Accelerometers

4.1.2 Accelerometers

Accelerometers are fundamental inertial sensors used in spacecraft, launch vehicles, aircraft, drones, planetary vehicles, and inertial navigation systems.

Their most important physical interpretation is that an accelerometer does not directly measure total inertial acceleration. Instead, it measures the non-gravitational acceleration acting on the sensor, known as specific force.

This distinction can be expressed as

\[ \boxed{ \text{Accelerometer does not directly measure inertial acceleration} } \]

The specific-force vector is

\[ \boxed{ \mathbf f = \mathbf a-\mathbf g } \]

where

• \(\mathbf f\) is specific force,
• \(\mathbf a\) is inertial acceleration,
• \(\mathbf g\) is gravitational acceleration.

For a three-axis accelerometer,

\[ \mathbf f = \begin{bmatrix} f_x\\ f_y\\ f_z \end{bmatrix}. \]

A simple accelerometer measurement model is

\[ \boxed{ \mathbf f_m = \mathbf f_{\text{true}} + \mathbf b_a + \mathbf n_a } \]

where \(\mathbf b_a\) is accelerometer bias and \(\mathbf n_a\) represents random measurement noise.

A more complete three-axis model includes scale-factor and alignment errors:

\[ \boxed{ \mathbf f_m = \left( \mathbf I+\mathbf S_a+\mathbf M_a \right) \mathbf f_{\text{true}} + \mathbf b_a + \mathbf n_a } \]

where \(\mathbf S_a\) represents scale-factor error and \(\mathbf M_a\) represents axis misalignment and cross-axis sensitivity.

4.1.2.1 Role of Accelerometers in GNC

Accelerometers provide information about translational motion produced by non-gravitational forces acting on a vehicle.

These measurements are central to inertial navigation because specific force can be transformed into a navigation frame, combined with a gravity model, and integrated to estimate velocity and position.

Typical GNC applications include

• inertial navigation,
• velocity propagation,
• position propagation,
• launch-vehicle navigation,
• spacecraft manoeuvre reconstruction,
• thrust monitoring,
• drag estimation,
• propulsion-system performance monitoring,
• IMU/GNSS sensor fusion,
• entry, descent, and landing,
• planetary navigation,
• rendezvous and proximity operations,
• fault detection and isolation.

The basic information chain is

\[ \boxed{\text{Applied Forces}} \rightarrow \boxed{\text{Vehicle Motion}} \rightarrow \boxed{\text{Accelerometer}} \rightarrow \boxed{\mathbf f_m} \rightarrow \boxed{\text{Navigation}} \rightarrow \boxed{\hat{\mathbf v},\hat{\mathbf r}}. \]

4.1.2.2 What an Accelerometer Actually Measures

Saying that an accelerometer “measures acceleration” is incomplete. The quantity measured by an ideal accelerometer is specific force.

The fundamental relationship is

\[ \boxed{ \mathbf f = \mathbf a-\mathbf g } \]

or, equivalently,

\[ \boxed{ \mathbf a = \mathbf f+\mathbf g }. \]

Therefore, if the navigation system knows the local gravitational acceleration and measures specific force, the inertial acceleration can be reconstructed.

Units

Specific force may be expressed in

\[ \mathrm{m/s^2} \]

or in multiples of standard gravity,

\[ g_0 \approx 9.80665~\mathrm{m/s^2}. \]

4.1.2.3 Specific Force and the Gravity Relationship

The meaning of specific force can be derived directly from Newton's second law.

Suppose the vehicle is acted upon by gravitational force \(\mathbf F_g\) and non-gravitational force \(\mathbf F_{ng}\):

\[ m\mathbf a = \mathbf F_g + \mathbf F_{ng}. \]

Since

\[ \mathbf F_g = m\mathbf g, \]

we obtain

\[ m\mathbf a = m\mathbf g + \mathbf F_{ng}. \]

Rearranging,

\[ \mathbf a-\mathbf g = \frac{\mathbf F_{ng}}{m}. \]

Therefore,

\[ \boxed{ \mathbf f = \frac{\mathbf F_{ng}}{m} = \mathbf a-\mathbf g }. \]

Thus an ideal accelerometer responds to non-gravitational force per unit mass.

4.1.2.4 Why an Accelerometer Reads Approximately 1g While Stationary

Consider an accelerometer resting on a stationary horizontal table.

Its inertial acceleration is approximately

\[ \mathbf a=0. \]

Suppose gravity acts along the negative \(z\)-direction:

\[ \mathbf g = \begin{bmatrix} 0\\ 0\\ -g \end{bmatrix}. \]

The specific-force equation gives

\[ \mathbf f = \mathbf a-\mathbf g = \begin{bmatrix} 0\\ 0\\ g \end{bmatrix}. \]

The sensor therefore reports approximately \(1g\) along the upward sensing axis.

Physically, the table provides a normal force that prevents the sensor from falling. The accelerometer responds to this support force.

4.1.2.5 Why an Accelerometer Reads Approximately Zero in Free Fall

Consider an ideal object undergoing gravitational free fall.

Its inertial acceleration is

\[ \mathbf a=\mathbf g. \]

Therefore,

\[ \mathbf f = \mathbf a-\mathbf g = 0. \]

Hence,

\[ \boxed{ \mathbf f=0 } \]

for ideal gravitational free fall.

This is particularly important for spacecraft. An orbiting spacecraft may have substantial inertial acceleration, but it is still in free fall. Therefore, an ideal onboard accelerometer may read close to zero during unforced orbital coast.

Thus,

\[ \boxed{ \text{Orbital acceleration} \neq \text{accelerometer output} }. \]

4.1.2.6 Physical Operating Principle

Most accelerometers contain a proof mass suspended inside a sensor housing.

When the housing experiences non-gravitational acceleration, the proof mass moves relative to the housing. The resulting displacement or force is measured and converted into an electrical output.

A simplified mass-spring-damper model is

\[ m_p\ddot{x} + c\dot{x} + kx = -m_pa, \]

where

• \(m_p\) is proof-mass mass,
• \(c\) is damping,
• \(k\) is suspension stiffness,
• \(x\) is proof-mass displacement.

At sufficiently low frequencies,

\[ kx \approx -m_pa, \]

so

\[ x\propto a. \]

MEMS Accelerometers

Modern MEMS accelerometers commonly use capacitive sensing.

Proof-mass displacement changes the differential capacitance:

\[ \Delta C\propto x. \]

Since

\[ x\propto f, \]

the electrical signal can be calibrated directly in units of specific force.

Other accelerometer technologies include piezoresistive, piezoelectric, resonant, force-rebalance, servo, and optical accelerometers.

4.1.2.7 Single-Axis and Three-Axis Accelerometers

A single-axis accelerometer measures specific force along one sensitive axis.

For example,

\[ f_x. \]

A three-axis accelerometer measures

\[ \boxed{ \mathbf f_m = \begin{bmatrix} f_{mx}\\ f_{my}\\ f_{mz} \end{bmatrix} }. \]

Ideally,

\[ f_{mx}=f_x, \qquad f_{my}=f_y, \qquad f_{mz}=f_z. \]

In practice, each axis has its own bias, scale-factor error, noise, alignment error, bandwidth, and saturation limit.

4.1.2.8 Accelerometer Coordinate Frames

The accelerometer measures specific force in its own sensor frame, denoted here by \(A\).

Let \(B\) denote the spacecraft body frame.

The specific-force vector in the accelerometer frame is

\[ \boxed{ \mathbf f^A = C_B^A \mathbf f^B }. \]

For perfect sensor alignment,

\[ C_B^A=I. \]

If the navigation solution is expressed in frame \(N\),

\[ \boxed{ \mathbf f^N = C_B^N C_A^B \mathbf f^A }. \]

Thus accelerometer modelling depends directly on the attitude transformations introduced earlier using DCMs and quaternions.

4.1.2.9 Lever-Arm Effects

If the accelerometer is not located at the vehicle centre of mass, rotational motion can create additional acceleration at the sensor location.

Let

\[ \mathbf r_{BA} \]

denote the vector from the body reference point or centre of mass to the accelerometer.

For a rigidly mounted sensor,

\[ \boxed{ \mathbf a_A = \mathbf a_B + \dot{\boldsymbol{\omega}} \times \mathbf r_{BA} + \boldsymbol{\omega} \times \left( \boldsymbol{\omega} \times \mathbf r_{BA} \right) }. \]

The term

\[ \dot{\boldsymbol{\omega}} \times \mathbf r_{BA} \]

represents tangential acceleration.

The term

\[ \boldsymbol{\omega} \times \left( \boldsymbol{\omega} \times \mathbf r_{BA} \right) \]

represents centripetal acceleration.

Lever-arm effects become especially important during rapid rotations or when the accelerometer is mounted far from the vehicle centre of mass.

4.1.2.10 Ideal Accelerometer Model

For an ideal accelerometer,

\[ \boxed{ \mathbf f_m = \mathbf f_{\text{true}} } \]

where

\[ \mathbf f_{\text{true}} = \mathbf a-\mathbf g. \]

The ideal model assumes

• zero bias,
• perfect scale factor,
• perfect alignment,
• zero cross-axis sensitivity,
• zero noise,
• infinite resolution,
• unlimited measurement range,
• zero latency,
• perfect thermal stability.

4.1.2.11 Constant Bias

Accelerometer bias is an additive offset in the measured specific-force signal.

For a three-axis accelerometer,

\[ \boxed{ \mathbf f_m = \mathbf f_{\text{true}} + \mathbf b_a } \]

where

\[ \mathbf b_a = \begin{bmatrix} b_{ax}\\ b_{ay}\\ b_{az} \end{bmatrix}. \]

For one axis,

\[ f_m = f_{\text{true}} + b_a. \]

If the true specific force is zero,

\[ f_{\text{true}}=0, \]

a biased accelerometer may still report

\[ f_m=b_a. \]

4.1.2.12 Why Accelerometer Bias Is Dangerous for Navigation

Accelerometer bias is particularly damaging in inertial navigation because acceleration is integrated once to estimate velocity and twice to estimate position.

For a constant acceleration bias \(b_a\),

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

Thus,

\[ \boxed{ \delta v(t)\propto t }. \]

Position error grows as

\[ \delta r(t) \approx \frac{1}{2} b_a t^2. \]

Therefore,

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

Even a small accelerometer bias can therefore cause a rapidly increasing navigation error when no external correction is available.

4.1.2.13 Bias Drift and Bias Random Walk

Real accelerometer bias changes slowly with time because of thermal effects, electronic drift, aging, mechanical stress, radiation, and stochastic sensor processes.

A random-walk bias model is

\[ \boxed{ \dot{\mathbf b}_a = \mathbf n_{ba} } \]

where \(\mathbf n_{ba}\) is a stochastic driving process.

In discrete time,

\[ \boxed{ \mathbf b_{a,k+1} = \mathbf b_{a,k} + \mathbf w_{ba,k} }. \]

A bounded first-order Gauss-Markov model can instead be used:

\[ \boxed{ \dot{\mathbf b}_a = -\frac{1}{\tau_a}\mathbf b_a + \mathbf n_{ba} }. \]

Here, \(\tau_a\) represents the bias correlation time.

4.1.2.14 Scale-Factor Error

Scale-factor error represents an incorrect sensor sensitivity.

For one axis,

\[ \boxed{ f_m = (1+s_a) f_{\text{true}} } \]

where \(s_a\) is the fractional scale-factor error.

For a three-axis accelerometer,

\[ S_a = \begin{bmatrix} s_x&0&0\\ 0&s_y&0\\ 0&0&s_z \end{bmatrix}. \]

The measurement becomes

\[ \boxed{ \mathbf f_m = (I+S_a) \mathbf f_{\text{true}} }. \]

Scale-factor error becomes increasingly important as the magnitude of applied specific force increases.

4.1.2.15 Axis Misalignment

Ideally, the three accelerometer sensing axes are mutually orthogonal and aligned with the nominal sensor coordinate frame.

A small misalignment matrix may be written as

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

The measurement becomes

\[ \boxed{ \mathbf f_m = (I+M_a) \mathbf f_{\text{true}} }. \]

For the \(x\)-axis,

\[ f_{mx} = f_x + m_{xy}f_y + m_{xz}f_z. \]

Thus acceleration along another axis can appear as a false component in the \(x\)-axis measurement.

4.1.2.16 Cross-Axis Sensitivity

Cross-axis sensitivity describes the unwanted response of one sensing axis to acceleration applied along another axis.

Ideally,

\[ f_x \rightarrow f_{mx}. \]

In practice,

\[ f_y,f_z \rightarrow f_{mx} \]

may also occur.

Sources include manufacturing tolerances, sensor non-orthogonality, packaging, mounting errors, and structural deformation.

Calibration matrices can be used to compensate for these effects.

4.1.2.17 White Noise and Noise Density

Accelerometer outputs contain stochastic high-frequency noise.

A simple model is

\[ \boxed{ \mathbf f_m = \mathbf f_{\text{true}} + \mathbf n_a }. \]

Assuming zero-mean noise,

\[ E[\mathbf n_a]=0. \]

Ideal white noise may be represented as

\[ E \left[ \mathbf n_a(t) \mathbf n_a^T(t') \right] = Q_a\delta(t-t'). \]

Noise Density

Datasheets may specify accelerometer noise density in

\[ \frac{\mathrm{m/s^2}}{\sqrt{\mathrm{Hz}}} \]

or

\[ \frac{\mu g}{\sqrt{\mathrm{Hz}}}. \]

For effective bandwidth \(B\),

\[ \boxed{ \sigma_a \propto N_a\sqrt{B} } \]

where \(N_a\) is the acceleration noise density.

4.1.2.18 Velocity Random Walk

White acceleration noise accumulates when accelerometer measurements are integrated to obtain velocity.

The resulting effect is called Velocity Random Walk, or VRW.

Conceptually,

\[ \boxed{ \sigma_v(t) \propto N_a\sqrt{t} } \]

for white acceleration noise.

VRW is the translational analogue of gyro Angle Random Walk.

4.1.2.19 Bias Instability

Even under constant environmental and mechanical conditions, the accelerometer zero-input output may vary slowly.

Conceptually,

\[ f_m(t) = b_a(t) + n_a(t) \]

when the true specific-force input is zero.

This slowly varying offset is referred to as bias instability.

It is particularly important for inertial navigation because a persistent low-frequency acceleration error is integrated into velocity and position.

4.1.2.20 Allan Variance / Allan Deviation

Allan variance and Allan deviation provide a useful way to identify stochastic sensor-error processes from long-duration stationary accelerometer measurements.

Different slopes on an Allan-deviation plot can indicate different stochastic mechanisms.

These may include

• quantisation noise,
• white acceleration noise,
• Velocity Random Walk,
• bias instability,
• acceleration random walk,
• long-term drift.

The practical workflow is

\[ \boxed{\text{Raw Accelerometer Data}} \rightarrow \boxed{\text{Allan Deviation}} \rightarrow \boxed{\text{Noise Parameters}} \rightarrow \boxed{\text{Simulation / Filter Model}}. \]

This is especially useful when building realistic IMU models from actual sensor data.

4.1.2.21 Quantisation and Resolution

Digital accelerometers represent measurements using a finite number of discrete levels.

If the smallest output increment is

\[ \Delta f, \]

the quantised measurement is approximately

\[ \boxed{ f_q = \Delta f \operatorname{round} \left( \frac{f}{\Delta f} \right) }. \]

The corresponding quantisation error lies approximately within

\[ -\frac{\Delta f}{2} \leq e_q \leq \frac{\Delta f}{2}. \]

Therefore, physical changes smaller than the sensor resolution may not produce a distinguishable digital output change.

4.1.2.22 Saturation and Dynamic Range

Accelerometers can measure only a finite specific-force range.

Suppose

\[ -f_{\max} \leq f \leq f_{\max}. \]

The saturated measurement can be represented as

\[ f_m = \begin{cases} -f_{\max}, & f<-f_{\max}\\[4pt] f, & |f|\leq f_{\max}\\[4pt] +f_{\max}, & f>f_{\max}. \end{cases} \]

The required dynamic range strongly depends on mission type.

Launch vehicles may experience large accelerations, whereas precision orbital sensors may need to resolve extremely small non-gravitational forces.

This creates a design trade-off between

\[ \boxed{\text{Large Dynamic Range}} \qquad \text{and} \qquad \boxed{\text{High Sensitivity}}. \]

4.1.2.23 Temperature Effects

Accelerometer bias and scale factor may vary with temperature.

A simple linear bias-temperature model is

\[ \boxed{ b_a(T) = b_0+k_T(T-T_0) }. \]

A higher-order model may be

\[ b_a(T) = b_0 + k_1(T-T_0) + k_2(T-T_0)^2. \]

Scale factor may also be temperature-dependent:

\[ s_a=s_a(T). \]

Spacecraft thermal cycling can therefore change accelerometer characteristics unless appropriate calibration and thermal compensation are applied.

4.1.2.24 Nonlinearity

An ideal accelerometer has a linear relationship between physical input and measured output.

Ideally,

\[ f_m = Kf_{\text{true}}. \]

A real sensor may contain higher-order terms:

\[ \boxed{ f_m = K_1f + K_2f^2 + K_3f^3 + \cdots }. \]

Nonlinearity may become more significant near the limits of the sensor's dynamic range.

4.1.2.25 Bandwidth, Sampling and Latency

Real accelerometers do not produce continuous, instantaneous, infinite-bandwidth measurements.

Sampling occurs at

\[ t_k=kT_s \]

with

\[ \boxed{ f_s = \frac{1}{T_s} }. \]

A typical measurement chain may contain

\[ \boxed{\text{Mechanical Sensor}} \rightarrow \boxed{\text{Analog Filter}} \rightarrow \boxed{\text{ADC}} \rightarrow \boxed{\text{Digital Filter}} \rightarrow \boxed{\text{Sample Output}}. \]

This introduces

• finite bandwidth,
• phase lag,
• sampling effects,
• aliasing,
• latency,
• digital-filter delay.

Measurement latency can be particularly important in high-bandwidth control loops.

4.1.2.26 Complete Three-Axis Accelerometer Measurement Model

Combining the major deterministic and stochastic effects gives

\[ \boxed{ \mathbf f_m = (\mathbf I+\mathbf S_a+\mathbf M_a) C_B^A \mathbf f_{\text{true}}^B + \mathbf b_a + \mathbf n_a }. \]

A more complete simulation-level model is

\[ \boxed{ \mathbf f_m = Q \left[ \operatorname{sat} \left( (\mathbf I+\mathbf S_a+\mathbf M_a) C_B^A \mathbf f_{\text{true}}^B + \mathbf b_a(T) + \mathbf n_a \right) \right] }. \]

where

• \(C_B^A\): body-to-accelerometer mounting transformation,
• \(\mathbf S_a\): scale-factor error matrix,
• \(\mathbf M_a\): misalignment/cross-axis matrix,
• \(\mathbf b_a(T)\): temperature-dependent bias,
• \(\mathbf n_a\): measurement noise,
• \(\operatorname{sat}(\cdot)\): saturation operator,
• \(Q[\cdot]\): quantisation operator.

If lever-arm effects are important, the true specific force at the accelerometer location should be calculated before this sensor model is applied.

4.1.2.27 Continuous-Time Accelerometer Model

A basic continuous-time accelerometer model is

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

Bias dynamics may be represented as a random walk:

\[ \boxed{ \dot{\mathbf b}_a(t) = \mathbf n_{ba}(t) }. \]

Alternatively,

\[ \boxed{ \dot{\mathbf b}_a = -\frac{1}{\tau_a} \mathbf b_a + \mathbf n_{ba} }. \]

These models are useful for continuous spacecraft simulation and sensor dynamics studies.

4.1.2.28 Discrete-Time Accelerometer Model

For onboard digital processing,

\[ t_k=kT_s. \]

The measurement model becomes

\[ \boxed{ \mathbf f_{m,k} = \mathbf f_k + \mathbf b_{a,k} + \mathbf v_{a,k} }. \]

A random-walk bias model is

\[ \boxed{ \mathbf b_{a,k+1} = \mathbf b_{a,k} + \mathbf w_{a,k} }. \]

A discrete first-order Gauss-Markov model is

\[ \boxed{ \mathbf b_{a,k+1} = e^{-T_s/\tau_a} \mathbf b_{a,k} + \mathbf w_{a,k} }. \]

This form is suitable for Kalman-filter and EKF implementation.

4.1.2.29 Simulink Accelerometer Model

A practical Simulink model should reproduce the complete signal chain from true vehicle motion to the measurement supplied to the navigation algorithm.

True Vehicle Acceleration
          |
          v
    Subtract Gravity
          |
          v
      Specific Force
          |
          v
  Lever-Arm Correction
     (if required)
          |
          v
   Body-to-Sensor Frame
          |
          v
    Scale-Factor Error
          |
          v
      Misalignment
          |
          v
        SUM  <----- Constant Bias
         ^   <----- Bias Drift
         ^   <----- White Noise
          |
          v
      Saturation
          |
          v
      Quantisation
          |
          v
 Sampling / ZOH / Delay
          |
          v
 Measured Specific Force
          |
          v
     Navigation / EKF
    

Recommended Blocks

• Sum block for \(\mathbf a-\mathbf g\),
• Matrix Gain for frame transformation,
• Matrix Gain for scale factor and misalignment,
• Constant block for fixed bias,
• Band-Limited White Noise block,
• Integrator for bias random walk,
• Saturation block,
• Quantizer block,
• Zero-Order Hold,
• Discrete Delay or Transport Delay,
• To Workspace for logging.

4.1.2.30 Example Accelerometer Measurement

Suppose the true specific force is

\[ \mathbf f_{\text{true}} = \begin{bmatrix} 0.20\\ -0.10\\ 0.05 \end{bmatrix} \mathrm{m/s^2}. \]

Let the accelerometer bias be

\[ \mathbf b_a = \begin{bmatrix} 0.01\\ -0.005\\ 0.002 \end{bmatrix} \mathrm{m/s^2}. \]

Assume one noise realization

\[ \mathbf n_a = \begin{bmatrix} 0.003\\ 0.002\\ -0.001 \end{bmatrix} \mathrm{m/s^2}. \]

Ignoring scale-factor and alignment errors,

\[ \mathbf f_m = \mathbf f_{\text{true}} + \mathbf b_a + \mathbf n_a. \]

Therefore,

\[ \boxed{ \mathbf f_m = \begin{bmatrix} 0.213\\ -0.103\\ 0.051 \end{bmatrix} \mathrm{m/s^2} }. \]

The navigation system receives the measured quantity, not the ideal physical specific force.

4.1.2.31 Acceleration → Velocity → Position Propagation

The accelerometer measurement is first corrected for estimated bias.

In the navigation frame,

\[ \hat{\mathbf f}^N = C_B^N \left( \mathbf f_m^B - \hat{\mathbf b}_a \right). \]

Gravity is then restored:

\[ \boxed{ \hat{\mathbf a}^N = \hat{\mathbf f}^N + \mathbf g^N }. \]

Velocity is propagated using

\[ \boxed{ \dot{\hat{\mathbf v}} = \hat{\mathbf a} }. \]

Position is propagated using

\[ \boxed{ \dot{\hat{\mathbf r}} = \hat{\mathbf v} }. \]

Thus the fundamental navigation chain is

\[ \boxed{\mathbf f_m} \rightarrow \boxed{\text{Bias Correction}} \rightarrow \boxed{\text{Frame Transformation}} \rightarrow \boxed{+\mathbf g} \rightarrow \boxed{\mathbf a} \rightarrow \boxed{\mathbf v} \rightarrow \boxed{\mathbf r}. \]

4.1.2.32 Why Attitude Error Affects Accelerometer Navigation

Accelerometer measurements are usually obtained in the body frame and rotated into the navigation frame using an estimated attitude.

Ideally,

\[ \mathbf f^N = C_B^N \mathbf f^B. \]

If the estimated attitude is incorrect,

\[ \hat C_B^N \neq C_B^N, \]

then the transformed specific force is also incorrect.

A small tilt error can project part of gravity into a horizontal axis.

For a small angular error \(\delta\theta\),

\[ \boxed{ \delta a \approx g\,\delta\theta } \]

where \(\delta\theta\) is expressed in radians.

This produces an important GNC coupling:

\[ \boxed{\text{Gyro Error}} \rightarrow \boxed{\text{Attitude Error}} \rightarrow \boxed{\text{Incorrect Accelerometer Rotation}} \rightarrow \boxed{\text{Velocity Error}} \rightarrow \boxed{\text{Position Error}}. \]

4.1.2.33 Accelerometer + Gyroscope = IMU

An Inertial Measurement Unit generally combines three gyroscope axes and three accelerometer axes.

Its six-axis measurement vector can be written as

\[ \boxed{ \mathbf z_{\mathrm{IMU}} = \begin{bmatrix} \boldsymbol{\omega}_m\\ \mathbf f_m \end{bmatrix} }. \]

The six measurement channels are

\[ \omega_x,\omega_y,\omega_z, f_x,f_y,f_z. \]

The gyroscope provides information for attitude propagation:

\[ \boldsymbol{\omega}_m \rightarrow \hat{\mathbf q}, \]

while the accelerometer supports translational propagation:

\[ \mathbf f_m \rightarrow \hat{\mathbf v} \rightarrow \hat{\mathbf r}. \]

Together,

\[ \boxed{\text{Gyroscope + Accelerometer}} \rightarrow \boxed{\text{IMU}} \rightarrow \boxed{\text{Inertial Navigation}}. \]

4.1.2.34 Strapdown Inertial Navigation

Modern IMUs are generally implemented as strapdown systems, meaning that the sensors are rigidly attached to the vehicle.

The accelerometer therefore measures

\[ \mathbf f^B. \]

The navigation computer must rotate this quantity into the navigation frame:

\[ \boxed{ \mathbf f^N = C_B^N \mathbf f^B }. \]

The attitude matrix \(C_B^N\) is itself obtained using gyroscope measurements.

Gyroscope
    |
    v
Angular Rate
    |
    v
Attitude Propagation
    |
    v
   C_B^N
    |
    +----------------------+
                           |
Accelerometer              |
    |                      |
    v                      v
Specific Force ------> Rotate to Navigation Frame
                           |
                           v
                       Add Gravity
                           |
                           v
                       Acceleration
                           |
                           v
                        Velocity
                           |
                           v
                        Position
    

This is the fundamental architecture of a strapdown inertial-navigation system.

4.1.2.35 Accelerometer → EKF Bias Estimation

Accelerometer bias can be included directly in the navigation estimator state.

A representative error-state vector is

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

Here,

• \(\delta\mathbf r\) is position error,
• \(\delta\mathbf v\) is velocity error,
• \(\delta\boldsymbol{\theta}\) is attitude error,
• \(\mathbf b_g\) is gyro bias,
• \(\mathbf b_a\) is accelerometer bias.

The corrected accelerometer measurement is

\[ \boxed{ \mathbf f_{\mathrm{corr}} = \mathbf f_m - \hat{\mathbf b}_a }. \]

External navigation measurements such as GNSS position and velocity can be compared with the inertially propagated position and velocity.

The resulting innovation allows the EKF to update position, velocity, attitude errors, gyro bias, and accelerometer bias.

Accelerometer
      |
      v
Measured Specific Force
      |
      v
Subtract Estimated Bias
      |
      v
Navigation Propagation
      |
      v
Predicted Position / Velocity
      |
      +---------------------------+
                                  |
GNSS                              |
 |                                |
 v                                v
Measured Position / Velocity --> Innovation
                                  |
                                  v
                                 EKF
                                  |
                    +-------------+-------------+
                    |             |             |
                    v             v             v
              Position       Velocity      Accelerometer
             Correction     Correction      Bias Estimate
                                              |
                                              +----> Bias Correction
    

Thus,

\[ \boxed{\text{Accelerometer}} + \boxed{\text{GNSS}} + \boxed{\text{EKF}} \rightarrow \boxed{\text{Navigation + Bias Estimation}}. \]

4.1.2.36 Reading an Accelerometer Datasheet

Accelerometer selection requires interpretation of both performance specifications and spacecraft resource constraints.

A very large measurement range does not automatically imply good precision-navigation performance.

The design trade should consider

\[ \boxed{\text{Required Range}} + \boxed{\text{Bias Stability}} + \boxed{\text{Noise}} + \boxed{\text{Bandwidth}} + \boxed{\text{Environment}}. \]

4.1.2.37 MEMS vs High-Performance Accelerometers

Technology should be selected according to the required mission accuracy, environment, mass, power, bandwidth, range, and reliability.

4.1.2.38 Spacecraft Applications

Propulsive Manoeuvres

During a thruster firing,

\[ \mathbf a_{\text{thrust}} = \frac{\mathbf F_T}{m}. \]

The accelerometer can observe the resulting non-gravitational acceleration.

This can support

• burn monitoring,
• achieved-\(\Delta v\) estimation,
• thrust anomaly detection,
• propulsion characterization.

Atmospheric Drag

In LEO,

\[ \mathbf a_D = \frac{\mathbf F_D}{m}. \]

Sensitive accelerometers can support estimation of non-gravitational disturbances such as atmospheric drag.

Rendezvous and Docking

During rendezvous and proximity operations, accelerometers can support manoeuvre reconstruction, translational control monitoring, inertial propagation, and fusion with GNSS or relative-navigation sensors.

Launch Vehicles

Launch vehicles use accelerometers extensively because thrust-driven acceleration is large and persistent during powered flight.

Planetary Entry and Landing

Accelerometers can measure deceleration during

• atmospheric entry,
• parachute deployment,
• powered descent,
• touchdown.

4.1.2.39 Spacecraft Free Fall vs Thruster Firing

Orbital Coast

During ideal orbital free fall,

\[ \mathbf f \approx 0. \]

Thruster Firing

If a thruster produces force \(\mathbf F_T\),

\[ \boxed{ \mathbf f \approx \frac{\mathbf F_T}{m} } \]

when other non-gravitational forces are neglected.

Thrusters OFF
     |
     v
Orbital Free Fall
     |
     v
Accelerometer ≈ 0


Thrusters ON
     |
     v
Non-Gravitational Force
     |
     v
Specific Force
     |
     v
Accelerometer ≠ 0
    

This provides a direct connection between accelerometer measurements, spacecraft propulsion, and manoeuvre execution.

4.1.2.40 Accelerometer Calibration

A practical calibration model can be written as

\[ \mathbf f_m = K_a \mathbf f_{\text{true}} + \mathbf b_a \]

where

\[ K_a = I+S_a+M_a. \]

Therefore,

\[ \boxed{ \hat{\mathbf f}_{true} = K_a^{-1} \left( \mathbf f_m-\mathbf b_a \right) }. \]

Accelerometer calibration may seek to estimate

• bias,
• scale factor,
• axis misalignment,
• cross-axis sensitivity,
• temperature coefficients,
• nonlinear coefficients.

The calibrated output is then closer to the physical specific force required by the navigation algorithm.

4.1.2.41 Accelerometer Error Budget

For engineering design, it is useful to separate the total accelerometer error into individual components.

A conceptual error budget can be written as

\[ \delta\mathbf f = \delta\mathbf f_{\text{bias}} + \delta\mathbf f_{\text{scale}} + \delta\mathbf f_{\text{misalign}} + \delta\mathbf f_{\text{noise}} + \delta\mathbf f_{\text{quant}} + \delta\mathbf f_{\text{thermal}} + \cdots \]

This separation is useful for requirements allocation, calibration, sensitivity studies, and Monte Carlo simulation.

Each error source can be assigned a deterministic value, stochastic model, or uncertainty distribution depending on the available sensor characterization data.

4.1.2.42 Accelerometer → Navigation → Sensor Fusion

The complete accelerometer-based navigation chain begins with the true spacecraft dynamics.

The accelerometer senses specific force,

\[ \mathbf f = \mathbf a-\mathbf g. \]

The measured signal is then calibrated, corrected for estimated bias, transformed into the appropriate navigation frame, and combined with a gravity model.

Conceptually,

\[ \boxed{\text{True Vehicle Dynamics}} \]

\[ \downarrow \]

\[ \boxed{\mathbf a-\mathbf g} \]

\[ \downarrow \]

\[ \boxed{\text{Accelerometer}} \]

\[ \downarrow \]

\[ \boxed{\mathbf f_m} \]

\[ \downarrow \]

\[ \boxed{\text{Bias / Calibration Correction}} \]

\[ \downarrow \]

\[ \boxed{\text{Body-to-Navigation Transformation}} \]

\[ \downarrow \]

\[ \boxed{+\mathbf g} \]

\[ \downarrow \]

\[ \boxed{\hat{\mathbf a}} \]

\[ \downarrow \]

\[ \boxed{\hat{\mathbf v}} \]

\[ \downarrow \]

\[ \boxed{\hat{\mathbf r}}. \]

In a practical navigation system, accelerometer measurements are combined with gyroscopes and external navigation sensors.

The complete sensor-fusion architecture is therefore

\[ \boxed{\text{Accelerometer + Gyroscope + External Sensors}} \rightarrow \boxed{\text{EKF / UKF}} \rightarrow \boxed{\text{Navigation State Estimate}}. \]

This provides the direct connection from accelerometer physics and sensor modelling into inertial navigation and the sensor-fusion algorithms developed later in Section 4.2.