable output.

\[ z_m= \begin{cases} 0,& |z| Dead-zone behaviour reduces sensitivity near zero.

4.1.7.21 Saturation

Every physical sensor has a finite operating range.

If

\[ z_{min}\le z\le z_{max}, \]

the measured value may be represented as

\[ \boxed{ z_m = \operatorname{sat} (z,z_{min},z_{max}). } \]

When the input exceeds the measurement range, information is lost.

Examples include:

A realistic simulation should generate a saturation flag when appropriate.

4.1.7.22 Quantisation

Digital sensors represent continuous measurements using discrete levels.

For quantisation step

\[ \Delta, \] \[ \boxed{ z_q = \Delta \operatorname{round} \left( \frac{z}{\Delta} \right). } \]

Quantisation error is

\[ e_q=z_q-z. \]

For an ideal uniform quantiser,

\[ -\frac{\Delta}{2} \le e_q < \frac{\Delta}{2}. \]

Under suitable assumptions,

\[ \boxed{ \sigma_q^2\approx\frac{\Delta^2}{12}. } \]

This approximation should not be blindly applied when the signal is highly correlated with the quantisation process.

4.1.7.23 Resolution vs Quantisation

Resolution and quantisation are related but conceptually different.

Resolution describes the smallest distinguishable input increment.

Quantisation describes the process of mapping a continuous value onto discrete output levels.

A sensor may also have effective resolution poorer than its digital bit depth because of analog noise.

Thus,

\[ \boxed{ \text{Digital Bits} \neq \text{Effective Measurement Resolution}. } \]

4.1.7.24 White Noise

A common sensor-noise model is zero-mean Gaussian white noise:

\[ \boxed{ n(t)\sim\mathcal N(0,\sigma^2). } \]

In discrete form,

\[ z_k=z_{true,k}+n_k. \]

with

\[ n_k\sim\mathcal N(0,R). \]

For vector measurements,

\[ \boxed{ \mathbf n_k\sim\mathcal N(\mathbf0,R). } \]

The covariance matrix need not be diagonal.

4.1.7.25 Measurement Noise Covariance

For a three-axis sensor,

\[ R= \begin{bmatrix} \sigma_x^2&R_{xy}&R_{xz}\\ R_{yx}&\sigma_y^2&R_{yz}\\ R_{zx}&R_{zy}&\sigma_z^2 \end{bmatrix}. \]

If axis errors are independent,

\[ R= \operatorname{diag} (\sigma_x^2,\sigma_y^2,\sigma_z^2). \]

Assuming diagonal covariance without justification can underestimate filter uncertainty when measurement errors are correlated.

4.1.7.26 Noise Density

Sensor noise is often specified as an amplitude spectral density.

Typical units include

\[ \frac{^\circ/s}{\sqrt{\mathrm{Hz}}}, \qquad \frac{m/s^2}{\sqrt{\mathrm{Hz}}}, \qquad \frac{nT}{\sqrt{\mathrm{Hz}}}. \]

A generic relationship is

\[ \boxed{ \sigma^2 = \int S_n(f)\,df. } \]

For approximately white noise over effective bandwidth \(B\),

\[ \boxed{ \sigma \propto N_d\sqrt{B}. } \]

The exact numerical conversion depends on whether the specification uses one-sided or two-sided PSD conventions and on the actual filter bandwidth.

4.1.7.27 Power Spectral Density

The power spectral density describes how noise power is distributed with frequency.

\[ \boxed{ S_x(f). } \]

White noise approximately has

\[ S_x(f)=\text{constant}. \]

Coloured noise has frequency-dependent PSD.

PSD analysis helps distinguish different stochastic processes that cannot be understood from RMS noise alone.

4.1.7.28 Coloured Noise

Real sensor noise is frequently temporally correlated.

Unlike white noise,

\[ E[n_k n_{k-j}] \neq0 \]

for some \(j\neq0\).

Examples include:

A filter that assumes white measurement noise may become inconsistent if the real noise is strongly correlated.

4.1.7.29 Flicker Noise

Flicker noise is often associated with approximately

\[ \boxed{ S(f)\propto\frac1f. } \]

It dominates many sensors and electronic systems at low frequencies.

It is related conceptually to the slowly varying behaviour observed as bias instability.

4.1.7.30 Random Walk

Random walk arises when white noise is integrated.

\[ \boxed{ x_{k+1}=x_k+w_k. } \]

Its variance grows with time.

This behaviour is fundamental in inertial navigation.

4.1.7.31 Angle Random Walk

Gyroscope white rate noise integrates into attitude uncertainty.

Conceptually,

\[ \boxed{ \text{Gyro White Noise} \xrightarrow{\int} \text{Angle Random Walk}. } \]

Attitude uncertainty therefore increases with integration time.

4.1.7.32 Velocity Random Walk

Accelerometer white noise integrates into velocity uncertainty:

\[ \boxed{ \text{Acceleration Noise} \xrightarrow{\int} \text{Velocity Random Walk}. } \]

Further integration propagates the error into position.

4.1.7.33 Rate Random Walk

Rate random walk describes a lower-frequency stochastic process in gyroscope output, distinct from ordinary white rate noise.

It is one of several noise mechanisms that can appear in Allan-deviation analysis.

4.1.7.34 Allan Variance

Allan variance is widely used to characterize inertial-sensor stochastic errors.

For averaging interval \(\tau\),

\[ \boxed{ \sigma_A^2(\tau) = \frac12 E [ (\bar y_{k+1}-\bar y_k)^2 ]. } \]

The Allan deviation is

\[ \boxed{ \sigma_A(\tau) = \sqrt{\sigma_A^2(\tau)}. } \]

Plotting

\[ \log\sigma_A \quad\text{against}\quad \log\tau \]

reveals different stochastic regimes.

4.1.7.35 Interpreting an Allan-Deviation Plot

Different slope regions may correspond to different processes such as:

Conceptually,

\[ \boxed{ \text{Time-Series Data} \rightarrow \text{Allan Deviation} \rightarrow \text{Noise Identification} \rightarrow \text{Stochastic Sensor Model}. } \]

Coefficient extraction depends on the exact Allan-variance convention and units being used, so numerical conversion factors should be applied carefully.

4.1.7.36 Temperature Effects

Temperature affects nearly every spacecraft sensor.

A bias model can be written as

\[ \boxed{ b(T) = b_0 + k_{b1}(T-T_0) + k_{b2}(T-T_0)^2 +\cdots. } \]

Similarly,

\[ s=s(T). \]

A complete measurement may therefore be

\[ \boxed{ z_m = [1+s(T)]z_{true} + b(T) + n. } \]

Temperature effects may arise from:

4.1.7.37 Thermal Gradient Effects

Uniform temperature and thermal gradients are different problems.

A sensor may have acceptable bias stability at uniform temperature but experience alignment errors when different parts of the spacecraft structure are at different temperatures.

For precision attitude determination,

\[ \boxed{ \text{Thermal Gradient} \rightarrow \text{Structural Deformation} \rightarrow \text{Sensor-to-Payload Misalignment}. } \]

This is especially important for star trackers and precision payloads.

4.1.7.38 Ageing

Sensor characteristics may change over mission life:

\[ b=b(t_{life}), \qquad s=s(t_{life}). \]

Ageing can affect:

Long-duration missions may therefore require in-orbit recalibration.

4.1.7.39 Radiation Effects

The space radiation environment can produce:

Conceptually,

\[ \boxed{ \text{Radiation} \rightarrow \text{Sensor Parameter Drift / Transient Fault / Permanent Degradation}. } \]

Radiation should therefore be considered both as an error source and as a reliability issue.

4.1.7.40 Vibration and Mechanical Effects

Launch and on-orbit vibration can influence sensor measurements.

Sources include:

A vibration signal may enter the sensor bandwidth and appear as real measurement content.

Thus,

\[ \boxed{ \text{Mechanical Disturbance} \rightarrow \text{Sensor Motion} \rightarrow \text{Measured Disturbance}. } \]

This is not always sensor noise—the sensor may be correctly measuring actual structural motion.

4.1.7.41 g-Sensitivity of Gyroscopes

Some gyroscopes exhibit sensitivity to linear acceleration:

\[ \boxed{ \boldsymbol{\omega}_m = \boldsymbol{\omega} + K_g\mathbf a + \mathbf b_g + \mathbf n_g. } \]

This may matter during:

4.1.7.42 Lever-Arm Effects

A sensor mounted away from the spacecraft centre of mass experiences rotational acceleration.

For sensor offset \(\mathbf r\),

\[ \boxed{ \mathbf a_s = \mathbf a_O + \dot{\boldsymbol\omega}\times\mathbf r + \boldsymbol\omega\times (\boldsymbol\omega\times\mathbf r) } \]

for a rigid body-fixed sensor.

Therefore an accelerometer away from the reference point measures both translational and rotational effects.

GNSS antennas also require proper treatment of antenna location relative to the spacecraft reference point.

4.1.7.43 Electromagnetic Interference

Spacecraft electronics can interfere with sensors.

Potential sources include:

Magnetometers are particularly sensitive.

Conceptually,

\[ \boxed{ \mathbf B_m = \mathbf B_{Earth} + \mathbf B_{spacecraft} + \mathbf n. } \]

4.1.7.44 Optical Contamination and Stray Light

Optical sensors are affected by environmental optical errors.

Examples include:

These effects apply particularly to:

The measurement may become biased or completely invalid.

Therefore a realistic model requires a validity state, not merely increased Gaussian noise.

4.1.7.45 GNSS-Specific Environmental Errors

GNSS contains several error sources that differ from conventional inertial sensors:

Thus GNSS error cannot generally be represented only by

\[ \mathbf r_m=\mathbf r+\mathbf n. \]

That simple model is appropriate only at solution-level simulation fidelity.

4.1.7.46 Sampling

Digital sensors provide measurements at discrete times:

\[ \boxed{ t_k=kT_s. } \]

The sampled measurement is

\[ z[k]=z(t_k). \]

Sampling introduces important GNC considerations:

4.1.7.47 Aliasing

If the sampling frequency is insufficient, high-frequency signal content can appear as false lower-frequency content.

The Nyquist condition for a band-limited signal is

\[ \boxed{ f_s>2f_{max}. } \]

In practice, anti-aliasing filters are normally required before sampling.

Aliasing is especially important when spacecraft vibration frequencies overlap sensor bandwidth.

4.1.7.48 Sensor Bandwidth

A real sensor cannot respond instantaneously.

A simple first-order model is

\[ \boxed{ G(s)=\frac{1}{\tau s+1}. } \]

The cutoff frequency is approximately

\[ \boxed{ f_c=\frac{1}{2\pi\tau}. } \]

Bandwidth affects:

A higher bandwidth is not automatically better because it can also admit more noise.

4.1.7.49 Latency

A sensor measurement may correspond to an earlier physical state.

\[ \boxed{ z_m(t)=h[x(t-\tau)]+n(t). } \]

For a rotating spacecraft, attitude error caused by timing error is approximately

\[ \boxed{ \delta\boldsymbol\theta \approx \boldsymbol\omega\,\delta t. } \]

Latency therefore becomes increasingly important during fast maneuvers.

4.1.7.50 Latency vs Timestamp Error

These should be distinguished.

Known latency

If

\[ \tau \]

is known accurately, the estimator can often compensate for it.

Timestamp error

If the measurement epoch itself is uncertain,

\[ \delta t\neq0, \]

the measurement is associated with the wrong spacecraft state.

Therefore,

\[ \boxed{ \text{Known Latency} \neq \text{Timing Uncertainty}. } \]

4.1.7.51 Sampling Jitter

The actual sample time may differ from its nominal value:

\[ \boxed{ t_k=kT_s+\delta t_k. } \]

For rapidly changing signals,

\[ \delta z \approx \dot z\,\delta t. \]

Sampling jitter can therefore become a measurable sensor error.

4.1.7.52 Synchronization Error Between Sensors

Sensor fusion assumes measurements are associated with consistent epochs.

Suppose

\[ z_1=z_1(t) \]

but

\[ z_2=z_2(t-\Delta t). \]

Directly combining them as simultaneous measurements introduces inconsistency.

This matters for:

4.1.7.53 Dropout

Sensors do not always provide measurements.

Define

\[ \boxed{ valid_k\in\{0,1\}. } \]

Then

\[ valid_k=0 \]

means the measurement should not be used.

Dropout can arise from:

4.1.7.54 Outliers

Not every erroneous measurement follows a Gaussian distribution.

An outlier may be represented conceptually as

\[ \boxed{ z_m=z_{true}+n+o } \]

where \(o\) is a large occasional error.

Sources include:

Outliers require robust validation rather than merely increasing \(R\).

4.1.7.55 Stuck Measurements

A sensor can fail while continuing to output a plausible-looking constant value:

\[ \boxed{ z_k=z_{k-1}. } \]

A simple range check may not detect such a fault.

Detection may require:

4.1.7.56 Spike Errors

A transient sensor fault may produce

\[ z_k=z_{true,k}+A_{spike}. \]

Spike detection can use:

Care is needed not to reject genuine rapid spacecraft motion.

4.1.7.57 Noise vs Fault

A fundamental distinction is

\[ \boxed{ \text{Expected Random Variation} \neq \text{Sensor Fault}. } \]

Noise belongs within the expected statistical measurement model.

Faults correspond to abnormal behaviour outside the assumed model.

The GNC architecture should distinguish these cases because the response is different.

4.1.7.58 Measurement Validity Logic

A realistic sensor output should include more than a numerical value.

A measurement packet can contain

\[ \boxed{ \mathcal Z = \{ \mathbf z,\, t,\, R,\, valid,\, quality,\, status \}. } \]

This allows the estimator to know:

4.1.7.59 Innovation Monitoring

For an estimator,

\[ \boxed{ \mathbf y_k = \mathbf z_k - h(\hat{\mathbf x}_k^-) } \]

is the innovation.

Its covariance is

\[ \boxed{ S_k = H_kP_k^-H_k^T+R_k. } \]

Large innovations may indicate:

4.1.7.60 Mahalanobis Gating

A normalized innovation statistic is

\[ \boxed{ d^2 = \mathbf y^TS^{-1}\mathbf y. } \]

A measurement can be rejected when

\[ \boxed{ d^2>\gamma } \]

where \(\gamma\) is selected from an appropriate statistical threshold.

This is preferable to applying the same raw-error threshold to measurements with different covariance.

4.1.7.61 Residual Monitoring

With redundant measurements, residuals can be formed between sensors.

For two sensors measuring the same quantity,

\[ \boxed{ r=z_1-z_2. } \]

A growing residual may indicate sensor degradation.

However, the expected residual covariance must include uncertainty from both sensors.

4.1.7.62 Redundancy

Spacecraft often carry redundant sensors.

Possible configurations include:

Redundancy provides:

\[ \boxed{ \text{Fault Detection} + \text{Fault Isolation} + \text{Continued Operation}. } \]

4.1.7.63 Common-Mode Errors

Redundant sensors do not guarantee independent errors.

If both sensors experience the same disturbance,

\[ e_1\approx e_2, \]

comparison may not reveal the problem.

Examples include:

This is a major consideration in redundancy architecture.

4.1.7.64 Correlated Sensor Errors

For two measurements,

\[ E[e_1e_2^T] = P_{12}. \]

If

\[ P_{12}\neq0, \]

their errors are correlated.

The combined covariance is therefore not simply diagonal.

Ignoring correlation can cause an estimator to become overconfident.

4.1.7.65 Calibration

Calibration estimates parameters relating sensor output to the physical truth.

Conceptually,

\[ \boxed{ \text{Known Reference} + \text{Sensor Measurement} \rightarrow \text{Parameter Estimation} \rightarrow \text{Calibration Model}. } \]

Typical calibration parameters include:

4.1.7.66 One-Point Calibration

A one-point calibration can estimate an offset.

If the reference value is \(z_{ref}\),

\[ \boxed{ \hat b=z_m-z_{ref}. } \]

This does not independently determine scale factor.

4.1.7.67 Two-Point Calibration

With two known inputs,

\[ z_{ref,1},\qquad z_{ref,2}, \]

and outputs

\[ z_{m,1},\qquad z_{m,2}, \]

a linear model

\[ z_m=az+b \]

can estimate both scale factor and offset.

4.1.7.68 Multi-Position Calibration

Three-axis inertial and magnetic sensors frequently require measurements in many orientations.

The calibration problem becomes

\[ \boxed{ \min_{A,b} \sum_k \| \mathbf z_{m,k} - A\mathbf z_{ref,k} - \mathbf b \|^2. } \]

This can estimate:

4.1.7.69 Temperature Calibration

Sensor parameters can be measured over a thermal range:

\[ \boxed{ \{T_i,b_i,s_i\} \rightarrow b(T),s(T). } \]

The flight software can then apply temperature compensation.

This requires an accurate sensor temperature measurement and an appropriate calibration model.

4.1.7.70 Ground Calibration vs In-Orbit Calibration

Ground calibration

Allows controlled reference inputs and environmental testing.

In-orbit calibration

Can account for:

Therefore,

\[ \boxed{ \text{Ground Calibration} + \text{In-Orbit Estimation} } \]

is often more powerful than either alone.

4.1.7.71 Calibration Residual Error

Calibration does not eliminate uncertainty completely.

If

\[ \hat b=b+\delta b, \]

then after correction,

\[ z_{corr}=z_m-\hat b \]

contains

\[ -\delta b. \]

Therefore calibration parameters themselves require uncertainty models.

4.1.7.72 Sensor Error States in a Kalman Filter

Slowly varying errors can be estimated as states.

For an INS,

\[ \boxed{ \delta\mathbf x = \begin{bmatrix} \delta\mathbf r\\ \delta\mathbf v\\ \delta\boldsymbol\theta\\ \mathbf b_a\\ \mathbf b_g \end{bmatrix}. } \]

Additional states may include:

The appropriate state vector depends on mission requirements and observability.

4.1.7.73 Process Noise vs Measurement Noise

These are fundamentally different.

Process noise

Represents uncertainty in state evolution:

\[ \boxed{ \mathbf x_{k+1} = f(\mathbf x_k)+\mathbf w_k } \]

with

\[ \mathbf w_k\sim\mathcal N(0,Q). \]

Measurement noise

Represents uncertainty in sensor observations:

\[ \boxed{ \mathbf z_k = h(\mathbf x_k)+\mathbf v_k } \]

with

\[ \mathbf v_k\sim\mathcal N(0,R). \]

Thus,

\[ \boxed{ Q\neq R. } \]

Confusing the two produces incorrect filter tuning.

4.1.7.74 Observability of Sensor Errors

Including a bias state in an EKF does not guarantee that the filter can estimate it.

A parameter must influence measurements in a distinguishable way.

Conceptually,

\[ \boxed{ \text{State Included in Filter} \not\Rightarrow \text{State Observable}. } \]

Observability depends on:

4.1.7.75 Error Propagation Through Integration

Some sensor errors become increasingly severe because GNC algorithms integrate their measurements.

Gyro:

\[ \boxed{ \delta\theta = \int\delta\omega\,dt. } \]

Accelerometer:

\[ \boxed{ \delta v = \int\delta a\,dt } \]

and

\[ \boxed{ \delta r = \int\delta v\,dt. } \]

This explains why inertial navigation requires external aiding.

4.1.7.76 Attitude Error Coupling into Acceleration

Accelerometer measurements must be transformed using the estimated attitude.

If the attitude estimate contains

\[ \delta\boldsymbol\theta, \]

then the transformed acceleration also contains error.

For a small attitude error,

\[ \boxed{ \delta\mathbf a \approx -[\mathbf f\times] \delta\boldsymbol\theta } \]

up to the adopted frame/error convention.

Thus gyro errors can indirectly produce position errors.

4.1.7.77 Frame-Transformation Errors

Sensor measurements often require transformations:

\[ \boxed{ \mathbf z^B = C_S^B\mathbf z^S. } \]

If

\[ C_S^B \]

is uncertain, the transformed measurement contains additional error.

This applies to:

4.1.7.78 Error Covariance Transformation

Suppose measurement covariance is known in frame \(A\):

\[ P^A. \]

After rotation into frame \(B\),

\[ \boxed{ P^B = C_A^B P^A (C_A^B)^T. } \]

Covariance must be transformed with the measurement.

Rotating the vector but leaving its covariance in the original frame is inconsistent.

4.1.7.79 Linear Error Propagation

For

\[ \mathbf y=f(\mathbf x), \]

small perturbations satisfy

\[ \boxed{ \delta\mathbf y \approx J\delta\mathbf x } \]

where

\[ J=\frac{\partial f}{\partial\mathbf x}. \]

Therefore covariance propagates as

\[ \boxed{ P_y \approx JP_xJ^T. } \]

This is one of the fundamental tools of GNC uncertainty analysis.

4.1.7.80 Root-Sum-Square Error Budget

If scalar error contributors are independent,

\[ \boxed{ \sigma_{total} = \sqrt{ \sigma_1^2+ \sigma_2^2+ \cdots+ \sigma_n^2 }. } \]

This is commonly used in engineering error budgets.

However, RSS is not valid without modification when errors are correlated.

4.1.7.81 Covariance-Based Error Budget

For correlated vector errors,

\[ \boxed{ P_{total} = \sum_iP_i + \sum_{i\neq j}P_{ij}. } \]

This is more general than simply adding independent \(1\sigma\) values by RSS.

4.1.7.82 1σ, 2σ and 3σ

For a one-dimensional Gaussian distribution,

\[ x\sim\mathcal N(\mu,\sigma^2), \]

the intervals

\[ \mu\pm\sigma, \qquad \mu\pm2\sigma, \qquad \mu\pm3\sigma \]

represent progressively larger probability regions.

However, the corresponding probabilities for multidimensional error ellipsoids are not identical to the one-dimensional interpretation.

4.1.7.83 RMS Error

Root-mean-square error is

\[ \boxed{ RMSE = \sqrt{ \frac1N \sum_{k=1}^{N} (z_k-z_{true,k})^2 }. } \]

RMSE combines bias and random variation.

For a scalar error with mean bias \(b\) and variance \(\sigma^2\),

\[ \boxed{ RMSE^2=b^2+\sigma^2. } \]

4.1.7.84 Monte Carlo Sensor Error Modelling

Monte Carlo simulation is essential for evaluating combined sensor uncertainties.

For run \(j\), sample uncertain parameters such as

\[ \boxed{ b^{(j)},\; s^{(j)},\; \delta\theta^{(j)},\; \sigma^{(j)},\; \tau^{(j)}. } \]

Then simulate

\[ \boxed{ \text{Truth} \rightarrow \text{Sensor Errors} \rightarrow \text{Estimator} \rightarrow \text{Guidance} \rightarrow \text{Controller} \rightarrow \text{Mission Outcome}. } \]

This captures nonlinear interaction between errors that cannot always be understood from an analytical RSS budget.

4.1.7.85 Fixed vs Run-to-Run vs Time-Varying Errors

This distinction is critical in Monte Carlo simulation.

Fixed parameter

Constant for every simulation:

\[ p=p_0. \]

Run-to-run uncertainty

Randomly sampled once per Monte Carlo run:

\[ p^{(j)}\sim\mathcal D_p. \]

Examples:

Time-varying uncertainty

Changes throughout a simulation:

\[ p=p(t). \]

Examples:

These should not be modelled identically.

4.1.7.86 Worst-Case Analysis vs Monte Carlo

Worst-case analysis asks whether requirements remain satisfied under extreme parameter combinations.

Monte Carlo estimates statistical performance.

Therefore,

\[ \boxed{ \text{Worst Case} \neq \text{Monte Carlo}. } \]

Both may be required for safety-critical spacecraft functions.

4.1.7.87 Sensor Error Budget

A complete sensor error budget can contain:

Error Source Type Typical Model Calibration Possible? Time Dependence
Bias Additive \(b\) Yes Constant / varying
Scale factor Multiplicative \(1+s\) Yes Slow
Misalignment Geometric \(C\) Yes Slow
White noise Stochastic \(n\) No Fast
Bias drift Stochastic \(b(t)\) Partly Slow
Quantisation Digital \(Q(z)\) No Sample dependent
Saturation Nonlinear \(\operatorname{sat}(z)\) No Input dependent
Temperature Environmental \(b(T),s(T)\) Often Slow
Latency Timing \(z(t-\tau)\) Often Fixed / varying
Dropout Availability \(valid=0\) No Event dependent

4.1.7.88 Error Sources by Sensor Type

Error Source Gyro Accelerometer Magnetometer Star Tracker Sun Sensor GNSS
Bias ✓ ✓ ✓ ✓ ✓ ✓
Scale factor ✓ ✓ ✓ —/limited ✓ —
Misalignment ✓ ✓ ✓ ✓ ✓ ✓
White noise ✓ ✓ ✓ ✓ ✓ ✓
Random walk ✓ ✓ possible — — clock/process dependent
Temperature ✓ ✓ ✓ ✓ ✓ ✓
Quantisation ✓ ✓ ✓ ✓ ✓ ✓
Saturation ✓ ✓ ✓ detector detector RF/tracking
Environmental interference vibration vibration magnetic stray light albedo/light RF/multipath
Dropout possible possible possible common eclipse visibility
Timing error ✓ ✓ ✓ critical relevant critical

The exact relevance and magnitude are sensor- and implementation-dependent.

4.1.7.89 Gyroscope Error Model

A useful complete model is

\[ \boxed{ \boldsymbol\omega_m = (I+S_g+M_g)\boldsymbol\omega + \mathbf b_g + K_{ga}\mathbf a + \mathbf n_g. } \]

with

\[ \dot{\mathbf b}_g = -\frac1{\tau_g}\mathbf b_g + \mathbf w_{bg} \]

or a random-walk model where appropriate.

Important gyro errors include:

4.1.7.90 Accelerometer Error Model

A representative model is

\[ \boxed{ \mathbf f_m = (I+S_a+M_a)\mathbf f + \mathbf b_a + \mathbf n_a. } \]

Additional effects may include:

4.1.7.91 Magnetometer Error Model

A representative model is

\[ \boxed{ \mathbf B_m = A_m\mathbf B_{true} + \mathbf b_h + \mathbf B_{spacecraft} + \mathbf n_m. } \]

where \(A_m\) can represent soft-iron, scale-factor and alignment effects, while \(\mathbf b_h\) represents hard-iron offset.

Magnetic contamination is often as important as the intrinsic sensor error.

4.1.7.92 Star-Tracker Error Model

A high-level model is

\[ \boxed{ q_m = \delta q_{err} \otimes q_{true}(t-\tau). } \]

The error quaternion can contain contributions from:

A validity flag should represent blinding and loss of tracking.

4.1.7.93 Sun-Sensor Error Model

For a vector-level Sun sensor,

\[ \boxed{ \mathbf s_m = C_{err}\mathbf s_{true} + \mathbf b_s + \mathbf n_s. } \]

For photodiode-based coarse Sun sensors, errors can additionally arise from:

During eclipse,

\[ \boxed{ valid_{Sun}=0. } \]

4.1.7.94 GNSS Error Model

At solution level,

\[ \boxed{ \mathbf r_m = \mathbf r + \mathbf b_r + \mathbf n_r } \]

and

\[ \boxed{ \mathbf v_m = \mathbf v + \mathbf b_v + \mathbf n_v. } \]

At raw-observable level,

\[ \boxed{ \rho_i = R_i + c(\delta t_r-\delta t_i) + I_i+T_i + \Delta\rho_{orbit} + \Delta\rho_{mp} + b_i + n_i. } \]

For spacecraft applications, geometry, antenna orientation, timing and signal availability are essential components of the error model.

4.1.7.95 Sensor Error and Control-System Performance

Measurement noise enters the feedback loop.

Conceptually,

\[ \boxed{ \text{Sensor Noise} \rightarrow \text{Estimated State Noise} \rightarrow \text{Control Command Variation} \rightarrow \text{Actuator Activity}. } \]

Excessive sensor noise may therefore increase:

Filtering must balance noise rejection against latency and dynamic response.

4.1.7.96 Sensor Error and Guidance Performance

Guidance operates on estimated states rather than truth.

\[ \boxed{ \hat{\mathbf x} = \mathbf x + \delta\mathbf x. } \]

Therefore,

\[ \mathbf u_G = G(\hat{\mathbf x}) \]

differs from the ideal guidance command.

For RPO, relative-navigation uncertainty affects:

4.1.7.97 Sensor Error and Autonomous Mode Logic

Sensor errors also influence discrete spacecraft decisions.

Examples include:

\[ \boxed{ \text{Sun Detected?} } \] \[ \boxed{ \text{Star Tracker Valid?} } \] \[ \boxed{ \text{GNSS Fix Valid?} } \] \[ \boxed{ \text{Capture Conditions Satisfied?} } \]

Thus sensor uncertainty affects not only continuous estimation but also spacecraft state-machine transitions.

4.1.7.98 Requirement Allocation

Mission-level performance requirements should be flowed down to sensor requirements.

Conceptually,

\[ \boxed{ \text{Mission Requirement} \rightarrow \text{GNC Requirement} \rightarrow \text{Navigation Error Budget} \rightarrow \text{Sensor Requirement}. } \]

For example,

\[ \sigma_{position,mission} \]

may be allocated among:

Sensor selection should therefore begin from system requirements rather than datasheet comparison alone.

4.1.7.99 Calibration vs Estimation vs Filtering

These terms solve different problems.

Calibration

Determines systematic sensor parameters.

\[ \boxed{ \text{Calibration} \rightarrow b,S,M,\ldots } \]

Estimation

Determines unknown spacecraft states and sometimes sensor errors.

\[ \boxed{ \text{Estimator} \rightarrow \hat x,\hat b,\ldots } \]

Filtering

Reduces unwanted measurement variation and estimates state using models and measurements.

They complement one another.

4.1.7.100 Error Mitigation Strategies

Sensor errors can be addressed using several mechanisms:

  1. better sensor hardware,
  2. mechanical/thermal design,
  3. careful placement,
  4. electromagnetic cleanliness,
  5. preflight calibration,
  6. temperature compensation,
  7. in-orbit calibration,
  8. filtering,
  9. sensor fusion,
  10. bias estimation,
  11. redundancy,
  12. FDIR,
  13. measurement gating,
  14. adaptive covariance,
  15. operational constraints.

No single technique removes all sensor errors.

4.1.7.101 Sensor Error Simulation Architecture

A useful generic simulation chain is 

Truth State
    |
    v
Ideal Measurement h(x)
    |
    v
Frame Transformation
    |
    v
Scale Factor
    |
    v
Misalignment / Cross-Axis Error
    |
    v
Nonlinearity
    |
    v
Bias
    |
    v
Bias Drift
    |
    v
Environmental Effects
    |
    v
White / Coloured Noise
    |
    v
Bandwidth / Sensor Dynamics
    |
    v
Saturation
    |
    v
Quantisation
    |
    v
Sampling
    |
    v
Latency / Timestamp
    |
    v
Dropout / Fault Logic
    |
    v
Measurement + Covariance + Validity
    |
    v
Navigation Filter

The exact ordering should reflect the physics of the particular sensor; this chain is a modelling framework rather than a universal physical sequence.

4.1.7.102 Simulink Implementation

A reusable Simulink sensor-error subsystem can contain: 

Truth Signal
     |
     v
Coordinate Transform
     |
     v
Calibration Matrix
     |
     +------ Bias
     |
     +------ Bias Drift
     |
     +------ Temperature Model
     |
     +------ Noise
     |
     v
Sensor Dynamics
     |
     v
Saturation
     |
     v
Quantizer
     |
     v
Zero-Order Hold
     |
     v
Transport Delay
     |
     v
Validity / Fault Logic
     |
     v
Sensor Output

Outputs should ideally include

\[ \boxed{ \mathbf z_m,\quad R,\quad t_m,\quad valid,\quad status. } \]

4.1.7.103 Low-, Medium- and High-Fidelity Models

Level 1 — Simple Measurement Model

\[ \boxed{ z_m=z_{true}+b+n. } \]

Suitable for early controller development.

Level 2 — Engineering Sensor Model

\[ \boxed{ \mathbf z_m = (I+S+M)\mathbf z_{true} + \mathbf b(t,T) + \mathbf n. } \]

Add:

Suitable for GNC design and Monte Carlo analysis.

Level 3 — Physics-Based Sensor Model

Model:

Suitable for high-fidelity SIL/HIL and verification.

4.1.7.104 Sensor Datasheet Parameters

When reading a datasheet, do not look only at “accuracy.”

Important parameters include:

Sensor-specific pages then add parameters such as star-tracker exclusion angles or GNSS signal/tracking capabilities.

4.1.7.105 Datasheet Value vs Simulation Parameter

A datasheet specification is not always directly usable as a simulation standard deviation.

For example,

\[ \boxed{ \text{Noise Density} \neq \text{Per-Sample }\sigma } \]

without knowing bandwidth and convention.

Similarly,

\[ \boxed{ \text{Maximum Error} \neq 1\sigma. } \]

And

\[ \boxed{ \text{Bias Stability} \neq \text{Constant Bias}. } \]

The statistical meaning of each specification must be understood before using it in Monte Carlo simulation.

4.1.7.106 Units and Conversions

Sensor error modelling is especially vulnerable to unit mistakes.

Examples include:

\[ ^\circ/s \leftrightarrow rad/s \] \[ \mu g \leftrightarrow m/s^2 \] \[ nT \leftrightarrow T \] \[ arcsec \leftrightarrow rad. \]

For angular conversion,

\[ \boxed{ 1^\circ=\frac{\pi}{180}\;rad } \]

and

\[ \boxed{ 1\;arcsec = \frac{\pi}{180\times3600}\;rad. } \]

Every simulation parameter should therefore carry an explicit unit.

4.1.7.107 Error Correlation Time

The magnitude of an error is not enough to characterize it.

Two noise processes may have the same RMS but very different time behaviour.

Thus a stochastic model may require both

\[ \boxed{ \sigma } \]

and

\[ \boxed{ \tau_c } \]

where \(\tau_c\) is correlation time.

This strongly affects estimator performance.

4.1.7.108 Sensor Error State Machine

Some sensor errors are mode-dependent.

A sensor can transition through

\[ \boxed{ \text{OFF} \rightarrow \text{INITIALIZING} \rightarrow \text{ACQUIRING} \rightarrow \text{VALID} \rightarrow \text{DEGRADED} \rightarrow \text{FAILED}. } \]

Examples:

This is often more realistic than assuming continuous availability.

4.1.7.109 Adaptive Measurement Covariance

Sensor quality can change with operating conditions.

Instead of constant

\[ R=R_0, \]

use

\[ \boxed{ R=R(\text{quality},\text{geometry},\text{environment}). } \]

Examples:

This allows the estimator to weight measurements according to their actual quality.

4.1.7.110 Sensor Fault Injection

Verification should deliberately inject faults.

Examples include:

\[ b\rightarrow b+\Delta b \] \[ s\rightarrow s+\Delta s \] \[ valid\rightarrow0 \] \[ z\rightarrow z_{stuck} \] \[ z\rightarrow z+A_{spike} \] \[ \tau\rightarrow\tau+\Delta\tau. \]

This allows verification of:

4.1.7.111 Verification Against Truth

Simulation provides access to truth.

Therefore measurement error can be calculated as

\[ \boxed{ \mathbf e_k = \mathbf z_{m,k} - \mathbf z_{true,k}. } \]

Useful statistics include:

\[ \bar{\mathbf e}, \] \[ \sigma_e, \] \[ RMSE, \] \[ e_{max}, \]

and error PSD/Allan deviation where appropriate.

Estimator consistency can additionally be evaluated using covariance-normalized metrics.

4.1.7.112 Sensor Error Budget to Navigation Error Budget

Individual sensor specifications are not the final engineering objective.

The important chain is

\[ \boxed{ \text{Sensor Errors} } \] \[ \downarrow \] \[ \boxed{ \text{Measurement Covariance} } \] \[ \downarrow \] \[ \boxed{ \text{Navigation State Covariance} } \] \[ \downarrow \] \[ \boxed{ \text{Guidance / Control Uncertainty} } \] \[ \downarrow \] \[ \boxed{ \text{Mission Performance}. } \]

This is the connection between sensor engineering and spacecraft GNC.

4.1.7.113 Complete Generalized Sensor Model

A useful high-level model combining the major error classes is

\[ \boxed{ \begin{aligned} \mathbf z_m(t_k) ={}& \mathcal Q \Big[ \mathcal D \big( A(T,t)\, C_{B}^{S}(t)\, h[\mathbf x(t_k-\tau)] \\ &+ \mathbf b(T,t) + \mathbf e_{env}(t) + \mathbf e_{dyn}(t) + \mathbf n(t) \big) \Big] \end{aligned} } \]

subject to

\[ \boxed{ z_{min}\le z_m\le z_{max} } \]

and

\[ \boxed{ valid_k\in\{0,1\}. } \]

Here:

Not every sensor requires every term.

4.1.7.114 Complete Error-Management Chain

At spacecraft-system level:

\[ \boxed{ \text{Sensor Physics} \rightarrow \text{Error Characterization} \rightarrow \text{Calibration} \rightarrow \text{Measurement Model} } \] \[ \downarrow \] \[ \boxed{ \text{Covariance Modelling} \rightarrow \text{Sensor Fusion} \rightarrow \text{Innovation Monitoring} } \] \[ \downarrow \] \[ \boxed{ \text{FDIR} \rightarrow \text{Navigation Estimate} \rightarrow \text{Guidance} \rightarrow \text{Control}. } \]

4.1.7.115 Final Engineering Perspective

The most important point of this chapter is:

\[ \boxed{ \text{Sensor Error} \neq \text{White Noise Only}. } \]

A realistic spacecraft sensor can contain

\[ \boxed{ \text{Bias} + \text{Drift} + \text{Scale Factor} + \text{Misalignment} + \text{Nonlinearity} + \text{Noise} + \text{Environment} + \text{Timing} + \text{Digital Effects} + \text{Availability}. } \]

These errors also behave differently:

\[ \boxed{ \text{Constant} \neq \text{Run-to-Run} \neq \text{Time-Varying} \neq \text{Fault}. } \]

The complete spacecraft engineering philosophy should therefore be

\[ \boxed{ \text{Characterize} \rightarrow \text{Calibrate} \rightarrow \text{Model} \rightarrow \text{Estimate} \rightarrow \text{Monitor} \rightarrow \text{Mitigate}. } \]

And ultimately,

\[ \boxed{ \text{Sensor Error Model} \rightarrow \text{Navigation Uncertainty} \rightarrow \text{GNC Performance} \rightarrow \text{Mission Safety and Success}. } \]

This gives 4.1.7 a different and important role from the individual sensor pages: the gyro, accelerometer, magnetometer, star-tracker, Sun-sensor, and GNSS pages explain the errors of each instrument, while Sensor Error Sources explains the common mathematical and engineering framework connecting all of those errors to calibration, stochastic modelling, Kalman filtering, Monte Carlo analysis, FDIR, and spacecraft-level GNC performance.