Fundamentals of Sensor Fusion

State estimation, prediction and correction, uncertainty, observability, timing, sensor confidence, and the mathematical foundations required before developing spacecraft Kalman-filter-based navigation systems.

4.2.1 Fundamentals of Sensor Fusion

A spacecraft does not directly know its own position, velocity, attitude, angular velocity, acceleration, or relative state with respect to another spacecraft.

These quantities exist as part of the true physical state of the spacecraft, but the onboard flight computer cannot directly observe that truth.

Instead, the flight computer receives measurements from sensors. Different sensors observe different physical quantities, and each measurement contains uncertainty.

A representative spacecraft state may be written as

\[ \boxed{ \mathbf{x} = \begin{bmatrix} \mathbf r \\ \mathbf v \\ \mathbf q \\ \boldsymbol{\omega} \\ \mathbf b_g \\ \mathbf b_a \\ \cdots \end{bmatrix} } \]

where

Additional states may be included depending on the mission. These may include GNSS clock bias, sensor alignment errors, target-relative position, target-relative velocity, relative attitude, target angular rate, or slowly varying environmental parameters.

Different Sensors Observe Different Quantities

No single spacecraft sensor normally measures the complete state.

Instead,

\[ \boxed{\text{Gyroscope}} \rightarrow \boldsymbol{\omega}_m \]

\[ \boxed{\text{Accelerometer}} \rightarrow \mathbf f_m \]

\[ \boxed{\text{Star Tracker}} \rightarrow \mathbf q_m \]

\[ \boxed{\text{GNSS}} \rightarrow \mathbf r_m, \mathbf v_m \]

while relative-navigation sensors may provide measurements such as

\[ \boxed{\text{Camera}} \rightarrow \text{bearing, features, line of sight, relative pose} \]

and

\[ \boxed{\text{LiDAR}} \rightarrow \text{range, geometry, point cloud, relative pose}. \]

Measurements Are Not Perfect Truth

Every real sensor contains imperfections. Measurements may contain bias, noise, drift, scale-factor error, axis misalignment, sampling effects, latency, quantisation, saturation, or temporary loss of validity.

A general sensor measurement can therefore be represented as

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

where

The navigation problem is therefore not simply:

What did the sensor measure?

The deeper question is:

Given several imperfect measurements and knowledge of how the spacecraft moves, what is the best available estimate of the spacecraft state?

This is the fundamental problem addressed by sensor fusion and state estimation.

What Is Sensor Fusion?

Sensor fusion is the process of combining information from several sensors, measurements, mathematical models, or previous state estimates to obtain a state estimate that is more useful than the information supplied by any individual source.

A useful conceptual representation is

\[ \boxed{ \text{Sensor Fusion} = \text{Measurements} + \text{System Model} + \text{Uncertainty Information} }. \]

The output is therefore not normally only a state estimate.

A probabilistic estimator conceptually produces

\[ \boxed{ \hat{\mathbf x}_k, \mathbf P_k }. \]

Here,

Measurement Path

One information path begins with the physical spacecraft:

\[ \boxed{\text{True State}} \rightarrow \boxed{\text{Sensors}} \rightarrow \boxed{\text{Measurements}}. \]

Prediction Path

Model-based estimation introduces another path:

\[ \boxed{\text{Previous Estimate}} \rightarrow \boxed{\text{Dynamic Model}} \rightarrow \boxed{\text{Predicted State}}. \]

The estimator then combines both sources:

\[ \boxed{\text{Predicted State}} + \boxed{\text{New Measurements}} \rightarrow \boxed{\text{Updated State Estimate}}. \]

This continuous cycle of prediction, measurement, comparison, and correction forms the basis of many spacecraft estimation algorithms.

Sensor Fusion Is Not Simple Averaging

A common misconception is that sensor fusion simply means averaging measurements from several sensors.

Consider two sensors measuring the same scalar quantity \(x\):

\[ z_1=x+v_1 \]

and

\[ z_2=x+v_2. \]

A simple arithmetic average would be

\[ \boxed{ \hat{x} = \frac{z_1+z_2}{2} }. \]

This gives both measurements equal influence.

But suppose sensor 1 has standard deviation

\[ \sigma_1=0.01 \]

while sensor 2 has

\[ \sigma_2=1. \]

Sensor 1 is substantially more precise than sensor 2. Giving the two measurements equal influence ignores useful information about their measurement quality.

Uncertainty-Weighted Fusion

A better approach is to assign greater influence to measurements with lower uncertainty.

Conceptually,

\[ \boxed{\text{High Confidence}} \Rightarrow \boxed{\text{Greater Influence}} \]

and

\[ \boxed{\text{Low Confidence}} \Rightarrow \boxed{\text{Smaller Influence}}. \]

For two independent scalar measurements with variances \(\sigma_1^2\) and \(\sigma_2^2\), an inverse-variance-weighted estimate is

\[ \boxed{ \hat{x} = \frac{ z_1/\sigma_1^2 + z_2/\sigma_2^2 }{ 1/\sigma_1^2 + 1/\sigma_2^2 } }. \]

Therefore,

\[ w_i \propto \frac{1}{\sigma_i^2}. \]

This simple result introduces one of the most important principles in state estimation:

\[ \boxed{ \text{Sensor Fusion} \neq \text{Simple Averaging} }. \]

Instead,

\[ \boxed{ \text{Sensor Fusion} \approx \text{Information Combination According to Confidence} }. \]

The Kalman Filter later extends this idea to multidimensional dynamic systems in which both the predicted state and sensor measurements contain uncertainty.

Why Multiple Sensors Are Required

Different spacecraft sensors possess different strengths, weaknesses, bandwidths, update rates, accuracy levels, failure modes, and environmental limitations.

Multiple sensors are therefore used for two broad reasons:

Complementary Measurement Characteristics

Consider a gyroscope and star tracker.

The gyroscope can provide high-rate angular-rate information and is therefore excellent for short-term attitude propagation.

However, even small bias and noise errors accumulate when angular-rate measurements are integrated.

A star tracker provides an absolute attitude reference that does not suffer from the same integration drift, but it usually operates at a lower measurement rate.

Therefore,

\[ \boxed{\text{Gyroscope}} + \boxed{\text{Star Tracker}} \rightarrow \boxed{\text{Improved Attitude Estimate}}. \]

The gyro contributes short-term propagation, while the star tracker provides long-term correction.

Different Sensors May Observe Different Physical Quantities

Sensor fusion does not require two sensors to measure the same quantity.

An IMU may provide

\[ \boldsymbol{\omega}_m, \qquad \mathbf f_m \]

while GNSS provides

\[ \mathbf r_m, \qquad \mathbf v_m. \]

These quantities are different, but spacecraft kinematics and dynamics establish relationships between them.

The estimator uses those relationships to combine the measurements into a common state estimate.

Redundancy Versus Complementarity

Redundant and complementary sensors serve different purposes and should not be confused.

Redundant Sensors

Redundant sensors provide similar or overlapping information.

For example,

\[ \boxed{\text{Star Tracker 1}} + \boxed{\text{Star Tracker 2}}. \]

Redundancy can improve

If one sensor becomes invalid or fails, another may continue supplying the required information.

Complementary Sensors

Complementary sensors provide different types of information or possess different error characteristics.

Examples include

\[ \boxed{\text{Gyroscope}} + \boxed{\text{Star Tracker}} \]

and

\[ \boxed{\text{IMU}} + \boxed{\text{GNSS}}. \]

The first combination provides high-rate angular propagation and absolute attitude correction.

The second provides high-rate inertial propagation and external position/velocity correction.

Therefore,

\[ \boxed{\text{Redundancy}} \rightarrow \boxed{\text{Similar Information from Multiple Sources}} \]

whereas

\[ \boxed{\text{Complementarity}} \rightarrow \boxed{\text{Different Information with Complementary Strengths}}. \]

True State, Measured Quantity, and Estimated State

A fundamental distinction in navigation must be maintained between

\[ \boxed{\mathbf x} \qquad \boxed{\mathbf z} \qquad \boxed{\hat{\mathbf x}}. \]

These quantities represent different levels of knowledge.

True State

The true state

\[ \mathbf x_k \]

represents the actual physical state of the spacecraft.

For a simple translational problem,

\[ \mathbf x_k = \begin{bmatrix} \mathbf r_k \\ \mathbf v_k \end{bmatrix}. \]

The true state exists physically but is normally unknown to the flight computer.

Measurement

A sensor supplies a measurement

\[ \mathbf z_k. \]

For example, GNSS may provide

\[ \mathbf z_{GNSS} = \begin{bmatrix} \mathbf r_m \\ \mathbf v_m \end{bmatrix}. \]

Because the measurements contain error,

\[ \mathbf r_m \neq \mathbf r_{true} \]

and

\[ \mathbf v_m \neq \mathbf v_{true}. \]

Estimated State

The estimator combines measurements and models to produce

\[ \boxed{\hat{\mathbf x}_k}. \]

Ideally,

\[ \boxed{ \hat{\mathbf x}_k \approx \mathbf x_k }. \]

The difference between truth and estimate is the estimation error:

\[ \boxed{ \mathbf e_k = \mathbf x_k - \hat{\mathbf x}_k }. \]

Importance in Closed-Loop Simulation

In a realistic GNC simulation, guidance and control should normally use the estimated state rather than the perfect truth state.

Therefore,

\[ \boxed{\mathbf x_{truth}} \rightarrow \boxed{\text{Sensor Models}} \rightarrow \boxed{\mathbf z} \rightarrow \boxed{\text{Estimator}} \rightarrow \boxed{\hat{\mathbf x}} \rightarrow \boxed{\text{Guidance / Control}}. \]

If the controller is accidentally supplied with \(\mathbf x_{truth}\), navigation errors are bypassed and the simulation no longer represents a realistic closed-loop system.

What Does the Estimator Actually Estimate?

The estimated state vector is selected according to the navigation problem.

There is no single universal state vector for every spacecraft estimator.

Simple Translational State

\[ \boxed{ \mathbf x = \begin{bmatrix} \mathbf r \\ \mathbf v \end{bmatrix} }. \]

Attitude State

A spacecraft attitude estimator may include

\[ \boxed{ \mathbf x = \begin{bmatrix} \mathbf q \\ \mathbf b_g \end{bmatrix} }. \]

Here, the estimator determines both spacecraft attitude and gyroscope bias.

Inertial Navigation State

A more complete inertial-navigation state may contain

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

Relative Navigation State

During rendezvous and proximity operations, a relative-navigation estimator may use

\[ \boxed{ \mathbf x_{rel} = \begin{bmatrix} \mathbf r_{rel} \\ \mathbf v_{rel} \\ \mathbf q_{rel} \\ \boldsymbol{\omega}_{rel} \end{bmatrix} }. \]

Additional states may include

The state should therefore contain quantities that are both important to the mission and sufficiently observable from the available measurements.

Directly Measured and Indirectly Estimated States

An important principle of state estimation is that a state variable does not necessarily require a sensor that measures it directly.

Consider the simple state

\[ \mathbf x = \begin{bmatrix} r\\ v \end{bmatrix} \]

while only position is measured:

\[ z = r + v_m. \]

The measurement equation is

\[ z = \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} r\\ v \end{bmatrix} + v_m. \]

Therefore,

\[ \boxed{ \mathbf H = \begin{bmatrix} 1 & 0 \end{bmatrix} }. \]

Velocity does not appear directly in the sensor measurement.

However, the dynamics contain the relationship

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

Measurements of position over time therefore contain information about velocity.

The estimator may consequently reconstruct velocity even though the sensor does not measure velocity directly.

This gives an important distinction:

\[ \boxed{ \text{Not Directly Measured} \neq \text{Not Estimable} }. \]

Whether indirect estimation is actually possible depends on observability.

The Two Fundamental Models in State Estimation

Most model-based estimators require two mathematical descriptions:

\[ \boxed{\text{Process / Dynamic Model}} \]

and

\[ \boxed{\text{Measurement Model}}. \]

The process model describes how the spacecraft state evolves.

The measurement model describes how that state appears to a particular sensor.

Process or Dynamic Model

The process model describes how the state evolves as time advances.

For a nonlinear continuous-time system,

\[ \boxed{ \dot{\mathbf x} = f(\mathbf x,\mathbf u,t) + \mathbf w }. \]

In discrete time,

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

For a linear discrete system,

\[ \boxed{ \mathbf x_{k+1} = \mathbf F_k\mathbf x_k + \mathbf B_k\mathbf u_k + \mathbf w_k }. \]

Here,

What Can the Process Model Contain?

Depending on the mission, the process model may include

Simple Constant-Velocity Example

Consider the scalar state

\[ \mathbf x_k = \begin{bmatrix} r_k\\ v_k \end{bmatrix}. \]

If velocity is assumed constant during the short interval \(\Delta t\),

\[ r_{k+1} = r_k + v_k\Delta t \]

and

\[ v_{k+1} = v_k. \]

Therefore,

\[ \boxed{ \mathbf x_{k+1} = \begin{bmatrix} 1 & \Delta t\\ 0 & 1 \end{bmatrix} \mathbf x_k }. \]

Thus,

\[ \boxed{ \mathbf F = \begin{bmatrix} 1 & \Delta t\\ 0 & 1 \end{bmatrix} }. \]

The dynamic model tells the estimator that position and velocity are physically connected.

Measurement Model

The measurement model describes the measurement that a sensor should produce for a particular spacecraft state.

The general nonlinear form is

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

For a linear measurement,

\[ \boxed{ \mathbf z_k = \mathbf H_k\mathbf x_k + \mathbf v_k }. \]

Position-Only Measurement

For

\[ \mathbf x = \begin{bmatrix} r\\ v \end{bmatrix}, \]

if the sensor measures position only,

\[ z = \begin{bmatrix} 1 & 0 \end{bmatrix} \mathbf x + v_m. \]

Velocity-Only Measurement

If the sensor measures velocity,

\[ z = \begin{bmatrix} 0 & 1 \end{bmatrix} \mathbf x + v_m. \]

Position and Velocity Measurement

If both quantities are measured,

\[ \mathbf z = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \mathbf x + \mathbf v. \]

The measurement model therefore maps information from state space into measurement or sensor space.

Prediction: What Should the State Be Now?

Assume that the estimator possesses an updated state estimate at time \(k-1\):

\[ \hat{\mathbf x}_{k-1}^{+}. \]

The process model is used to propagate this state to the next measurement time.

For a nonlinear system,

\[ \boxed{ \hat{\mathbf x}_{k}^{-} = f( \hat{\mathbf x}_{k-1}^{+}, \mathbf u_{k-1} ) }. \]

For a linear system,

\[ \boxed{ \hat{\mathbf x}_{k}^{-} = \mathbf F_{k-1} \hat{\mathbf x}_{k-1}^{+} + \mathbf B_{k-1}\mathbf u_{k-1} }. \]

The notation

\[ (-) \]

denotes a prior or predicted estimate.

The notation

\[ (+) \]

denotes a posterior or updated estimate.

Prediction therefore answers the question:

Based on the previous estimate, known inputs, and spacecraft dynamics, where should the state be now?

Predicted Measurement

Once the state has been predicted, the estimator must determine what the sensor should observe if the predicted state were correct.

For a nonlinear measurement model,

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

For a linear model,

\[ \boxed{ \hat{\mathbf z}_k = \mathbf H_k \hat{\mathbf x}_k^-. } \]

This is the predicted measurement.

The estimator should compare quantities expressed in the same measurement space.

Therefore,

\[ \hat{\mathbf x}_k^- \xrightarrow{h(\cdot)} \hat{\mathbf z}_k \]

and then

\[ \boxed{\mathbf z_k} \quad\text{is compared with}\quad \boxed{\hat{\mathbf z}_k}. \]

This distinction becomes particularly important for nonlinear sensors, such as cameras, star trackers, Sun sensors, range sensors, and relative-navigation systems.

Innovation or Measurement Residual

The difference between the actual sensor measurement and predicted measurement is

\[ \boxed{ \mathbf y_k = \mathbf z_k - \hat{\mathbf z}_k } \]

or equivalently,

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

This quantity is commonly called the measurement residual.

In Kalman-filter terminology it is usually called the innovation.

Simple Example

Suppose the estimator predicts

\[ \hat r^- = 10.0~\text{m} \]

while the sensor measures

\[ r_m = 10.3~\text{m}. \]

The innovation is therefore

\[ \boxed{ y = 10.3 - 10.0 = 0.3~\text{m} }. \]

The estimator now knows that the sensor and prediction disagree by \(0.3\) m.

However, the innovation alone cannot determine how much the state estimate should change.

The estimator must also know how uncertain the prediction and measurement are.

This leads directly to covariance.

State-Estimation Uncertainty

A state estimate alone does not completely describe what the estimator knows.

Consider two navigation systems that both report

\[ \hat r = 1.00~\text{m}. \]

System A may have uncertainty

\[ \sigma_r = 0.01~\text{m}, \]

while system B may have

\[ \sigma_r = 1.0~\text{m}. \]

Although both state estimates have the same numerical value, their confidence is very different.

Therefore, state estimation must represent both

\[ \boxed{\text{Estimated Value}} \]

and

\[ \boxed{\text{Estimated Uncertainty}}. \]

Error Covariance Matrix

Define the estimation error as

\[ \mathbf e_k = \mathbf x_k - \hat{\mathbf x}_k. \]

The uncertainty associated with this error can be represented by the covariance matrix

\[ \boxed{ \mathbf P_k = E \left[ \mathbf e_k \mathbf e_k^T \right] }. \]

Two-State Example

For

\[ \mathbf x = \begin{bmatrix} r\\ v \end{bmatrix}, \]

the covariance matrix may be written as

\[ \boxed{ \mathbf P = \begin{bmatrix} \sigma_r^2 & P_{rv}\\ P_{vr} & \sigma_v^2 \end{bmatrix} }. \]

The diagonal terms describe the variance associated with the individual state components.

Therefore,

\[ P_{11} = \sigma_r^2 \]

and

\[ P_{22} = \sigma_v^2. \]

Off-Diagonal Terms

The off-diagonal terms describe correlations between state-estimation errors.

For example,

\[ P_{rv} \neq 0 \]

means that position and velocity estimation errors are statistically related.

This occurs naturally because spacecraft dynamics couple different state variables.

Standard Deviation and Covariance

If a diagonal covariance element is

\[ P_{ii} = \sigma_i^2, \]

then the corresponding standard deviation is

\[ \boxed{ \sigma_i = \sqrt{P_{ii}} }. \]

For example,

\[ P_{rr} = 10^{-4}~\text{m}^2 \]

corresponds to

\[ \sigma_r = 0.01~\text{m}. \]

Standard deviation therefore provides a physically intuitive way to interpret individual covariance terms.

Process Noise

No mathematical spacecraft model perfectly reproduces the real physical system.

Even a detailed model may contain uncertainty associated with

These effects are represented through the process-noise term

\[ \mathbf w_k. \]

Its covariance is

\[ \boxed{ \mathbf Q_k = E \left[ \mathbf w_k \mathbf w_k^T \right] }. \]

Conceptually,

\[ \boxed{\mathbf Q} \rightarrow \boxed{\text{Uncertainty Introduced by the Process Model}}. \]

A common intuitive interpretation is that \(Q\) influences how much confidence the estimator places in its state-propagation model.

Formally, however, \(Q\) is the covariance assigned to the process disturbance rather than a purely subjective tuning parameter.

Measurement Noise

Sensor measurements also contain uncertainty.

In the measurement equation

\[ \mathbf z_k = h(\mathbf x_k) + \mathbf v_k, \]

the term

\[ \mathbf v_k \]

represents measurement uncertainty.

Its covariance is

\[ \boxed{ \mathbf R_k = E \left[ \mathbf v_k \mathbf v_k^T \right] }. \]

Scalar Example

Suppose a position sensor has measurement standard deviation

\[ \sigma_r = 0.02~\text{m}. \]

The corresponding measurement variance is

\[ \boxed{ R = \sigma_r^2 = (0.02)^2 = 4\times10^{-4}~\text{m}^2 }. \]

For multi-dimensional measurements, \(\mathbf R\) becomes a matrix.

P, Q, and R — Three Covariances to Distinguish

These three matrices are central to Kalman-filter-based estimation and are frequently confused.

State-Estimate Covariance — P

\[ \boxed{ \mathbf P = \text{Uncertainty of the Estimated State} }. \]

Process-Noise Covariance — Q

\[ \boxed{ \mathbf Q = \text{Uncertainty Introduced by the Process Model} }. \]

Measurement-Noise Covariance — R

\[ \boxed{ \mathbf R = \text{Uncertainty Associated with the Measurement} }. \]

Compact Interpretation

\[ \boxed{\mathbf Q} \rightarrow \boxed{\text{Prediction / Model Uncertainty}} \]

\[ \boxed{\mathbf R} \rightarrow \boxed{\text{Measurement Uncertainty}} \]

while

\[ \boxed{\mathbf P} \rightarrow \boxed{\text{Estimated-State Uncertainty}}. \]

Understanding these three quantities is essential before developing the Kalman Filter.

Prediction Also Propagates Uncertainty

State prediction changes not only the estimated state but also the uncertainty associated with that state.

For a linear system,

\[ \hat{\mathbf x}_k^- = \mathbf F \hat{\mathbf x}_{k-1}^+ \]

and the covariance is propagated as

\[ \boxed{ \mathbf P_k^- = \mathbf F \mathbf P_{k-1}^+ \mathbf F^T + \mathbf Q }. \]

Physical Interpretation

The previous uncertainty is transformed by the state dynamics:

\[ \mathbf F \mathbf P^+ \mathbf F^T. \]

Additional uncertainty from imperfect modelling is then added:

\[ +\mathbf Q. \]

Consequently, when no correcting measurements are available, uncertainty generally grows.

Conceptually,

\[ \boxed{\text{Propagation}} \Rightarrow \boxed{\text{Uncertainty Accumulation}}. \]

This is one reason inertial-navigation systems cannot normally propagate indefinitely without external corrections.

Measurement Updates and Uncertainty Reduction

When an informative independent measurement becomes available, the estimator receives new information about the spacecraft state.

Conceptually,

\[ \boxed{\text{Prediction}} + \boxed{\text{Measurement}} \rightarrow \boxed{\text{Updated Estimate}}. \]

The measurement can reduce uncertainty in states that are directly observed and, through state correlations and dynamics, may also reduce uncertainty in indirectly observed states.

A typical conceptual behaviour is therefore

\[ \mathbf P \nearrow \qquad \text{during propagation} \]

and

\[ \mathbf P \searrow \qquad \text{after informative measurement updates}. \]

This is a useful conceptual picture, although not every covariance element must decrease after every measurement update.

Prediction Versus Measurement Trust

Suppose the dynamic model predicts

\[ \hat{x}^- = 10.0 \]

while a sensor measures

\[ z = 10.5. \]

The updated estimate should not automatically become \(10.5\), nor should the measurement automatically be ignored.

Accurate Model, Noisy Sensor

If the predicted state has low uncertainty and the measurement has high uncertainty, the updated estimate should remain relatively close to \(10.0\).

Uncertain Model, Accurate Sensor

If the model prediction is uncertain while the sensor measurement is precise, the updated estimate should move more strongly toward \(10.5\).

Therefore,

\[ \boxed{ \text{Correction Strength} = f( \text{Prediction Uncertainty}, \text{Measurement Uncertainty} ) }. \]

This principle ultimately produces the Kalman gain.

Conceptually,

\[ \boxed{ \hat{\mathbf x}^{+} = \hat{\mathbf x}^{-} + \text{Gain} \times \text{Innovation} }. \]

Later this becomes

\[ \boxed{ \hat{\mathbf x}^{+} = \hat{\mathbf x}^{-} + \mathbf K\mathbf y }. \]

The full derivation of \(\mathbf K\) is developed in the Kalman Filter section rather than here.

High-Rate Propagation and Low-Rate Correction

Many spacecraft navigation systems combine high-rate inertial sensors with lower-rate absolute reference sensors.

Attitude Example

\[ \boxed{\text{Gyroscope}} \rightarrow \boxed{\text{High-Rate Attitude Propagation}} \]

while

\[ \boxed{\text{Star Tracker}} \rightarrow \boxed{\text{Absolute Attitude Correction}}. \]

Translational Navigation Example

\[ \boxed{\text{IMU}} \rightarrow \boxed{\text{High-Rate Navigation Propagation}} \]

while

\[ \boxed{\text{GNSS}} \rightarrow \boxed{\text{Absolute Position / Velocity Correction}}. \]

The estimator may therefore operate as

\[ \boxed{\text{Predict}} \rightarrow \boxed{\text{Predict}} \rightarrow \boxed{\text{Predict}} \rightarrow \boxed{\text{Predict}} \rightarrow \boxed{\text{Update}}. \]

Prediction and correction therefore do not need to occur at the same frequency.

Multi-Rate Sensor Fusion

Consider an IMU operating at

\[ f_{IMU} = 100~\text{Hz} \]

and GNSS operating at

\[ f_{GNSS} = 10~\text{Hz}. \]

One GNSS measurement is available every

\[ 0.1~\text{s}, \]

while the IMU provides measurements every

\[ 0.01~\text{s}. \]

Therefore, approximately ten IMU-driven propagation steps may occur between two GNSS updates.

Conceptually,

\[ \boxed{\text{IMU}} \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots \rightarrow P_{10} \rightarrow \boxed{\text{GNSS Update}}. \]

Here \(P_1,P_2,\ldots\) represent successive propagation operations, not the covariance matrix \(\mathbf P\).

The estimator must maintain both state and uncertainty continuously between absolute sensor updates.

Asynchronous Measurements

Real spacecraft sensors do not necessarily operate from the same clock or at integer multiples of one another.

A representative system might contain

These measurements may arrive at different times.

A practical estimator therefore often follows the sequence

\[ \boxed{\text{Propagate State to Measurement Time}} \]

followed by

\[ \boxed{\text{Apply the Available Measurement Update}}. \]

Accurate timestamping is therefore part of the navigation problem, not merely a communication-system detail.

Sensor Latency

Measurement arrival time and measurement observation time are not necessarily the same.

Suppose a camera measurement becomes available at

\[ t = 10.10~\text{s} \]

but the image was actually captured at

\[ t = 10.00~\text{s}. \]

The measurement represents the spacecraft state at \(10.00\) s rather than \(10.10\) s.

If the estimator incorrectly applies the measurement as though it were current, a timing-induced navigation error is introduced.

Therefore, a fusion algorithm should distinguish

\[ \boxed{\text{Measurement Time}} \]

from

\[ \boxed{\text{Measurement Arrival Time}}. \]

This becomes especially important during fast manoeuvres, close-range RPO, docking, and high-rate attitude motion.

Coordinate Frames Must Be Consistent

Different sensors often express their measurements in different coordinate frames.

For example,

Measurements must therefore be transformed consistently before they are compared or fused.

Conceptually,

\[ \boxed{\text{Sensor Measurement}} \rightarrow \boxed{\text{Frame Transformation}} \rightarrow \boxed{\text{Estimator-Compatible Measurement}}. \]

For a vector transformed from frame \(A\) to frame \(B\),

\[ \boxed{ \mathbf z^B = \mathbf C_A^B \mathbf z^A }. \]

A mathematically correct estimator can still produce physically incorrect results if frame definitions, axis directions, quaternion conventions, or transformation directions are inconsistent.

Sensor Biases as Estimated States

Many sensor errors are not purely instantaneous random noise.

Consider a gyroscope measurement:

\[ \boxed{ \boldsymbol{\omega}_m = \boldsymbol{\omega}_{true} + \mathbf b_g + \mathbf n_g }. \]

If the gyro bias \(\mathbf b_g\) is ignored, it accumulates during attitude propagation and produces growing attitude error.

Instead, the estimator may augment the state:

\[ \boxed{ \mathbf x = \begin{bmatrix} \mathbf q\\ \mathbf b_g \end{bmatrix} }. \]

The estimator then determines both spacecraft attitude and gyroscope bias.

Similarly, an inertial-navigation state may contain

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

Bias estimation is therefore an important link between the sensor-error models developed in Section 4.1 and the state-estimation algorithms developed in Section 4.2.

Bias Dynamics

Sensor bias may itself evolve with time.

Constant Bias Model

The simplest model assumes the bias remains constant:

\[ \boxed{ \dot{\mathbf b} = 0 }. \]

Random-Walk Bias Model

Slowly varying bias may be represented as

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

Here \(\mathbf w_b\) is a random driving process.

First-Order Gauss-Markov Model

A correlated bias process may be represented as

\[ \boxed{ \dot{\mathbf b} = - \frac{1}{\tau} \mathbf b + \mathbf w_b }. \]

The parameter \(\tau\) is the correlation time constant.

The estimator process model must therefore be selected consistently with the physical error behaviour assumed for the sensor.

Observability

Adding a quantity to the estimator state vector does not guarantee that the estimator can actually determine that quantity.

The measurements and dynamics must contain enough information to distinguish changes in the state.

This property is called observability.

Linear Observability

Consider the linear system

\[ \mathbf x_{k+1} = \mathbf F \mathbf x_k \]

with measurements

\[ \mathbf z_k = \mathbf H \mathbf x_k. \]

The observability matrix is

\[ \boxed{ \mathcal O = \begin{bmatrix} \mathbf H\\ \mathbf H\mathbf F\\ \mathbf H\mathbf F^2\\ \vdots\\ \mathbf H\mathbf F^{n-1} \end{bmatrix} }. \]

If

\[ \boxed{ \operatorname{rank}(\mathcal O) = n }, \]

the linear system is observable.

This means the complete state can, in principle, be reconstructed from measurements collected over time.

Observability Depends on More Than Sensor Count

Simply adding more sensors does not automatically guarantee that every state becomes observable.

Observability depends on

\[ \boxed{\text{Measurement Type}} + \boxed{\text{Geometry}} + \boxed{\text{Dynamics}} + \boxed{\text{Vehicle Motion}}. \]

Relative Navigation Example

A monocular camera may provide accurate line-of-sight information to a target but relatively weak instantaneous information about absolute range.

A LiDAR can directly provide range information.

Therefore,

\[ \boxed{\text{Camera Bearing}} + \boxed{\text{LiDAR Range}} \rightarrow \boxed{\text{Improved Relative-State Observability}}. \]

This demonstrates why sensor geometry and measurement type are as important as nominal sensor accuracy.

Measurement Geometry

Navigation uncertainty is generally directional rather than identical in every axis.

A measurement may strongly constrain one direction while providing weak information in another.

The resulting estimator uncertainty might therefore satisfy

\[ \sigma_x \ll \sigma_y. \]

Geometry plays an important role in

This is another reason why navigation uncertainty must be represented using covariance matrices rather than a single accuracy number.

Measurement Validation

A practical estimator should not assume that every received measurement is valid.

A measurement may be corrupted by

The estimator already computes the innovation

\[ \mathbf y_k = \mathbf z_k - \hat{\mathbf z}_k. \]

This residual provides a natural basis for determining whether the incoming measurement agrees with the predicted spacecraft state.

Conceptually,

\[ \boxed{\text{Measurement}} \rightarrow \boxed{\text{Validation}} \rightarrow \begin{cases} \text{Accept}\\ \text{Reject}\\ \text{Down-weight} \end{cases}. \]

This process is commonly called measurement gating.

Innovation Covariance

The magnitude of the innovation alone is not sufficient for determining whether a measurement is reasonable.

For example, a residual of

\[ 0.5~\text{m} \]

would be very large for a centimetre-level measurement system but may be perfectly reasonable for a sensor with metre-level uncertainty.

The innovation must therefore be interpreted relative to its expected uncertainty.

In the linear Kalman Filter, this uncertainty is represented by the innovation covariance

\[ \boxed{\mathbf S}. \]

For the linear filter,

\[ \boxed{ \mathbf S = \mathbf H \mathbf P^- \mathbf H^T + \mathbf R }. \]

This combines

Innovation covariance later becomes important for

Sensor Dropout

Spacecraft sensors may temporarily become unavailable.

A star tracker may lose tracking, GNSS may experience a temporary loss of navigation solution, or a camera may lose the target.

Star Tracker Loss

\[ \boxed{\text{Star Tracker Lost}} \]

does not necessarily mean attitude estimation stops immediately.

The estimator may continue using

\[ \boxed{\text{Gyro Propagation}} \rightarrow \boxed{\text{Temporary Attitude Estimate}}. \]

However, the uncertainty generally grows because the absolute correction is no longer available.

GNSS Loss

Similarly,

\[ \boxed{\text{GNSS Loss}} \rightarrow \boxed{\text{IMU Propagation}} \]

can maintain a navigation solution temporarily, but inertial errors accumulate with time.

A robust fusion system must therefore be designed to operate not only when measurements are available but also when they disappear.

Sensor Availability and Mission Phase

The most useful sensor combination may change during different mission phases.

Rendezvous and proximity operations provide a clear example.

Far Range

\[ \boxed{\text{GNSS / Relative GNSS}} \]

may provide the primary relative translational navigation information.

Mid Range

\[ \boxed{\text{GNSS + Camera}} \]

may provide both absolute/relative translational information and optical line-of-sight observations.

Close Range

\[ \boxed{\text{Camera + LiDAR}} \]

may become more important as range, target geometry, and relative pose must be determined accurately.

Capture Phase

The navigation solution may need accurate estimates of

\[ \boxed{ \mathbf r_{rel}, \mathbf v_{rel}, \mathbf q_{rel}, \boldsymbol{\omega}_{rel} }. \]

Sensor fusion may therefore need to change dynamically as the spacecraft progresses through the mission.

This concept is developed later in Sensor Handover and Mode-Dependent Fusion.

Filter Initialization

Every recursive estimator requires an initial state estimate.

The estimator begins with

\[ \boxed{ \hat{\mathbf x}_0 } \]

and probabilistic filters also require an initial covariance

\[ \boxed{ \mathbf P_0 }. \]

These quantities describe

Poor initialization can result in

Initialization is therefore part of estimator design rather than only a software startup detail.

Estimator Convergence

The initial estimate may differ from the true state:

\[ \hat{\mathbf x}_0 \neq \mathbf x_0. \]

As valid measurements are processed, the estimate should ideally move toward the true state:

\[ \boxed{ \hat{\mathbf x}_k \rightarrow \mathbf x_k }. \]

This process is called convergence.

Convergence depends on

A flight estimator must therefore answer not only

Is the estimator stable?

but also

Does the estimator converge sufficiently quickly for the mission phase?

Accuracy, Precision, and Consistency

Accuracy, precision, and filter consistency are related but different concepts.

Accuracy

Accuracy describes how close the estimated state is to the true state.

\[ \boxed{ \mathbf e = \mathbf x - \hat{\mathbf x} }. \]

Smaller estimation error generally means greater accuracy.

Precision

Precision describes the spread of repeated estimates or measurements.

A system may be highly precise but biased, meaning repeated estimates cluster tightly around the wrong value.

Consistency

Filter consistency asks whether the estimator's reported uncertainty realistically represents its actual estimation error.

A filter may report very small covariance even though the true estimation error is large.

Such a filter is overconfident.

Good estimation therefore requires both

\[ \boxed{\text{Small Estimation Error}} \]

and

\[ \boxed{\text{Realistic Estimated Uncertainty}}. \]

Fusion at Different Information Levels

Sensor fusion can occur at different stages of the measurement processing chain.

Raw or Data-Level Fusion

Low-level sensor measurements are combined directly.

Examples include

Feature-Level Fusion

Features extracted from sensor data are fused.

For example,

\[ \boxed{\text{Camera Features}} + \boxed{\text{LiDAR Features}} \rightarrow \boxed{\text{Relative Pose}}. \]

State-Level Fusion

Individual navigation subsystems may each produce their own state estimates:

\[ \hat{\mathbf x}_1, \qquad \hat{\mathbf x}_2. \]

These state estimates can then be fused:

\[ \boxed{\hat{\mathbf x}_1} + \boxed{\hat{\mathbf x}_2} \rightarrow \boxed{\hat{\mathbf x}_{fused}}. \]

These levels of fusion lead directly to the centralized, decentralized, federated, loosely coupled, tightly coupled, and deeply coupled architectures developed in Section 4.2.2.

Central Idea of Recursive State Estimation

The complete estimator process can now be expressed as a repeated sequence of propagation, measurement processing, validation, and correction.

Begin with the previous updated estimate

\[ \boxed{ \hat{\mathbf x}_{k-1}^{+}, \mathbf P_{k-1}^{+} }. \]

Step 1 — Propagate the State

\[ \hat{\mathbf x}_{k-1}^{+} \rightarrow \hat{\mathbf x}_k^{-}. \]

Step 2 — Propagate the Uncertainty

\[ \mathbf P_{k-1}^{+} \rightarrow \mathbf P_k^{-}. \]

Step 3 — Receive a Measurement

\[ \mathbf z_k. \]

Step 4 — Predict the Measurement

\[ \hat{\mathbf z}_k = h(\hat{\mathbf x}_k^-). \]

Step 5 — Compute the Innovation

\[ \mathbf y_k = \mathbf z_k - \hat{\mathbf z}_k. \]

Step 6 — Validate the Measurement

\[ \mathbf y_k \rightarrow \boxed{\text{Measurement Gate}}. \]

Step 7 — Correct the State

\[ \hat{\mathbf x}_k^- \rightarrow \hat{\mathbf x}_k^+. \]

Step 8 — Correct the Covariance

\[ \mathbf P_k^- \rightarrow \mathbf P_k^+. \]

Step 9 — Repeat

\[ k \rightarrow k+1. \]

Therefore,

\[ \boxed{\text{Predict}} \rightarrow \boxed{\text{Measure}} \rightarrow \boxed{\text{Compare}} \rightarrow \boxed{\text{Validate}} \rightarrow \boxed{\text{Correct}} \rightarrow \boxed{\text{Repeat}}. \]

Example: Attitude Sensor Fusion

Consider a spacecraft equipped with

The gyroscope measurement may be represented as

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

The estimator subtracts its current bias estimate:

\[ \hat{\boldsymbol{\omega}} = \boldsymbol{\omega}_m - \hat{\mathbf b}_g. \]

This angular-rate estimate can be used to propagate the quaternion:

\[ \boxed{ \dot{\mathbf q} = \frac{1}{2} \mathbf\Omega \left( \boldsymbol{\omega}_m - \hat{\mathbf b}_g \right) \mathbf q }. \]

The propagated attitude is

\[ \hat{\mathbf q}^{-}. \]

The star tracker then provides an absolute attitude observation

\[ \mathbf q_{ST}. \]

The disagreement between propagated and measured attitude provides information about

Conceptually,

\[ \boxed{\text{Gyroscope}} \rightarrow \boxed{\text{Quaternion Propagation}} \]

and

\[ \boxed{\text{Star Tracker}} \rightarrow \boxed{\text{Attitude Correction}}. \]

This is the basic architecture later formalised through the Multiplicative Extended Kalman Filter.

Example: IMU/GNSS Sensor Fusion

Consider an inertial-navigation estimator with the state

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

The IMU supplies specific-force measurements

\[ \mathbf f_m \]

and angular-rate measurements

\[ \boldsymbol{\omega}_m. \]

These measurements are used to propagate

\[ \mathbf q, \qquad \mathbf v, \qquad \mathbf r. \]

However, accelerometer bias, gyro bias, attitude error, and inertial sensor noise accumulate with time.

GNSS provides external navigation observations:

\[ \mathbf r_{GNSS}, \qquad \mathbf v_{GNSS}. \]

Position Innovation

\[ \boxed{ \mathbf y_r = \mathbf r_{GNSS} - \hat{\mathbf r}^{-} }. \]

Velocity Innovation

\[ \boxed{ \mathbf y_v = \mathbf v_{GNSS} - \hat{\mathbf v}^{-} }. \]

These residuals can directly correct position and velocity.

Through covariance coupling and observability, they can also contribute information about

This demonstrates why sensor fusion is more powerful than processing each sensor independently.

Example: Relative Navigation Sensor Fusion for RPO

During rendezvous and proximity operations, the chaser spacecraft may need an estimate of the target-relative state.

A representative state is

\[ \boxed{ \mathbf x_{rel} = \begin{bmatrix} \mathbf r_{rel}\\ \mathbf v_{rel}\\ \mathbf q_{rel}\\ \boldsymbol{\omega}_{rel} \end{bmatrix} }. \]

GNSS Contribution

At longer range, GNSS or relative GNSS may provide relative translational information.

Camera Contribution

A camera may provide

\[ \text{bearing}, \qquad \text{features}, \qquad \text{relative orientation}. \]

LiDAR Contribution

LiDAR may provide

\[ \text{range}, \qquad \text{geometry}, \qquad \text{relative pose}. \]

The sensor-fusion architecture may therefore evolve with range:

\[ \boxed{\text{GNSS}} \rightarrow \boxed{\text{GNSS + Vision}} \rightarrow \boxed{\text{Vision + LiDAR}} \rightarrow \boxed{\text{Capture Pose Estimate}}. \]

The estimator must maintain continuity of

\[ \hat{\mathbf x}_{rel} \]

and its covariance while sensor availability and relative-navigation geometry change.

This continuity is particularly important because guidance and control should not receive abrupt discontinuities when navigation sensors are handed over.

Sensor Fusion and the GNC Loop

Sensor fusion is not an isolated algorithm.

It is part of the complete closed-loop guidance, navigation, and control architecture.

\[ \boxed{\text{Spacecraft Dynamics}} \]

\[ \downarrow \]

\[ \boxed{\text{True State}} \]

\[ \downarrow \]

\[ \boxed{\text{Sensors}} \]

\[ \downarrow \]

\[ \boxed{\text{Measurements}} \]

\[ \downarrow \]

\[ \boxed{\text{Sensor Fusion / Navigation}} \]

\[ \downarrow \]

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

\[ \downarrow \]

\[ \boxed{\text{Guidance}} \]

\[ \downarrow \]

\[ \boxed{\text{Control}} \]

\[ \downarrow \]

\[ \boxed{\text{Actuators}} \]

\[ \downarrow \]

\[ \boxed{\text{Spacecraft Dynamics}}. \]

Because the controller operates using the estimated state, navigation error directly affects control behaviour.

If

\[ \hat{\mathbf x} \neq \mathbf x, \]

the controller is acting on an imperfect representation of the real spacecraft.

Navigation requirements should therefore ultimately be connected to guidance and control performance requirements.

Truth, Navigation, and Control in Simulation

A well-structured spacecraft GNC simulation should maintain three logically separate information paths.

Truth Path

\[ \boxed{\mathbf x_{truth}} \]

represents the actual simulated spacecraft motion.

Navigation Path

\[ \boxed{\mathbf x_{truth}} \rightarrow \boxed{\text{Sensor Models}} \rightarrow \boxed{\text{Measurements}} \rightarrow \boxed{\text{Estimator}} \rightarrow \boxed{\hat{\mathbf x}}. \]

Control Path

\[ \boxed{\hat{\mathbf x}} \rightarrow \boxed{\text{Guidance / Control}}. \]

The controller should not receive

\[ \mathbf x_{truth} \]

unless the simulation is intentionally testing an ideal-navigation case.

Keeping truth, navigation, and control separate is essential for evaluating realistic closed-loop GNC behaviour.

Fundamental Sensor-Fusion Architecture

The concepts developed throughout this page can now be combined into a complete estimator architecture.

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

\[ \downarrow \]

\[ \boxed{\text{Sensor Physics and Error Models}} \]

\[ \downarrow \]

\[ \boxed{\mathbf z_1,\mathbf z_2,\ldots,\mathbf z_n} \]

\[ \downarrow \]

\[ \boxed{\text{Time Synchronization}} \]

\[ \downarrow \]

\[ \boxed{\text{Coordinate-Frame Transformation}} \]

\[ \downarrow \]

\[ \boxed{\text{Measurement Validation}} \]

\[ \downarrow \]

\[ \boxed{ \begin{array}{c} \text{State Prediction}\\ +\\ \text{Measurement Correction} \end{array} } \]

\[ \downarrow \]

\[ \boxed{\text{State Estimator}} \]

\[ \downarrow \]

\[ \boxed{\hat{\mathbf x},\mathbf P} \]

\[ \downarrow \]

\[ \boxed{\text{Guidance and Control}}. \]

This architecture forms the foundation for the individual fusion methods developed throughout Section 4.2.

The Five Questions Every Sensor-Fusion System Must Answer

Almost every spacecraft state-estimation problem can be reduced to five fundamental questions.

1. What Are We Trying to Estimate?

\[ \boxed{\mathbf x=?} \]

The required state may contain position, velocity, attitude, sensor biases, relative pose, target motion, or other parameters.

2. How Does the State Evolve?

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

This defines the process or dynamic model.

3. What Does Each Sensor Actually Measure?

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

This defines the measurement model.

4. How Uncertain Are the Model, Measurement, and Estimate?

Process and measurement uncertainty are represented using

\[ \boxed{\mathbf Q,\mathbf R} \]

while state-estimate uncertainty is represented using

\[ \boxed{\mathbf P}. \]

5. Is There Enough Information to Estimate the Desired State?

\[ \boxed{\text{Observability}}. \]

If the available measurements and dynamics do not contain sufficient independent information, adding a variable to the filter state does not make it estimable.

Most advanced state-estimation methods are extensions or refinements of these five fundamental questions.

Key Mathematical Framework

The fundamental state-estimation problem can now be summarized using a compact set of equations.

Process Model

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

Measurement Model

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

Process-Noise Covariance

\[ \boxed{ \mathbf Q_k = E [ \mathbf w_k \mathbf w_k^T ] } \]

Measurement-Noise Covariance

\[ \boxed{ \mathbf R_k = E [ \mathbf v_k \mathbf v_k^T ] } \]

State Estimate

\[ \boxed{ \hat{\mathbf x}_k } \]

Estimation Error

\[ \boxed{ \mathbf e_k = \mathbf x_k - \hat{\mathbf x}_k } \]

State-Estimate Covariance

\[ \boxed{ \mathbf P_k = E [ \mathbf e_k \mathbf e_k^T ] } \]

Predicted Measurement

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

Innovation

\[ \boxed{ \mathbf y_k = \mathbf z_k - \hat{\mathbf z}_k } \]

Conceptual Bayesian Update

At the most general level,

\[ \boxed{\text{Prior Estimate}} + \boxed{\text{Measurement Information}} \rightarrow \boxed{\text{Posterior Estimate}}. \]

These equations form the mathematical language used throughout the remainder of the sensor-fusion chapter.

Concept Summary

The fundamental sensor-fusion problem can be understood as a repeating flow of information:

\[ \boxed{\text{Previous State Estimate}} \]

\[ \downarrow \]

\[ \boxed{\text{Dynamic Prediction}} \]

\[ \downarrow \]

\[ \boxed{\hat{\mathbf x}^{-},\mathbf P^{-}} \]

\[ + \]

\[ \boxed{\text{Sensor Measurements}} \]

\[ \downarrow \]

\[ \boxed{\text{Innovation / Measurement Residual}} \]

\[ \downarrow \]

\[ \boxed{\text{Measurement Validation}} \]

\[ \downarrow \]

\[ \boxed{\text{State Correction}} \]

\[ \downarrow \]

\[ \boxed{\hat{\mathbf x}^{+},\mathbf P^{+}} \]

\[ \downarrow \]

\[ \boxed{\text{Repeat at the Next Time Step}}. \]

More advanced filters differ mainly in how they represent probability, nonlinear dynamics, attitude errors, uncertainty, and measurement updates.

The fundamental architecture remains the same:

\[ \boxed{ \text{Predict} \rightarrow \text{Compare} \rightarrow \text{Correct} }. \]

Next: 4.2.2 Sensor Fusion Architectures

Once the fundamentals of state estimation are established, the next question is how multiple sensors and estimation subsystems should be organised.

For example:

Should raw IMU, GNSS, camera, and LiDAR measurements all enter one central estimator?

Or:

Should individual sensors or navigation subsystems first produce their own estimates, which are then combined at a higher level?

These choices lead to different sensor-fusion architectures.

The next section develops

\[ \boxed{\text{Centralized Fusion}} \]

versus

\[ \boxed{\text{Decentralized Fusion}} \]

versus

\[ \boxed{\text{Federated Fusion}} \]

together with

\[ \boxed{\text{Loose Coupling}} \rightarrow \boxed{\text{Tight Coupling}} \rightarrow \boxed{\text{Deep Coupling}}. \]

4.2.2 Sensor Fusion Architectures

Centralized, decentralized, federated, cascaded, loosely coupled, tightly coupled, and deeply coupled sensor-fusion architectures for spacecraft attitude, navigation, GNSS/IMU integration, and relative-navigation systems.

Continue to Sensor Fusion Architectures →