Information and Federated Filters

4.2.11 Information and Federated Filters

The Information Filter and Federated Filter are important extensions of Kalman-filter theory for spacecraft systems that contain multiple sensors, multiple local estimators, distributed processing units, or fault-tolerant navigation architectures.

The Kalman Filter developed earlier represents a Gaussian state estimate using

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

and its covariance

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

This is called the covariance form of the estimator.

The same Gaussian probability distribution can instead be represented using

\[ \boxed{ \mathbf Y = \mathbf P^{-1} } \]

and

\[ \boxed{ \mathbf y = \mathbf Y\hat{\mathbf x}. } \]

Here,

• \mathbf Y is the information matrix,
• \mathbf y is the information vector.

This representation is called the information form of the Kalman Filter.

The two forms describe exactly the same Gaussian state knowledge:

\[ \boxed{ (\hat{\mathbf x},\mathbf P) \quad\Longleftrightarrow\quad (\mathbf y,\mathbf Y) } \]

with

\[ \boxed{ \mathbf Y=\mathbf P^{-1} } \]

and

\[ \boxed{ \mathbf y=\mathbf P^{-1}\hat{\mathbf x}. } \]

The covariance form emphasizes

\[ \boxed{ \text{How Uncertain Is the State?} } \]

while the information form emphasizes

\[ \boxed{ \text{How Much Information Do We Have About the State?} } \]

This becomes especially powerful when information arrives from several independent sensors or local estimators.

4.2.11.1 Why Information Form Is Useful

Consider a spacecraft with

• GNSS,
• IMU,
• star tracker,
• camera,
• LiDAR,
• radar,
• relative-navigation sensors.

A centralized Kalman Filter could process all measurements in one estimator.

However, there are situations where it is desirable to maintain separate local estimators.

For example:

\[ \boxed{ \text{GNSS/IMU Filter} } \]

\[ \boxed{ \text{Camera Relative-Navigation Filter} } \]

\[ \boxed{ \text{LiDAR Relative-Navigation Filter} } \]

\[ \boxed{ \text{Attitude Filter} } \]

and then combine their information.

The information representation is attractive because independent measurement information can often be combined by addition.

Conceptually,

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

This is a direct algebraic expression of Bayesian fusion.

4.2.11.2 From Covariance to Information

Suppose the state estimate is

\[ \hat{\mathbf x} \]

with covariance

\[ \mathbf P. \]

Define

\[ \boxed{ \mathbf Y = \mathbf P^{-1}. } \]

The matrix

\[ \mathbf Y \]

is called the information matrix.

Then define

\[ \boxed{ \mathbf y = \mathbf Y\hat{\mathbf x}. } \]

The vector

\[ \mathbf y \]

is called the information vector.

The covariance-form estimate

\[ (\hat{\mathbf x},\mathbf P) \]

can therefore be converted to

\[ (\mathbf y,\mathbf Y). \]

4.2.11.3 Recovering the State Estimate

If the information form is known,

\[ \mathbf Y, \qquad \mathbf y, \]

the covariance is recovered using

\[ \boxed{ \mathbf P = \mathbf Y^{-1}. } \]

The state estimate is then

\[ \boxed{ \hat{\mathbf x} = \mathbf Y^{-1}\mathbf y } \]

or equivalently

\[ \boxed{ \hat{\mathbf x} = \mathbf P\mathbf y. } \]

Therefore no information is lost by changing representations.

4.2.11.4 Physical Meaning of the Information Matrix

Covariance describes uncertainty.

Large covariance means large uncertainty:

\[ \boxed{ \mathbf P\uparrow \Rightarrow \text{Confidence}\downarrow. } \]

Because

\[ \mathbf Y=\mathbf P^{-1}, \]

large information corresponds to small uncertainty:

\[ \boxed{ \mathbf Y\uparrow \Rightarrow \mathbf P\downarrow. } \]

Thus:

\[ \boxed{ \text{High Information} \Longleftrightarrow \text{High Confidence}. } \]

Conversely,

\[ \boxed{ \text{Low Information} \Longleftrightarrow \text{High Uncertainty}. } \]

4.2.11.5 Scalar Example

Suppose a scalar state has variance

\[ P=4. \]

Then

\[ Y=P^{-1}=\frac14. \]

If the variance decreases to

\[ P=0.25, \]

then

\[ Y=4. \]

Thus more accurate knowledge corresponds to larger information.

This simple scalar relationship is useful for understanding the matrix case.

4.2.11.6 Gaussian Distribution in Information Form

A Gaussian distribution may be written in covariance form as

\[ p(\mathbf x) \propto \exp \left[ -\frac12 (\mathbf x-\hat{\mathbf x})^T \mathbf P^{-1} (\mathbf x-\hat{\mathbf x}) \right]. \]

Because

\[ \mathbf Y=\mathbf P^{-1} \]

and

\[ \mathbf y=\mathbf Y\hat{\mathbf x}, \]

the same distribution can be written in canonical information form.

This is one reason information matrices appear naturally in probabilistic estimation and optimization.

4.2.11.7 Linear Measurement Model

Consider the linear measurement model

\[ \boxed{ \mathbf z = \mathbf H\mathbf x + \mathbf v } \]

with

\[ \boxed{ \mathbf v \sim \mathcal N( 0,\mathbf R ). } \]

In the standard Kalman Filter, the measurement update uses

\[ \mathbf H, \mathbf R, \mathbf z. \]

In information form, the same measurement contributes directly to the information matrix and information vector.

4.2.11.8 Measurement Information Matrix

The measurement contributes

\[ \boxed{ \mathbf I_z = \mathbf H^T \mathbf R^{-1} \mathbf H. } \]

This can be interpreted as

\[ \boxed{ \text{Information Added by the Measurement}. } \]

A highly accurate sensor has small

\[ \mathbf R, \]

therefore large

\[ \mathbf R^{-1}, \]

and contributes more information.

4.2.11.9 Measurement Information Vector

The corresponding measurement-information vector is

\[ \boxed{ \mathbf i_z = \mathbf H^T \mathbf R^{-1} \mathbf z. } \]

Therefore the measurement contributes the pair

\[ \boxed{ ( \mathbf i_z, \mathbf I_z ). } \]

4.2.11.10 Information-Form Measurement Update

Suppose the prior information state is

\[ \mathbf y^-, \qquad \mathbf Y^-. \]

The measurement update becomes remarkably simple:

\[ \boxed{ \mathbf Y^+ = \mathbf Y^- + \mathbf H^T \mathbf R^{-1} \mathbf H } \]

and

\[ \boxed{ \mathbf y^+ = \mathbf y^- + \mathbf H^T \mathbf R^{-1} \mathbf z. } \]

Therefore,

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

This additive property is the central attraction of the information form.

4.2.11.11 Connection to Bayesian Estimation

Recall Bayesian estimation:

\[ \boxed{ \text{Prior} \times \text{Likelihood} \rightarrow \text{Posterior}. } \]

For Gaussian distributions, multiplication of probability densities corresponds to addition of their information parameters.

Thus the information update

\[ \boxed{ Y^+ = Y^- + H^TR^{-1}H } \]

is a matrix version of Bayesian information accumulation.

4.2.11.12 Why Independent Sensor Information Can Be Added

Suppose two independent sensors provide

\[ z_1=H_1x+v_1 \]

and

\[ z_2=H_2x+v_2. \]

Their information contributions are

\[ I_1 = H_1^TR_1^{-1}H_1 \]

and

\[ I_2 = H_2^TR_2^{-1}H_2. \]

If the measurement noises are independent,

\[ \boxed{ Y^+ = Y^- + I_1 + I_2. } \]

Similarly,

\[ \boxed{ y^+ = y^- + i_1 + i_2. } \]

This makes multi-sensor fusion conceptually simple.

4.2.11.13 Independent Information Sources

Suppose three sensors contribute information:

\[ (Y_1,y_1), \]

\[ (Y_2,y_2), \]

and

\[ (Y_3,y_3). \]

Under appropriate independence assumptions,

\[ \boxed{ Y_{\text{combined}} = Y_1+Y_2+Y_3 } \]

and

\[ \boxed{ y_{\text{combined}} = y_1+y_2+y_3. } \]

Then

\[ \boxed{ P_{\text{combined}} = Y_{\text{combined}}^{-1} } \]

and

\[ \boxed{ \hat x_{\text{combined}} = P_{\text{combined}} y_{\text{combined}}. } \]

4.2.11.14 Information Addition Is Powerful—but Conditional

The simple addition rule is valid only when the information being added is not being counted twice.

This condition is crucial.

If two local filters share common measurements or common prior information, their estimation errors can become correlated.

Then simply adding their information matrices may produce an estimator that is too confident.

Therefore:

\[ \boxed{ \text{Information Addition Requires Correlation Awareness}. } \]

This becomes one of the central issues in distributed estimation.

4.2.11.15 Information Filter Prediction

The measurement update is very simple in information form.

The prediction step is less simple.

For

\[ x_k = F_{k-1}x_{k-1} + w_{k-1}, \]

the covariance prediction is

\[ \boxed{ P_k^- = F_{k-1} P_{k-1}^+ F_{k-1}^T + Q_{k-1}. } \]

Then

\[ \boxed{ Y_k^- = (P_k^-)^{-1}. } \]

The information vector becomes

\[ \boxed{ y_k^- = Y_k^-\hat x_k^-. } \]

Thus the prediction step often still requires covariance-like operations or equivalent matrix identities.

4.2.11.16 Why Measurement Update Is More Natural Than Prediction in Information Space

Measurement information is additive.

Process noise has the opposite effect: it removes certainty.

During propagation,

\[ Q \]

adds uncertainty to covariance.

Therefore,

\[ P \]

increases while

\[ Y=P^{-1} \]

typically decreases.

Conceptually,

\[ \boxed{ \text{Measurement} \rightarrow \text{Adds Information} } \]

while

\[ \boxed{ \text{Propagation with Process Noise} \rightarrow \text{Loses Information}. } \]

4.2.11.17 Covariance Form vs Information Form

A useful comparison is:

4.2.11.18 Sparse Information Matrices

In large distributed systems, the information matrix may become sparse when dependencies between state variables are local.

Sparse matrices can offer computational advantages.

This is especially important for large estimation problems involving

• distributed spacecraft,
• multi-agent systems,
• large sensor networks,
• map-based navigation,
• simultaneous localization and mapping.

However, whether sparsity is preserved depends strongly on system structure and marginalization operations.

4.2.11.19 Centralized Sensor Fusion

In a centralized architecture,

\[ \boxed{ \text{All Sensor Measurements} \rightarrow \text{One Central Estimator}. } \]

For example,

\[ \boxed{ \text{GNSS} + \text{IMU} + \text{Camera} + \text{LiDAR} + \text{Star Tracker} \rightarrow \text{Central Navigation Filter}. } \]

The central filter directly processes all measurements.

4.2.11.20 Advantages of Centralized Fusion

A centralized estimator can potentially achieve excellent performance because it has access to all raw measurement information.

Advantages include

• direct treatment of cross-correlations,
• globally consistent covariance,
• maximum use of measurement information,
• straightforward optimal fusion under correct modelling assumptions.

4.2.11.21 Limitations of Centralized Fusion

A single central estimator may become

• computationally large,
• difficult to maintain,
• sensitive to a single-point failure,
• difficult to reconfigure,
• tightly coupled to all sensor interfaces.

For spacecraft with many heterogeneous sensors, modularity can become important.

This motivates decentralized and federated structures.

4.2.11.22 Decentralized Estimation

In decentralized estimation, multiple processing units estimate parts or versions of the system state.

For example,

\[ \boxed{ \text{GNSS/IMU} \rightarrow \text{Local Filter 1} } \]

\[ \boxed{ \text{Camera} \rightarrow \text{Local Filter 2} } \]

\[ \boxed{ \text{LiDAR} \rightarrow \text{Local Filter 3}. } \]

The local estimates can then be shared or fused.

4.2.11.23 Local Filters

Each local filter may maintain

\[ \boxed{ \hat x_i, \qquad P_i } \]

or equivalently

\[ \boxed{ y_i, \qquad Y_i. } \]

A local filter may use only its assigned subset of sensors.

For example:

Local Filter 1

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

Local Filter 2

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

Local Filter 3

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

4.2.11.24 Master Filter

A master filter combines outputs from the local filters.

Conceptually,

\[ \boxed{ \text{Local Filter 1} } \]

\[ \boxed{ \text{Local Filter 2} } \]

\[ \boxed{ \text{Local Filter 3} } \]

\[ \downarrow \]

\[ \boxed{ \text{Master Fusion Filter} } \]

\[ \downarrow \]

\[ \boxed{ \text{Global Navigation Estimate}. } \]

This general structure forms the basis of many federated-filter architectures.

4.2.11.25 What Is a Federated Filter?

A Federated Filter is a distributed estimation architecture in which multiple local filters process subsets of sensor information and their outputs are combined through a master or fusion stage.

The objective is not merely estimation accuracy.

Federated architectures are often designed for

• modularity,
• redundancy,
• fault isolation,
• graceful degradation,
• distributed computation,
• sensor reconfiguration.

Thus:

\[ \boxed{ \text{Federated Filtering} = \text{Estimation} + \text{Architecture} + \text{Fault Tolerance}. } \]

4.2.11.26 Basic Federated Architecture

A generic structure is

\[ \boxed{\text{Sensor Group 1}} \rightarrow \boxed{\text{Local Filter 1}} \]

\[ \boxed{\text{Sensor Group 2}} \rightarrow \boxed{\text{Local Filter 2}} \]

\[ \boxed{\text{Sensor Group 3}} \rightarrow \boxed{\text{Local Filter 3}} \]

followed by

\[ \boxed{ \text{Local Filter Outputs} \rightarrow \text{Master Filter} \rightarrow \text{Global Estimate}. } \]

4.2.11.27 State Estimates from Local Filters

Suppose local filter i produces

\[ \boxed{ \hat x_i } \]

with covariance

\[ \boxed{ P_i. } \]

The information form is

\[ \boxed{ Y_i=P_i^{-1} } \]

and

\[ \boxed{ y_i=Y_i\hat x_i. } \]

If the local estimates are mutually independent, they can be fused as

\[ \boxed{ Y_M = \sum_iY_i } \]

and

\[ \boxed{ y_M = \sum_iy_i. } \]

Then

\[ \boxed{ P_M=Y_M^{-1} } \]

and

\[ \boxed{ \hat x_M=P_My_M. } \]

4.2.11.28 Weighted Fusion Interpretation

The information fusion equation is the multidimensional generalization of uncertainty-weighted averaging.

For two scalar estimates,

\[ x_1,\quad \sigma_1^2 \]

and

\[ x_2,\quad \sigma_2^2, \]

the fused estimate becomes

\[ \boxed{ \hat x = \frac{ x_1/\sigma_1^2 + x_2/\sigma_2^2 }{ 1/\sigma_1^2 + 1/\sigma_2^2 }. } \]

The more precise estimate contributes more strongly.

4.2.11.29 Information Distribution

Some federated architectures allocate the global information among local filters using coefficients

\[ \beta_i. \]

A common relationship is

\[ \boxed{ \sum_i\beta_i=1. } \]

Information may then be partitioned approximately according to

\[ \boxed{ P_i^{-1} = \beta_iP_g^{-1} } \]

or equivalently

\[ \boxed{ P_i = \frac{1}{\beta_i}P_g } \]

depending on architecture and reset formulation.

The interpretation is that each local filter receives only a fraction of the global information.

4.2.11.30 Why Information Distribution Is Needed

If every local filter begins with the full global prior and their outputs are later added as if independent, the same prior information is counted several times.

This produces an overly small covariance.

Therefore the architecture must account for shared prior information.

This is one reason federated-filter design is more subtle than simply averaging local estimates.

4.2.11.31 The Double-Counting Problem

Suppose two local filters both use the same previous estimate.

They therefore share common information.

If their outputs are fused assuming independence,

\[ \boxed{ Y_f=Y_1+Y_2 } \]

counts the shared prior twice.

Consequently,

\[ P_f \]

becomes unrealistically small.

The estimator appears more confident than the actual available information justifies.

Thus:

\[ \boxed{ \text{Shared Information} + \text{Naive Fusion} \rightarrow \text{Overconfidence}. } \]

4.2.11.32 Cross-Correlation Between Local Filters

Suppose local estimation errors are

\[ e_1=x-\hat x_1 \]

and

\[ e_2=x-\hat x_2. \]

Their cross covariance is

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

If

\[ P_{12}\neq0, \]

the estimates are correlated.

These correlations may arise from

• common process models,
• common prior states,
• shared IMU measurements,
• shared environmental models,
• shared GNSS measurements,
• previous information exchange.

4.2.11.33 Why Ignoring Correlation Is Dangerous

If two highly correlated estimates are fused as if independent, they appear to provide more information than they actually do.

For example, two filters may both be wrong in nearly the same direction because both rely on the same biased sensor.

Naively fusing them does not remove the common bias.

Therefore:

\[ \boxed{ \text{Two Estimates} \neq \text{Two Independent Pieces of Information}. } \]

4.2.11.34 Federated Filtering and Independence Management

A key architectural challenge is therefore to structure the local filters so that the master knows how information should be combined.

Possible strategies include

• explicit cross-covariance tracking,
• information distribution,
• reset architectures,
• no-reset architectures,
• covariance intersection,
• conservative fusion.

4.2.11.35 Federated Filter Reset Concept

In some architectures, after the master computes the global estimate,

\[ \hat x_g, \qquad P_g, \]

this global state is fed back to the local filters.

Thus:

\[ \boxed{ \text{Local Filters} \rightarrow \text{Master Fusion} \rightarrow \text{Global State} \rightarrow \text{Reset Local Filters}. } \]

This is called a reset federated architecture.

4.2.11.36 Why Reset Local Filters?

Resetting local filters helps maintain consistency between local and global estimates.

After fusion, local filters can restart from the global estimate, with information allocated according to the federation strategy.

This can prevent local filters from drifting too far apart.

However, reset also creates shared information, so correlation management remains essential.

4.2.11.37 No-Reset Federated Architecture

In a no-reset architecture, local filters continue independently after master fusion.

The master output is not fed back into each local estimator.

Conceptually,

\[ \boxed{ \text{Local Filter} \rightarrow \text{Master} } \]

but not

\[ \boxed{ \text{Master} \rightarrow \text{Local Filter}. } \]

This can improve independence between local estimators but may lead to different convergence and consistency characteristics.

4.2.11.38 Reset vs No-Reset

A simplified comparison is:

Neither is universally superior.

The correct architecture depends on mission requirements.

4.2.11.39 Federated Filter Modes

Federated filtering is often discussed in several architectural forms, such as

• centralized-equivalent structures,
• reset federated filters,
• no-reset federated filters,
• fault-tolerant federated filters,
• distributed information filters.

The exact nomenclature can vary among sources.

The important distinction is how

\[ \boxed{ \text{Information Is Allocated, Fused, and Fed Back}. } \]

4.2.11.40 Fault Detection

A major benefit of local filters is that each sensor branch can be monitored independently.

Suppose

\[ \boxed{ \text{Camera} \rightarrow \text{Camera Local Filter}. } \]

If its innovations become inconsistent, the camera branch can be flagged.

Meanwhile

\[ \boxed{ \text{GNSS Local Filter} } \]

and

\[ \boxed{ \text{LiDAR Local Filter} } \]

may remain healthy.

Thus federated architecture naturally supports fault detection and isolation.

4.2.11.41 Innovation-Based Fault Detection

Each local filter can calculate

\[ \boxed{ r_i = z_i-\hat z_i } \]

and

\[ \boxed{ S_i. } \]

Then

\[ \boxed{ NIS_i = r_i^TS_i^{-1}r_i. } \]

If one branch repeatedly produces abnormal NIS values, it may indicate

• sensor failure,
• incorrect measurement association,
• timing problems,
• model mismatch,
• frame errors,
• calibration errors.

4.2.11.42 Fault Isolation

Detection answers:

Is something wrong?

Isolation answers:

Which sensor or subsystem is causing the problem?

Because local filters separate sensor groups, disagreement between branches can provide fault-isolation capability.

For example,

\[ \boxed{ \text{GNSS Local Filter Healthy} } \]

\[ \boxed{ \text{Camera Local Filter Faulted} } \]

\[ \boxed{ \text{LiDAR Local Filter Healthy}. } \]

The master can temporarily remove the camera branch.

4.2.11.43 Graceful Degradation

Suppose one sensor fails completely.

A monolithic estimator may require substantial reconfiguration.

A federated system can instead disable the affected local branch while continuing navigation using the remaining branches.

Thus:

\[ \boxed{ \text{Sensor Failure} \rightarrow \text{Reduced Information} \neq \text{Complete Navigation Failure}. } \]

This is called graceful degradation.

4.2.11.44 Redundancy

Federated navigation allows different sensors to estimate overlapping quantities.

For example,

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

and

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

may both constrain relative position.

Likewise,

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

and

\[ \boxed{ \text{Vision Relative Pose} } \]

may both constrain attitude.

This redundancy supports both improved accuracy and fault detection.

4.2.11.45 Federated Spacecraft Navigation Example

Consider a spacecraft with

• IMU,
• GNSS,
• star tracker,
• camera,
• LiDAR.

One possible architecture is:

\[ \boxed{ \text{IMU + GNSS} \rightarrow \text{Navigation Local Filter} } \]

\[ \boxed{ \text{IMU + Star Tracker} \rightarrow \text{Attitude Local Filter} } \]

\[ \boxed{ \text{Camera} \rightarrow \text{Optical Relative-Navigation Filter} } \]

\[ \boxed{ \text{LiDAR} \rightarrow \text{Range/Pose Local Filter} } \]

followed by

\[ \boxed{ \text{Master Fusion Filter}. } \]

4.2.11.46 Federated Filter for RPO

For rendezvous and proximity operations, the architecture may change with range.

Far Range

\[ \boxed{ \text{Relative GNSS Local Filter} } \]

Mid Range

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

Close Range

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

Capture

\[ \boxed{ \text{Relative Pose + Attitude + Range Local Filters}. } \]

The master filter provides the global relative-navigation solution.

4.2.11.47 Relative-Navigation Master State

A master RPO state might be

\[ \boxed{ x_g = \begin{bmatrix} r_{rel}\\ v_{rel}\\ \theta_{rel}\\ \omega_{rel} \end{bmatrix}. } \]

Different local filters may estimate different subsets.

For example:

\[ \boxed{ \text{GNSS} \rightarrow r_{rel},v_{rel} } \]

\[ \boxed{ \text{Camera} \rightarrow \text{Bearing / Relative Pose} } \]

\[ \boxed{ \text{LiDAR} \rightarrow \text{Range / Pose} } \]

\[ \boxed{ \text{Attitude Sensors} \rightarrow \theta_{rel},\omega_{rel}. } \]

4.2.11.48 Full-State vs Partial-State Local Filters

Not every local filter must estimate the complete global state.

A local filter may estimate only

\[ \boxed{ x_i = \begin{bmatrix} r_{rel}\\ v_{rel} \end{bmatrix} } \]

while another estimates

\[ \boxed{ x_j = \begin{bmatrix} \theta_{rel}\\ \omega_{rel} \end{bmatrix}. } \]

A mapping matrix may be required when transferring information into the global state.

4.2.11.49 Common-State Fusion

If every local filter estimates the same global state vector

\[ x, \]

fusion is relatively straightforward conceptually.

If local states differ, information must first be mapped into a common coordinate/state space.

Therefore:

\[ \boxed{ \text{Fusion Requires Common State Meaning}. } \]

This includes consistent

• dimensions,
• frames,
• units,
• timestamps,
• state definitions.

4.2.11.50 Coordinate Frames in Federated Navigation

Suppose a camera filter reports relative position in the camera frame

\[ r^C \]

while GNSS reports it in the Hill frame

\[ r^H. \]

The estimates cannot be fused directly.

A transformation is required:

\[ \boxed{ r^H = C_{HC}r^C. } \]

Covariance must also be transformed:

\[ \boxed{ P^H = C_{HC} P^C C_{HC}^T. } \]

Frame consistency is essential before information fusion.

4.2.11.51 Covariance Transformation

More generally, if

\[ x_b=g(x_a), \]

then locally

\[ \boxed{ P_b = J P_a J^T } \]

where

\[ J = \frac{\partial g}{\partial x_a}. \]

This is important when local filters use different state coordinates.

Information matrices must be transformed consistently as well.

4.2.11.52 Timestamp Alignment

Suppose local filter 1 produces

\[ \hat x_1(t_k) \]

but local filter 2 provides

\[ \hat x_2(t_k-\tau). \]

Fusing them directly is inconsistent.

Therefore local estimates should be propagated or interpolated to a common fusion epoch.

Thus:

\[ \boxed{ \text{Common State} + \text{Common Frame} + \text{Common Time} } \]

are all required before fusion.

4.2.11.53 Communication Latency

Distributed estimators may run on separate processors or spacecraft.

Information may arrive with latency.

For example:

\[ \boxed{ \text{Local Filter} \rightarrow \text{Communication Link} \rightarrow \text{Master Filter}. } \]

The master must know the timestamp corresponding to each local estimate.

Delayed information may require

• propagation,
• buffering,
• out-of-sequence update techniques,
• delayed-state filters.

4.2.11.54 Distributed Spacecraft Navigation

Information filtering is particularly attractive for multi-spacecraft systems.

Suppose spacecraft A and B each maintain local navigation estimates.

They may exchange

\[ \boxed{ \text{Information Vector} } \]

and

\[ \boxed{ \text{Information Matrix} } \]

rather than raw measurements.

This can reduce communication requirements in some architectures.

4.2.11.55 Cooperative Navigation

In cooperative navigation, spacecraft can use inter-satellite measurements such as

• range,
• range rate,
• relative bearing,
• time-of-flight,
• inter-satellite GNSS,
• optical relative measurements.

Each spacecraft has local knowledge.

Information exchange enables cooperative state estimation.

Conceptually,

\[ \boxed{ \text{Local Navigation} + \text{Inter-Satellite Information} \rightarrow \text{Cooperative Navigation}. } \]

4.2.11.56 Common-Information Problem in Cooperative Networks

When information is exchanged repeatedly,

\[ A\rightarrow B \]

and later

\[ B\rightarrow A, \]

A may receive information that originally came from A itself.

If this lineage is not tracked, the same information can be counted repeatedly.

This is known as the common-information problem.

It is closely related to the cross-correlation problem in federated filters.

4.2.11.57 Covariance Intersection

When two estimates must be fused but their cross-correlation is unknown, one conservative technique is Covariance Intersection (CI).

Suppose

\[ (\hat x_1,P_1) \]

and

\[ (\hat x_2,P_2) \]

must be combined.

CI forms

\[ \boxed{ P_{CI}^{-1} = \omega P_1^{-1} + (1-\omega)P_2^{-1} } \]

with

\[ 0\le\omega\le1. \]

The corresponding state estimate is

\[ \boxed{ \hat x_{CI} = P_{CI} \left[ \omega P_1^{-1}\hat x_1 + (1-\omega) P_2^{-1}\hat x_2 \right]. } \]

4.2.11.58 Why Covariance Intersection Is Useful

CI does not require the cross-correlation between the estimates to be known explicitly.

It produces a conservative fused covariance under appropriate conditions.

This is attractive when

• communication histories are complex,
• local filters share unknown common information,
• distributed fusion makes correlation tracking difficult.

However, conservatism may reduce the amount of information gained.

4.2.11.59 Selecting the CI Weight

The parameter

\[ \omega \]

may be chosen to optimize a scalar measure of the resulting covariance.

Possible objectives include minimizing

\[ \boxed{ \operatorname{tr}(P_{CI}) } \]

or

\[ \boxed{ \det(P_{CI}). } \]

The optimum depends on the application and uncertainty geometry.

4.2.11.60 Information Filter for Independent Measurements

Suppose independent measurements are received sequentially.

Starting with

\[ Y^-, \qquad y^-, \]

sensor 1 adds

\[ H_1^TR_1^{-1}H_1 \]

and

\[ H_1^TR_1^{-1}z_1. \]

Sensor 2 adds

\[ H_2^TR_2^{-1}H_2 \]

and

\[ H_2^TR_2^{-1}z_2. \]

Therefore:

\[ \boxed{ Y^+ = Y^- + \sum_i H_i^TR_i^{-1}H_i } \]

and

\[ \boxed{ y^+ = y^- + \sum_i H_i^TR_i^{-1}z_i. } \]

This equation captures the elegance of information-based multi-sensor fusion.

4.2.11.61 Information Form and Measurement Availability

If a sensor is unavailable, it simply contributes no new information.

Thus:

\[ \boxed{ \text{Sensor Dropout} \rightarrow \text{No Information Increment From That Sensor}. } \]

Other sensors can continue contributing information.

This provides a natural modular interpretation.

4.2.11.62 Sensor Quality in Information Space

Suppose two sensors measure the same quantity.

Sensor A has

\[ R_A=0.01 \]

while sensor B has

\[ R_B=1. \]

Their information values are proportional to

\[ R_A^{-1}=100 \]

and

\[ R_B^{-1}=1. \]

Thus sensor A contributes much more information.

This makes sensor quality particularly intuitive in information space.

4.2.11.63 Faulted Sensor Information

A failed sensor should not continue adding incorrect information.

If a sensor becomes unreliable, the fusion architecture may

• reject measurements,
• increase R,
• isolate its local filter,
• remove its information contribution.

Therefore:

\[ \boxed{ \text{Fault Isolation} \rightarrow \text{Remove or Reduce Faulted Information}. } \]

4.2.11.64 Adaptive Information Sharing

The amount of information shared by a local filter can be changed according to

• sensor health,
• geometric quality,
• estimated covariance,
• mission phase,
• computational availability.

For example, close-range LiDAR may receive higher weighting during capture while GNSS is downweighted.

This creates a mission-adaptive federated navigation architecture.

4.2.11.65 Federated Fusion During RPO Sensor Handover

A representative RPO sequence may be

\[ \boxed{ \text{Far Range: GNSS Dominant} } \]

\[ \downarrow \]

\[ \boxed{ \text{Mid Range: GNSS + Optical} } \]

\[ \downarrow \]

\[ \boxed{ \text{Close Range: Optical + LiDAR} } \]

\[ \downarrow \]

\[ \boxed{ \text{Capture: High-Precision Relative Pose}. } \]

A federated architecture can adjust which local filters contribute to the master without restarting the global estimator.

4.2.11.66 Fault-Tolerant RPO Example

Suppose near capture the system has

\[ \boxed{ \text{Camera Local Filter} } \]

and

\[ \boxed{ \text{LiDAR Local Filter}. } \]

If camera feature tracking fails,

\[ \boxed{ \text{Camera Branch} \rightarrow \text{Rejected} } \]

while

\[ \boxed{ \text{LiDAR Branch} \rightarrow \text{Continue}. } \]

The master covariance should increase appropriately because total available navigation information has decreased.

This is the correct physical response.

4.2.11.67 Global Covariance Must Reflect Sensor Loss

A fault-tolerant estimator should not simply continue outputting the same covariance after losing a sensor.

If information is removed,

\[ \boxed{ \text{Total Information}\downarrow } \]

and therefore

\[ \boxed{ \text{Global Uncertainty}\uparrow. } \]

Thus fault isolation and covariance management must be linked.

4.2.11.68 Master-Filter Health Monitoring

The master can monitor

• local innovations,
• local NIS,
• local covariance growth,
• disagreement between estimates,
• sensor health flags,
• local-filter convergence,
• timestamps,
• communication status.

A local estimator that becomes inconsistent can be isolated before it corrupts the global solution.

4.2.11.69 Local Filter Disagreement

Suppose

\[ \hat x_1 \]

and

\[ \hat x_2 \]

differ significantly.

The difference can be evaluated relative to expected covariance.

A large disagreement may indicate

• sensor fault,
• incorrect noise model,
• calibration error,
• frame mismatch,
• timestamp mismatch,
• unmodelled common bias.

Thus local-filter disagreement itself becomes a health-monitoring signal.

4.2.11.70 Federated Navigation and Common Biases

Suppose GNSS and camera estimates both depend on an incorrect attitude transformation.

Their errors are then correlated through a common modelling error.

A federated architecture does not automatically remove such common-mode faults.

Therefore:

\[ \boxed{ \text{Redundant Filters} \neq \text{Independence From Common-Mode Errors}. } \]

Common-mode failure analysis is essential.

4.2.11.71 Local EKF / UKF / ESKF Combinations

The local filters do not all need to use the same estimator type.

For example:

\[ \boxed{ \text{IMU/GNSS} \rightarrow \text{ESKF} } \]

\[ \boxed{ \text{Camera Relative Navigation} \rightarrow \text{EKF or UKF} } \]

\[ \boxed{ \text{Attitude} \rightarrow \text{MEKF}. } \]

The master then fuses compatible local state information.

Therefore the term federated filter describes the architecture, not necessarily one specific local filter algorithm.

4.2.11.72 Nonlinear Federated Filters

If local systems are nonlinear, local estimators may use

• EKF,
• ESKF,
• MEKF,
• UKF.

Fusion still requires careful treatment of

• state coordinates,
• covariance,
• error definitions,
• correlations.

For attitude states in particular, naive averaging of quaternions should be avoided.

4.2.11.73 Federated Attitude Fusion

Suppose two attitude estimators provide

\[ q_1 \]

and

\[ q_2. \]

Because quaternions lie on a nonlinear manifold,

\[ \boxed{ q_f \neq w_1q_1+w_2q_2 } \]

in general.

One approach is to choose a nominal attitude and fuse small relative attitude errors in a local tangent space.

This is conceptually consistent with MEKF and ESKF principles.

4.2.11.74 Distributed Relative Navigation

In multi-spacecraft rendezvous, each vehicle may maintain its own estimate.

For chaser C and target T,

\[ x_C \]

and

\[ x_T \]

may be estimated separately.

Relative state is

\[ \boxed{ r_{rel} = r_T-r_C } \]

and

\[ \boxed{ v_{rel} = v_T-v_C. } \]

If the underlying absolute estimates are correlated, relative covariance must account for cross-correlation.

4.2.11.75 Relative Covariance with Correlation

For

\[ r_{rel}=r_T-r_C, \]

the covariance is

\[ \boxed{ P_{rel} = P_T + P_C - P_{TC} - P_{CT}. } \]

Ignoring cross covariance can produce incorrect relative-navigation uncertainty.

This is particularly important in cooperative GNSS and formation flying.

4.2.11.76 Communication-Constrained Navigation

In distributed spacecraft systems, bandwidth may be limited.

Instead of transmitting all raw sensor data, a spacecraft may transmit

• state estimate,
• covariance,
• information increment,
• compressed sufficient statistics.

This can reduce communication load.

However, information lineage and correlation must still be managed.

4.2.11.77 Federated Navigation and Autonomy

A modular federated estimator supports autonomous reconfiguration.

For example:

\[ \boxed{ \text{Sensor Health Monitoring} } \]

\[ \downarrow \]

\[ \boxed{ \text{Select Healthy Local Filters} } \]

\[ \downarrow \]

\[ \boxed{ \text{Fuse Available Information} } \]

\[ \downarrow \]

\[ \boxed{ \text{Update Global Navigation Solution}. } \]

This is valuable for autonomous spacecraft and servicing missions.

4.2.11.78 Information Matrix and Observability

If certain state directions receive little or no measurement information, the information matrix may remain weak along those directions.

Conceptually,

\[ \boxed{ \text{Weak Observability} \rightarrow \text{Low Information in Certain State Directions}. } \]

Thus the eigenstructure of covariance or information matrices can provide insight into observability.

4.2.11.79 Information Growth from Repeated Measurements

If several independent measurements observe the same quantity, information accumulates.

For a scalar state with measurement variance

\[ R, \]

each independent measurement contributes approximately

\[ R^{-1}. \]

Therefore repeated observations increase information.

However, if measurements are correlated, this simple accumulation is no longer valid.

4.2.11.80 Process Noise Reduces Information

While measurements add information, process noise increases uncertainty.

Therefore during propagation,

\[ \boxed{ P\uparrow } \]

and generally

\[ \boxed{ Y=P^{-1}\downarrow. } \]

Thus estimation can be viewed as a continual competition between

\[ \boxed{ \text{Information Loss Through Uncertain Dynamics} } \]

and

\[ \boxed{ \text{Information Gain Through Measurements}. } \]

4.2.11.81 Numerical Considerations

Information filtering requires matrix inverses or linear solves involving covariance and information matrices.

Important numerical issues include

• positive definiteness,
• symmetry,
• conditioning,
• state scaling,
• sparse matrix methods,
• factorization methods.

Explicit inverse operations should generally be replaced by numerically stable linear solves where possible.

4.2.11.82 Singular or Weakly Informed States

If a state direction has effectively infinite uncertainty,

\[ P\rightarrow\infty, \]

then information approaches zero:

\[ Y\rightarrow0. \]

This can be conceptually convenient in information space.

However, exactly singular covariance/information matrices require careful numerical treatment.

4.2.11.83 Square-Root Information Filters

Just as square-root covariance filters exist, one can work with matrix factors of the information matrix.

For example,

\[ \boxed{ Y = R_I^TR_I. } \]

Square-root information filters can offer improved numerical stability and are important in some large-scale estimation problems.

4.2.11.84 Federated Filter Initialization

Each local filter must be initialized consistently.

This includes

\[ \hat x_{i,0} \]

and

\[ P_{i,0}. \]

If local filters receive information fractions,

\[ \beta_i, \]

their covariances must reflect that allocation.

Incorrect initialization can produce

• inconsistent fusion,
• double counting,
• slow convergence,
• overconfidence.

4.2.11.85 Local Filter Covariance Must Be Credible

If one local filter reports an unrealistically small covariance, the master may give it excessive influence.

Therefore federated fusion depends heavily on local covariance consistency.

A local filter that is accurate but overconfident can still damage the master estimate.

4.2.11.86 NIS for Local Filters

Each local estimator should evaluate

\[ \boxed{ NIS_i = r_i^TS_i^{-1}r_i. } \]

This supports

• local consistency monitoring,
• sensor gating,
• fault detection,
• adaptive weighting.

The master may use these health metrics when deciding which branches to accept.

4.2.11.87 NEES for Federated Validation

When truth is available,

\[ \boxed{ NEES = e^TP^{-1}e } \]

can be calculated for

• each local filter,
• the master filter,
• the complete federated architecture.

This helps determine whether information fusion produces a statistically credible global covariance.

4.2.11.88 Monte Carlo Verification

A federated estimator should be tested under randomized conditions including

• local sensor noise,
• sensor biases,
• timing errors,
• communication delays,
• local-filter initialization,
• sensor dropout,
• common-mode errors,
• outliers,
• incorrect associations,
• local filter faults,
• master filter reconfiguration.

Evaluate

\[ \boxed{ RMSE } \]

\[ \boxed{ NIS } \]

\[ \boxed{ NEES } \]

\[ \boxed{ Fault~Detection~Rate } \]

\[ \boxed{ False~Alarm~Rate } \]

\[ \boxed{ Isolation~Time } \]

and

\[ \boxed{ Navigation~Continuity. } \]

4.2.11.89 Fault-Injection Testing

Deliberately inject faults such as

• camera bias,
• LiDAR range jump,
• GNSS outage,
• IMU bias drift,
• frozen sensor,
• stale timestamp,
• attitude-frame error,
• communication loss.

Verify that

\[ \boxed{ \text{Fault} \rightarrow \text{Detection} \rightarrow \text{Isolation} \rightarrow \text{Reconfiguration} } \]

occurs without catastrophic corruption of the global navigation estimate.

4.2.11.90 Example RPO Federated Architecture

A practical conceptual architecture can be

\[ \boxed{ \text{Relative GNSS} + \text{IMU} \rightarrow \text{Local Filter A} } \]

\[ \boxed{ \text{Camera} + \text{IMU} \rightarrow \text{Local Filter B} } \]

\[ \boxed{ \text{LiDAR} \rightarrow \text{Local Filter C} } \]

\[ \boxed{ \text{Star Tracker / Relative Attitude} \rightarrow \text{Local Filter D} } \]

then

\[ \boxed{ A+B+C+D \rightarrow \text{Master Fusion Filter}. } \]

The master outputs

\[ \boxed{ \hat r_{rel}, \hat v_{rel}, \hat q_{rel}, \hat\omega_{rel}, P_g. } \]

4.2.11.91 Navigation Output to Guidance and Control

The master estimator supplies the global navigation solution to guidance and control.

Thus:

\[ \boxed{ \text{Sensors} \rightarrow \text{Local Filters} \rightarrow \text{Master Filter} \rightarrow \hat x_g,P_g } \]

then

\[ \boxed{ \hat x_g,P_g \rightarrow \text{Guidance} \rightarrow \text{Control} \rightarrow \text{Actuators}. } \]

4.2.11.92 Capture Readiness with Federated Navigation

For RPO capture, the navigation system may evaluate

\[ \|\hat r_{rel}\| < r_{\text{capture}} \]

\[ \|\hat v_{rel}\| < v_{\text{capture}} \]

\[ \|\hat\theta_{rel}\| < \theta_{\text{capture}} \]

and

\[ \|\hat\omega_{rel}\| < \omega_{\text{capture}}. \]

But the master covariance

\[ P_g \]

must also indicate sufficient confidence.

Therefore:

\[ \boxed{ \text{Capture Readiness} = \text{State Within Limits} + \text{Navigation Confidence} + \text{Sensor Health}. } \]

4.2.11.93 Federated Filter vs Centralized Filter

A centralized filter may achieve better theoretical information efficiency because it explicitly handles all sensor relationships.

A federated filter sacrifices some simplicity or optimality in exchange for

• modularity,
• fault isolation,
• redundancy,
• scalability,
• reconfigurability.

Therefore estimator architecture is an engineering trade.

4.2.11.94 Federated Filter vs Sequential Measurement Updates

Sequential measurement updates within one Kalman filter are still centralized estimation.

Federated filtering is different because independent or semi-independent local filters are maintained.

Thus:

\[ \boxed{ \text{Multiple Measurements} \neq \text{Federated Filter}. } \]

Federation concerns estimator architecture, not simply the number of sensors.

4.2.11.95 Federated Filter vs Sensor-Level Fusion

In sensor-level fusion, raw measurements are combined inside one estimator.

In federated fusion, higher-level estimates or information from local estimators are combined.

Thus:

\[ \boxed{ \text{Raw Measurement Fusion} } \]

and

\[ \boxed{ \text{Estimate-Level Fusion} } \]

are different architectural choices.

4.2.11.96 Advantages of Information Filters

Information-form estimators provide several advantages:

• measurement information is naturally additive,
• multi-sensor fusion can be modular,
• sparse representations may be exploited,
• useful for distributed estimation,
• convenient interpretation of information contribution,
• good conceptual connection to Bayesian fusion.

4.2.11.97 Limitations of Information Filters

Important limitations include

• prediction is not as simple as measurement fusion,
• inversion/conditioning issues,
• correlations must still be handled,
• information matrices can become dense,
• nonlinear systems still require EKF/UKF-type treatment,
• manifold states such as quaternions require special care.

Information form does not automatically solve every distributed-estimation problem.

4.2.11.98 Advantages of Federated Filters

A well-designed federated architecture can provide

• fault isolation,
• redundancy,
• modular sensor integration,
• distributed computation,
• graceful degradation,
• easier sensor reconfiguration,
• mission-phase-dependent sensor use.

These features can be as important as raw estimation accuracy for autonomous spacecraft.

4.2.11.99 Limitations of Federated Filters

Potential challenges include

• inter-filter correlations,
• information double counting,
• complex initialization,
• communication latency,
• synchronization,
• master-filter design,
• reduced optimality compared with a fully centralized estimator,
• complex fault-management logic,
• state-mapping issues.

4.2.11.100 Common Implementation Errors

Typical mistakes include:

• adding information matrices from correlated estimates as if independent,
• double counting the global prior,
• inconsistent state ordering,
• inconsistent coordinate frames,
• inconsistent timestamps,
• incorrectly transforming covariance,
• assuming local filter covariance is always trustworthy,
• forgetting common-mode sensor errors,
• using different units between local filters,
• fusing quaternion components additively,
• failing to account for communication delay,
• not increasing uncertainty after sensor loss,
• continuing to fuse a faulted branch,
• resetting local filters inconsistently,
• ignoring common information after feedback.

4.2.11.101 Software Architecture

A clean federated software architecture may contain

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

followed by parallel local estimators:

\[ \boxed{\text{Local Filter A}} \]

\[ \boxed{\text{Local Filter B}} \]

\[ \boxed{\text{Local Filter C}} \]

Each produces

\[ \boxed{ \hat x_i,P_i,\text{health}_i. } \]

Then:

\[ \boxed{ \text{Time/Frame Alignment} } \]

\[ \downarrow \]

\[ \boxed{ \text{Correlation / Information Management} } \]

\[ \downarrow \]

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

\[ \downarrow \]

\[ \boxed{ \hat x_g,P_g. } \]

Finally:

\[ \boxed{ \text{Navigation Output} \rightarrow \text{Guidance and Control}. } \]

4.2.11.102 Recommended Health Outputs

Each local filter should ideally provide more than just

\[ \hat x_i. \]

Useful outputs include

\[ \boxed{ \hat x_i } \]

\[ \boxed{ P_i } \]

\[ \boxed{ r_i } \]

\[ \boxed{ S_i } \]

\[ \boxed{ NIS_i } \]

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

\[ \boxed{ \text{Filter Health Flag}. } \]

These allow intelligent master-filter decisions.

4.2.11.103 Mission-Phase Reconfiguration

The federation can be reconfigured according to mission phase.

For example:

Cruise

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

Far-RPO

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

Mid-RPO

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

Close-RPO

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

Capture

\[ \boxed{ \text{Precision Pose + Attitude Sensors}. } \]

The master architecture remains conceptually the same while the active information sources change.

4.2.11.104 Key Equations at a Glance

Information Matrix

\[ \boxed{ Y=P^{-1} } \]

Information Vector

\[ \boxed{ y=Y\hat x } \]

Recover Covariance

\[ \boxed{ P=Y^{-1} } \]

Recover State

\[ \boxed{ \hat x=Y^{-1}y } \]

Measurement Information

\[ \boxed{ I_z = H^TR^{-1}H } \]

Measurement Information Vector

\[ \boxed{ i_z = H^TR^{-1}z } \]

Information Update

\[ \boxed{ Y^+ = Y^-+H^TR^{-1}H } \]

Information-Vector Update

\[ \boxed{ y^+ = y^-+H^TR^{-1}z } \]

Independent Information Fusion

\[ \boxed{ Y_f = \sum_iY_i } \]

\[ \boxed{ y_f = \sum_iy_i } \]

Fused State

\[ \boxed{ P_f = Y_f^{-1} } \]

\[ \boxed{ \hat x_f = P_fy_f. } \]

Covariance Intersection

\[ \boxed{ P_{CI}^{-1} = \omega P_1^{-1} + (1-\omega)P_2^{-1} } \]

and

\[ \boxed{ \hat x_{CI} = P_{CI} \left[ \omega P_1^{-1}\hat x_1 + (1-\omega)P_2^{-1}\hat x_2 \right]. } \]

4.2.11.105 Information and Federated Filtering in One Diagram

The complete architecture can be remembered as

\[ \boxed{ \text{Sensor Group 1} \rightarrow \text{Local Filter 1} } \]

\[ \boxed{ \text{Sensor Group 2} \rightarrow \text{Local Filter 2} } \]

\[ \boxed{ \text{Sensor Group 3} \rightarrow \text{Local Filter 3} } \]

then

\[ \boxed{ \text{Local Estimates + Covariances + Health} } \]

\[ \downarrow \]

\[ \boxed{ \text{Frame and Time Alignment} } \]

\[ \downarrow \]

\[ \boxed{ \text{Correlation / Information Management} } \]

\[ \downarrow \]

\[ \boxed{ \text{Master Fusion Filter} } \]

\[ \downarrow \]

\[ \boxed{ \text{Global State + Global Covariance} } \]

\[ \downarrow \]

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

4.2.11.106 Key Takeaways

The Information and Federated Filter concepts should be understood through the following points:

• The covariance form represents a Gaussian estimate using

\[ \hat x,\;P. \]

• The information form represents the same knowledge using

\[ y,\;Y. \]

• The information matrix is

\[ Y=P^{-1}. \]

• The information vector is

\[ y=Y\hat x. \]

• Large information corresponds to small uncertainty.
• A linear measurement contributes

\[ H^TR^{-1}H \]

to the information matrix.

• Measurement information is naturally additive when measurements are independent.
• This makes the information form attractive for distributed and multi-sensor estimation.
• Local filters can independently process different sensor groups.
• A master filter combines local estimates into a global navigation solution.
• Federated filtering is an estimator architecture, not one specific Kalman-filter equation.
• Local filters may themselves be KF, EKF, ESKF, MEKF, or UKF implementations.
• Federated architectures support modularity and redundancy.
• They are useful for fault detection and isolation.
• A failed sensor branch can be removed while the remaining filters continue.
• Global covariance should increase appropriately when information is lost.
• Shared prior information can create correlations between local filters.
• Naively adding correlated information causes double counting.
• Cross-correlation management is therefore one of the most important problems in distributed estimation.
• Covariance Intersection provides a conservative fusion approach when cross-correlation is unknown.
• Frame consistency is mandatory before fusion.
• Time alignment is mandatory before fusion.
• Covariance must be transformed together with the state.
• Common-mode errors can affect several local filters simultaneously.
• Redundancy does not automatically protect against common-mode faults.
• RPO navigation is a natural application for federated filtering because different sensors dominate at different ranges.
• GNSS, camera, LiDAR, IMU, and attitude sensors can operate through separate local filters.
• Mission-phase sensor handover can be implemented by changing which local filters contribute to the master.
• Local NIS monitoring is useful for health assessment.
• Master-level NEES and Monte Carlo analysis should be used during verification.
• Fault-injection testing should include sensor failures, timing errors, communication loss, and common biases.
• The master should output both

\[ \hat x_g \]

and

\[ P_g. \]

• Guidance and control should use the global navigation state together with credible uncertainty and health information.

The most important information-filter idea is

\[ \boxed{ \text{Posterior Information} = \text{Prior Information} + \text{New Independent Measurement Information}. } \]

The most important federated-filter idea is

\[ \boxed{ \text{Multiple Local Estimators} + \text{Careful Information Management} \rightarrow \text{Robust Global Navigation Solution}. } \]

And the most important warning is

\[ \boxed{ \text{Never Treat Correlated Estimates as Independent Simply Because They Come From Different Filters}. } \]

For spacecraft navigation, the real value of a federated architecture is therefore not just combining more sensors. It is the ability to build a navigation system that is modular, distributed, uncertainty-aware, fault-tolerant, reconfigurable, and capable of continuing safely when individual information sources degrade or fail.

Next: 4.2.12 Particle Filter

Information and federated filtering retain a Gaussian mean-and-covariance description of uncertainty. The next section moves to particle-based Bayesian estimation, where the posterior is represented using a weighted set of samples rather than a single Gaussian approximation.