Estimator Design Workflow and Algorithm Selection

4.3.18 Estimator Design Workflow and Algorithm Selection

Estimator design is not simply about choosing an EKF, UKF, least-squares solver, or some advanced nonlinear filter. It is a system-level design problem. The estimator has to be selected according to the system physics, the measurements available, what must be estimated, how uncertainty evolves, computational limits, and whether the estimate is required in real time or can be produced later through post-processing.

The central design question is

\[ \boxed{ \text{What estimation architecture best matches the available information and the decision that must be made?} } \]

An estimator may therefore be static or dynamic, batch or sequential, linear or nonlinear, deterministic or probabilistic, filtering or smoothing, state-only or parameter-augmented, centralized or distributed, real-time or post-processed. There is no universally best estimator; the architecture must follow the estimation problem.

4.3.18.1 Start with the Estimation Problem, Not the Algorithm

The first mistake to avoid is beginning with:

“I will use an EKF.”

or:

“I will use batch least squares.”

The algorithm should come later.

First answer:

\[ \boxed{ \begin{array}{c} \text{What is unknown?}\\ \text{What is measured?}\\ \text{How do the unknowns evolve?}\\ \text{When is the estimate required?} \end{array}} \]

These four questions determine most of the architecture.

For example, estimating a constant calibration bias from an offline dataset is fundamentally different from estimating a moving spacecraft state continuously onboard.

4.3.18.2 Define the Estimation Objective

Before designing the estimator, define exactly what it must provide.

Possible outputs include:

• position,
• velocity,
• attitude,
• angular rate,
• orbital state,
• relative state,
• clock bias and drift,
• sensor biases,
• aerodynamic parameters,
• SRP parameters,
• thrust magnitude,
• burn timing,
• thrust direction,
• actuator scale factors,
• alignment errors,
• calibration parameters.

A general estimation vector may be written as

\[ \boxed{ \mathbf x= \begin{bmatrix} \mathbf x_{\rm physical}\\ \boldsymbol\theta_{\rm parameters}\\ \mathbf b_{\rm biases} \end{bmatrix}} \]

when physical states, model parameters, and biases are estimated together.

The purpose of explicitly writing the state is to make clear what the estimator is allowed to modify.

4.3.18.3 Define the State Vector Carefully

For basic translational orbital navigation,

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

may be sufficient.

A more complicated estimator might use

\[ \boxed{ \mathbf x= \begin{bmatrix} \mathbf r\\ \mathbf v\\ C_D\\ C_R\\ \mathbf b\\ \Delta\mathbf v_{\rm maneuver} \end{bmatrix}} \]

to estimate dynamic parameters and maneuver corrections simultaneously.

But every additional estimated quantity creates another direction the measurements must constrain.

Therefore,

\[ \boxed{ \text{More Estimated States} \neq \text{Better Estimator} } \]

unless those additional states are observable or identifiable.

4.3.18.4 Separate States, Parameters, and Disturbances

These quantities should not be treated as if they were the same.

Dynamic states

They evolve according to the system equations.

Examples:

\[ \mathbf r,\quad \mathbf v,\quad \mathbf q,\quad \boldsymbol\omega. \]

Parameters

They characterize the model and may be constant or slowly varying.

Examples:

\[ C_D,\quad C_R,\quad \text{sensor scale factor},\quad \text{alignment parameter}. \]

Disturbances

These represent forcing that is uncertain or not modeled explicitly.

Examples include:

• unmodeled accelerations,
• uncertain environmental forces,
• unresolved disturbances.

Estimator design should reflect these physical differences.

4.3.18.5 Define the Dynamic Model

A dynamic estimator requires a state-evolution model:

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

or in discrete form,

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

Here:

• \mathbf x = state,
• \mathbf u = known input,
• f = dynamic model,
• \mathbf w = process uncertainty.

A highly sophisticated filter does not compensate indefinitely for fundamentally incorrect dynamics. The estimator can only perform as well as the combined adequacy of the physics model and uncertainty model.

4.3.18.6 Define the Measurement Model

The general measurement model is

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

The estimator must know what each sensor physically measures.

Examples include:

Range

\[ \rho=\|\mathbf r\| \]

Range rate

\[ \dot\rho = \hat{\boldsymbol\rho}^{T}\mathbf v \]

LOS direction

\[ \hat{\boldsymbol\rho} = \frac{\mathbf r} {\|\mathbf r\|}. \]

The measurement equation is what connects sensor information to the estimated state.

4.3.18.7 Static versus Dynamic Estimation

A static estimator assumes the unknown quantity remains fixed:

\[ \boxed{ \mathbf z = h(\boldsymbol\theta)+\mathbf v. } \]

Typical uses include:

• fixed calibration parameters,
• bias estimation from a dataset,
• static parameter identification.

A dynamic estimator explicitly represents time evolution:

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

Hence,

\[ \boxed{ \text{Static} \rightarrow \text{Fixed Unknowns} } \]

and

\[ \boxed{ \text{Dynamic} \rightarrow \text{Time-Varying States}. } \]

Use dynamic estimation whenever the state evolves significantly over the observation interval.

4.3.18.8 Batch Estimation

Batch estimation processes an entire observation set together.

A representative cost function is

\[ \boxed{ J(\mathbf x_0) = \sum_{k=1}^{N} \mathbf r_k^T \mathbf R_k^{-1} \mathbf r_k } \]

with

\[ \mathbf r_k = \mathbf z_k- h_k(\mathbf x_k). \]

Possible unknowns include:

• initial/reference state,
• biases,
• maneuver parameters,
• dynamic parameters.

Batch processing is especially useful when the entire measurement arc is available.

4.3.18.9 Advantages of Batch Estimation

Batch estimation has several strengths:

• all observations are available simultaneously,
• complete measurement geometry can be exploited,
• long-arc parameters can be estimated,
• residuals can be examined globally,
• weak parameters and correlations can be diagnosed,
• the same data can be reprocessed,
• alternative dynamic or measurement models can be compared.

This makes batch estimation particularly useful for:

• orbit determination,
• trajectory reconstruction,
• calibration,
• parameter estimation,
• maneuver reconstruction,
• estimator validation.

4.3.18.10 Limitations of Batch Estimation

Batch estimation is usually less suitable for immediate real-time navigation because:

• the complete dataset may need to be available,
• large datasets require memory and computation,
• repeated nonlinear optimization may be expensive,
• a complete historical solution may need to be recomputed.

Thus if the current navigation state must always be available, sequential methods are usually more natural.

4.3.18.11 Sequential Estimation

Sequential estimation processes information as it becomes available:

\[ \boxed{ \text{Previous Estimate} \rightarrow \text{Prediction} \rightarrow \text{Measurement} \rightarrow \text{Correction}. } \]

The estimator maintains

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

and uncertainty

\[ \mathbf P_k. \]

This architecture is suited to continuous and real-time navigation.

4.3.18.12 Prediction and Measurement Update

Sequential estimation has two core stages.

Prediction

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

The dynamic model predicts the next state.

Measurement update

The predicted measurement is compared with the observation:

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

where \mathbf y_k is the innovation.

Thus sequential estimation repeatedly alternates:

\[ \boxed{ \text{Physics-Based Prediction} \leftrightarrow \text{Measurement-Based Correction}. } \]

4.3.18.13 Batch versus Sequential Estimation

The fundamental distinction is:

\[ \boxed{ \text{Batch} \rightarrow \text{Process an Observation Arc Together} } \]

versus

\[ \boxed{ \text{Sequential} \rightarrow \text{Update as Measurements Arrive}. } \]

Neither is universally superior.

Use batch processing when global arc information is valuable.

Use sequential estimation when the current state must be available continuously.

4.3.18.14 Hybrid Batch-Sequential Architecture

A very practical architecture is:

\[ \boxed{ \text{Batch Initialization} \rightarrow \text{Sequential Navigation} \rightarrow \text{Periodic Batch Recalibration}. } \]

Batch processing can determine initial states and difficult parameters.

A sequential filter can then maintain the real-time state.

Later batch processing can refine calibration and diagnose model deficiencies.

4.3.18.15 Linear Estimation

For a linear dynamic model:

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

and measurement model:

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

For correctly modeled Gaussian uncertainties, the Kalman filter provides the classical minimum-variance recursive estimator.

Linear estimation is attractive because it is:

• computationally efficient,
• mathematically transparent,
• comparatively easy to verify.

4.3.18.16 When Linear Estimation Is Appropriate

A linear method may be sufficient when:

• dynamics are naturally linear,
• measurement equations are linear,
• deviations from a reference trajectory are small,
• nonlinear effects are small compared with measurement and model uncertainty.

Some spacecraft problems are nonlinear globally but approximately linear locally.

That distinction is important.

4.3.18.17 Nonlinear Estimation

Many navigation problems satisfy

\[ \boxed{ \dot{\mathbf x} = f(\mathbf x,t) } \]

and

\[ \boxed{ \mathbf z = h(\mathbf x)+\mathbf v } \]

with nonlinear f and h.

Examples include:

• orbital dynamics,
• range,
• bearing angles,
• attitude dynamics,
• optical measurements,
• nonlinear disturbance forces.

A nonlinear or locally linearized estimator is then required.

4.3.18.18 Extended Kalman Filter

The EKF propagates the nonlinear system but locally linearizes it.

For the measurement:

\[ \boxed{ \mathbf H_k = \left. \frac{\partial h} {\partial\mathbf x} \right|_{\hat{\mathbf x}_k^-}. } \]

Dynamics are similarly linearized through a Jacobian or STM.

The EKF remains widely used because it provides a practical compromise between nonlinear modeling and computational cost.

4.3.18.19 EKF Limitations

The EKF may perform poorly when:

• initial errors are large,
• system nonlinearities are severe,
• geometry is weak,
• Jacobians are incorrect,
• uncertainty is strongly non-Gaussian,
• covariance becomes inconsistent.

Thus,

\[ \boxed{ \text{EKF Performance} \text{ depends on valid local linearization.} } \]

Changing to an EKF does not solve an unobservable problem.

4.3.18.20 Iterated EKF

The Iterated EKF performs repeated measurement linearization during one update:

\[ \boxed{ \text{Predict} \rightarrow \text{Linearize} \rightarrow \text{Correct} \rightarrow \text{Relinearize} \rightarrow \text{Correct Again}. } \]

It can improve nonlinear measurement updates, though at additional computational cost.

4.3.18.21 Unscented Kalman Filter

The UKF does not explicitly linearize using a Jacobian.

Instead it constructs sigma points around the mean and covariance:

\[ \boxed{ (\hat{\mathbf x},\mathbf P) \rightarrow \text{Sigma Points} \rightarrow f(\cdot),h(\cdot) \rightarrow (\hat{\mathbf x}^{+},\mathbf P^{+}). } \]

The transformed sigma points are used to approximate the propagated mean and covariance.

4.3.18.22 EKF versus UKF

The fundamental difference is:

\[ \boxed{ \text{EKF} \rightarrow \text{Local Derivatives} } \]

while

\[ \boxed{ \text{UKF} \rightarrow \text{Sigma-Point Propagation}. } \]

The UKF can represent certain nonlinear transformations better without analytical Jacobians, but normally requires more model evaluations.

Choice depends on:

• nonlinearity,
• state dimension,
• processor resources,
• availability and reliability of Jacobians,
• demonstrated performance.

4.3.18.23 Particle Filtering

Particle filters represent a posterior through samples:

\[ \boxed{ p(\mathbf x_k|\mathbf z_{1:k}) \approx \sum_{i=1}^{N} w_k^{(i)} \delta( \mathbf x_k-\mathbf x_k^{(i)} ). } \]

They are useful for:

• multimodal uncertainty,
• strongly non-Gaussian problems,
• severe nonlinearities,
• ambiguous observations.

The main limitation is computational cost, especially as the state dimension increases.

4.3.18.24 Gaussian versus Non-Gaussian Estimation

Kalman-family estimators primarily represent uncertainty using

\[ \boxed{ \text{Mean}+\text{Covariance}. } \]

This works well when the distribution is approximately Gaussian.

But if several different physical solutions are plausible, the posterior may be multimodal.

Examples include:

• ambiguous visual features,
• bearing-only ambiguity,
• uncertain data association.

A single mean and covariance may then hide the actual structure of the uncertainty.

4.3.18.25 Deterministic versus Probabilistic Estimation

A deterministic estimator may solve:

\[ \boxed{ \hat{\mathbf x} = \arg\min_{\mathbf x} J(\mathbf x). } \]

A probabilistic estimator represents

\[ \boxed{ p(\mathbf x|\mathbf z). } \]

Probabilistic treatment is valuable when uncertainty itself influences decisions, such as:

• guidance,
• collision assessment,
• autonomous navigation,
• fault detection.

4.3.18.26 Maximum Likelihood Estimation

Maximum likelihood estimates the quantity most likely to have produced the measurements:

\[ \boxed{ \hat{\boldsymbol\theta}_{ML} = \arg\max_{\boldsymbol\theta} p( \mathbf z| \boldsymbol\theta ). } \]

For Gaussian independent measurement errors, this leads to weighted least squares.

Thus classical least squares has a direct probabilistic interpretation.

4.3.18.27 Maximum A Posteriori Estimation

MAP estimation incorporates prior knowledge:

\[ \boxed{ \hat{\boldsymbol\theta}_{MAP} = \arg\max_{\boldsymbol\theta} p( \boldsymbol\theta| \mathbf z ). } \]

Using Bayes:

\[ p( \boldsymbol\theta| \mathbf z ) \propto p( \mathbf z| \boldsymbol\theta ) p( \boldsymbol\theta ). \]

Hence,

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

4.3.18.28 Least Squares and Bayesian Estimation

Weighted least squares corresponds to Gaussian maximum likelihood.

Including a Gaussian prior gives

\[ \boxed{ J= \mathbf r^T \mathbf R^{-1} \mathbf r + ( \mathbf x-\mathbf x_0 )^T \mathbf P_0^{-1} ( \mathbf x-\mathbf x_0 ). } \]

This links classical estimation directly to Bayesian MAP estimation.

4.3.18.29 Filtering versus Smoothing

Filtering estimates:

\[ \boxed{ p( \mathbf x_k| \mathbf z_{1:k} ). } \]

Only measurements up to the current time are used.

Smoothing estimates:

\[ \boxed{ p( \mathbf x_k| \mathbf z_{1:N} ), \qquad N>k. } \]

Future measurements are allowed to improve the historical state estimate.

This is why smoothing can outperform filtering for trajectory reconstruction.

4.3.18.30 Fixed-Interval Smoothing

A fixed-interval smoother processes a complete interval

\[ t_0\le t\le t_f. \]

All measurements in that interval are available.

Applications include:

• post-flight reconstruction,
• calibration,
• maneuver reconstruction,
• high-quality reference trajectories.

4.3.18.31 Fixed-Lag Smoothing

Fixed-lag smoothing deliberately waits a short time before finalizing the estimate:

\[ \boxed{ \hat{\mathbf x}_{k-L|k}. } \]

It gives

\[ \boxed{ \text{Improved State Estimate} \leftrightarrow \text{Additional Latency}. } \]

This is useful when real-time output is required but a small delay is acceptable.

4.3.18.32 Rauch-Tung-Striebel Smoother

The RTS smoother uses:

\[ \boxed{ \text{Forward Kalman Filter} \rightarrow \text{Backward Smoothing Pass} \rightarrow \text{Improved Historical Trajectory}. } \]

Nonlinear versions may be based on EKF or UKF formulations.

4.3.18.33 State Estimation versus Parameter Estimation

Dynamic states may be written as

\[ \boxed{ \mathbf x= [\mathbf r,\mathbf v]. } \]

Parameters may be

\[ \boxed{ \boldsymbol\theta= [C_D,C_R,b,\ldots]. } \]

States evolve dynamically.

Parameters characterize the system and may change slowly or remain approximately constant.

They may be estimated separately or simultaneously.

4.3.18.34 Augmented-State Estimation

Parameters can be appended to the navigation state:

\[ \boxed{ \mathbf x_a= \begin{bmatrix} \mathbf x\\ \boldsymbol\theta \end{bmatrix}. } \]

For a constant parameter:

\[ \boxed{ \dot{\boldsymbol\theta}=0. } \]

For a variable parameter, a stochastic dynamic model is more appropriate.

This enables a sequential estimator to estimate both physical state and unknown parameters.

4.3.18.35 Danger of Excessive State Augmentation

Too many estimated parameters can produce:

• weak observability,
• severe correlation,
• numerical ill-conditioning,
• slow convergence,
• growing covariance,
• physically meaningless parameter compensation.

Therefore:

\[ \boxed{ \text{Estimate Only Parameters Supported by Available Information}. } \]

This connects directly to the identifiability principles from 4.3.16.

4.3.18.36 Constant-Parameter Model

If a parameter is assumed fixed,

\[ \boxed{ \theta_{k+1} = \theta_k. } \]

No process uncertainty is assigned to its evolution.

This is suitable only when the parameter is physically expected to remain effectively constant over the estimation interval.

4.3.18.37 Random-Walk Parameter Model

For a slowly varying parameter:

\[ \boxed{ \theta_{k+1} = \theta_k+ w_{\theta,k}. } \]

The process noise controls allowed parameter variation.

Too little process noise causes the estimate to lag real changes.

Too much process noise makes the parameter chase measurement noise.

4.3.18.38 Gauss-Markov Parameter Model

A correlated parameter can be modeled using:

\[ \boxed{ \dot{\theta} = -\frac{1}{\tau} \theta+w. } \]

The time constant \tau determines how quickly the parameter forgets its previous value.

This model lies between a fixed parameter and unrestricted random walk.

4.3.18.39 Process Noise Is Part of the Physical Model

Process noise represents uncertainty in state evolution.

Examples include:

• unmodeled acceleration,
• uncertain environmental forcing,
• actuator uncertainty,
• unresolved disturbances,
• parameter variations.

Therefore Q should not merely be changed until the estimator plot looks attractive.

It should represent physical uncertainty.

4.3.18.40 Measurement Noise Model

Measurement covariance

\[ \boxed{\mathbf R} \]

may represent:

• random measurement noise,
• correlated channels,
• range-dependent uncertainty,
• geometry-dependent optical errors,
• measurement-quality variation.

If R is wrong, the estimator assigns the wrong amount of trust to the sensor.

4.3.18.41 Choosing Q and R

Broadly,

\[ \boxed{ Q\uparrow \rightarrow \text{Less Trust in Dynamics} } \]

and

\[ \boxed{ R\uparrow \rightarrow \text{Less Trust in Measurements}. } \]

But this interpretation should not encourage arbitrary tuning.

Both matrices should reflect realistic uncertainty and be checked through statistical consistency testing.

4.3.18.42 Continuous-to-Discrete Process Noise

Continuous disturbance covariance cannot always be converted simply by multiplying by \Delta t.

In general:

\[ \boxed{ \mathbf Q_d = \int_0^{\Delta t} \boldsymbol\Phi( \Delta t,\tau ) \mathbf G_w \mathbf Q_c \mathbf G_w^T \boldsymbol\Phi^T( \Delta t,\tau ) d\tau. } \]

The correct result depends on dynamics and where disturbance enters the state equations.

4.3.18.43 Measurement Rate versus Estimator Rate

A very important distinction is

\[ \boxed{ \text{Propagation Rate} \neq \text{Measurement Update Rate}. } \]

An estimator can propagate at a high internal frequency while sensors provide measurements less frequently.

The estimator then corrects the predicted state whenever a measurement is actually available.

4.3.18.44 Asynchronous Measurements

Sensors do not have to update simultaneously.

The sequence may be:

\[ \boxed{ \text{Propagate} \rightarrow \text{Sensor A Update} \rightarrow \text{Propagate} \rightarrow \text{Sensor B Update} \rightarrow \cdots } \]

Each measurement should be processed at its own measurement epoch.

4.3.18.45 Multi-Rate Estimation

A navigation architecture may combine:

• high-rate inertial sensors,
• medium-rate absolute navigation sensors,
• slower optical measurements,
• intermittent external measurements.

High-rate sensors help short-term propagation.

Lower-rate absolute sensors constrain accumulated drift.

This is one of the major strengths of sensor fusion.

4.3.18.46 Sensor Fusion

Sensor fusion is not simply averaging.

Its real benefit is that different sensors can observe different directions in state space.

Conceptually:

\[ \boxed{ \text{Dynamics} + \text{Sensor A} + \text{Sensor B} + \cdots \rightarrow \text{Fused State Estimate}. } \]

A range sensor and an angle sensor, for example, may provide complementary information.

4.3.18.47 Centralized Fusion

A centralized estimator processes all measurements using one shared state and covariance.

Advantages:

• direct treatment of correlations,
• consistent state updates,
• full exploitation of complementary measurements.

Disadvantages include:

• greater implementation complexity,
• central processor dependence.

4.3.18.48 Decentralized and Federated Estimation

Separate estimators can process different sensor groups and then fuse their results.

This can improve:

• modularity,
• fault isolation,
• subsystem independence.

But shared information creates correlations.

If those correlations are ignored, covariance may become artificially small and overconfident.

4.3.18.49 Relative Navigation Estimation

For general relative navigation:

\[ \boxed{ \mathbf x_{\rm rel} = \begin{bmatrix} \mathbf r_{\rm rel}\\ \mathbf v_{\rm rel} \end{bmatrix}. } \]

Measurements may include:

• relative range,
• range rate,
• LOS,
• optical bearings,
• LiDAR,
• radar,
• inter-spacecraft navigation observations.

The selection of estimator does not fundamentally depend on calling the problem “RPO.”

It depends on:

• geometry,
• nonlinearities,
• uncertainty,
• measurement type,
• update frequency,
• operational requirements.

4.3.18.50 Relative-Dynamics Model Selection

Possible relative-dynamics models include:

• linearized relative-motion equations,
• nonlinear relative dynamics,
• separate inertial propagation followed by differencing,
• perturbation-aware models.

A simple model may be sufficient over a limited region.

A higher-fidelity model becomes necessary when approximation errors become comparable with the required estimation accuracy.

4.3.18.51 Model Fidelity versus Estimator Complexity

More fidelity is not always better.

Higher-fidelity models may require:

• more computation,
• more uncertain parameters,
• more environmental data,
• more complex derivatives,
• additional calibration.

The objective is

\[ \boxed{ \text{Sufficient Physical Accuracy} } \]

rather than maximum possible model complexity.

4.3.18.52 Error-State Estimation

Rather than estimate the complete state directly, estimate a small error:

\[ \boxed{ \mathbf x_{\rm true} = \mathbf x_{\rm nominal} + \delta\mathbf x. } \]

The nonlinear nominal state is propagated separately.

The filter estimates only \delta\mathbf x.

Advantages include:

• small linearization region,
• better numerical scaling,
• natural integration with high-rate propagation.

4.3.18.53 Multiplicative Attitude Error

Quaternion attitude error should normally not be treated as an unconstrained additive four-state correction.

Instead:

\[ \boxed{ \mathbf q_{\rm true} = \delta\mathbf q \otimes \mathbf q_{\rm nominal}. } \]

A small three-component attitude-error vector can represent \delta\mathbf q.

This respects the unit-quaternion constraint and gives a minimal local error representation.

4.3.18.54 Error-State Kalman Filter

An ESKF typically:

1. propagates the nominal physical state,
2. propagates small-error covariance,
3. computes measurement residual,
4. estimates the state error,
5. injects the error correction,
6. resets the error state.

This is widely used for inertial and attitude-navigation problems.

4.3.18.55 Initial State Selection

A nonlinear estimator needs a reasonable starting estimate.

If

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

is too far from truth, a locally linearized estimator may fail.

Initialization may use:

• prior orbit,
• direct measurements,
• geometric initialization,
• external tracking,
• coarse navigation,
• batch preprocessing.

Initialization is therefore part of estimator design, not merely an input.

4.3.18.56 Initial Covariance Selection

The initial covariance

\[ \boxed{\mathbf P_0} \]

should represent actual initial uncertainty.

If it is too small, the estimator becomes overconfident in its initial state.

If too large, initial updates may become unnecessarily aggressive or numerically awkward.

The covariance should reflect what is genuinely known.

4.3.18.57 Observability before Filter Selection

This is one of the most important rules in the whole section.

If

\[ \boxed{ \text{Measurements contain no information about a state direction}, } \]

switching from an EKF to a UKF does not solve the problem.

Therefore,

\[ \boxed{ \text{Observability Problem} \neq \text{Filter Algorithm Problem}. } \]

Geometry and estimability must be checked first.

4.3.18.58 Numerical Conditioning before Solver Selection

Likewise, weak numerical conditioning should be understood before blaming the solver.

The correct order is:

\[ \boxed{ \text{Check Geometry} \rightarrow \text{Check Scaling} \rightarrow \text{Check Conditioning} \rightarrow \text{Choose Solver}. } \]

A solver cannot create physical information missing from the sensitivity matrix.

4.3.18.59 Residual Monitoring

Residuals are

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

They should be checked for:

• nonzero mean,
• trends,
• periodicity,
• outliers,
• wrong variance,
• temporal correlation.

Residuals provide information about both sensors and model adequacy.

4.3.18.60 Innovation Covariance

For Kalman filtering:

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

This tells us how large the innovation should statistically be.

A residual should therefore not be judged simply by its absolute magnitude.

4.3.18.61 Normalized Innovation

Normalize the innovation using

\[ \boxed{ \tilde{\mathbf y}_k = \mathbf S_k^{-1/2} \mathbf y_k. } \]

Persistent abnormal normalized innovations may indicate:

• wrong Q,
• wrong R,
• poor initialization,
• unmodeled dynamics,
• sensor bias,
• model deficiencies.

4.3.18.62 Innovation Gating

Use

\[ \boxed{ d_k^2 = \mathbf y_k^T \mathbf S_k^{-1} \mathbf y_k. } \]

This is the normalized innovation squared.

A measurement with a statistically excessive value may be:

• rejected,
• flagged,
• downweighted,
• investigated.

The gating threshold should depend on measurement dimension and the assumed probability model.

4.3.18.63 Outlier Handling

Possible causes include:

• corrupted measurement,
• incorrect association,
• saturation,
• communication error,
• timing error,
• temporary environmental effects.

Possible responses:

• reject,
• downweight,
• revalidate,
• classify as a fault.

However, repeated large residuals may indicate model error rather than isolated outliers.

4.3.18.64 Consistency Testing

Accuracy and consistency are different.

An estimate might be close to truth while its covariance is badly wrong.

Consistency means actual errors statistically agree with reported uncertainty.

Important diagnostics include:

• innovations,
• NIS,
• NEES,
• covariance envelopes,
• Monte Carlo consistency.

4.3.18.65 Normalized Estimation Error Squared

When truth is available:

\[ \boxed{ \epsilon_k = ( \mathbf x_k- \hat{\mathbf x}_k )^T \mathbf P_k^{-1} ( \mathbf x_k- \hat{\mathbf x}_k ). } \]

This compares actual state error with predicted covariance.

It is particularly valuable during simulation and Monte Carlo testing.

4.3.18.66 Covariance Is Not Actual Error

This distinction is essential:

\[ \boxed{ \mathbf P \neq ( \mathbf x- \hat{\mathbf x} ) ( \mathbf x- \hat{\mathbf x} )^T } \]

for one individual realization.

Covariance describes an expected statistical spread, not the instantaneous true error.

4.3.18.67 Filter Divergence

Possible causes of divergence include:

• wrong dynamics,
• wrong Jacobians,
• poor initialization,
• too-small process noise,
• too-small measurement noise,
• strong nonlinearities,
• numerical instability,
• bias,
• wrong data association.

Filter divergence should be diagnosed from the complete architecture rather than attributed automatically to one filter equation.

4.3.18.68 Covariance Collapse

If uncertainty becomes unrealistically small, the estimator becomes overconfident.

Then later measurements may be given too little influence.

This often indicates missing process uncertainty or repeated over-weighting of measurements.

4.3.18.69 Covariance Inflation

One corrective mechanism is

\[ \boxed{ \mathbf P \leftarrow \alpha\mathbf P, \qquad \alpha>1. } \]

This deliberately increases uncertainty.

But inflation should not be used to conceal a known modeling problem.

4.3.18.70 Adaptive Estimation

Adaptive methods modify quantities such as:

• Q,
• R,
• bias models,
• process parameters,

based on estimator behavior.

This may help when noise characteristics vary over time.

However, adaptation itself becomes part of the estimator dynamics and needs validation.

4.3.18.71 Multiple-Model Estimation

If several dynamic hypotheses are possible, maintain probabilities

\[ \boxed{ p( M_j|\mathbf z ). } \]

Examples include:

• different maneuver states,
• different disturbance models,
• different target-motion models.

This represents uncertainty in which model is active, not only uncertainty in continuous state variables.

4.3.18.72 Interacting Multiple Model Estimation

An IMM operates several filters corresponding to different models.

The filters exchange mixed initial conditions and update model probabilities based on measurement likelihood.

IMM is particularly useful when dynamics switch among a finite set of known modes.

4.3.18.73 Maneuver Detection and Estimation

Unexpected maneuvers generate residual patterns inconsistent with the nominal model.

Possible estimator responses include:

• increased Q,
• change-point detection,
• estimation of impulsive \Delta\mathbf v,
• finite-burn parameter estimation,
• model switching.

The architecture depends on how much is known about the maneuver.

4.3.18.74 Known versus Unknown Maneuvers

A known command can be modeled as an input:

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

Thus,

\[ \boxed{ \text{Known Maneuver} \rightarrow \text{Dynamic Input} } \]

whereas uncertain burn magnitude, direction, or time may give:

\[ \boxed{ \text{Unknown Maneuver} \rightarrow \text{Estimation Problem}. } \]

4.3.18.75 Parameter Estimation Architecture

Parameters can be estimated through:

• batch differential correction,
• augmented-state filtering,
• dual estimation,
• smoothing,
• dedicated calibration.

Selection depends on:

• parameter time variation,
• measurement sensitivity,
• real-time requirement,
• correlation,
• excitation.

4.3.18.76 Dual Estimation

Instead of one large augmented filter, use separate estimators for:

• physical state,
• unknown parameters.

They exchange information.

This can simplify some implementations but introduces coupled estimation errors and consistency challenges.

4.3.18.77 Sliding-Window Estimation

Use only the latest observations:

\[ \boxed{ \{ \mathbf z_{k-N+1}, \ldots, \mathbf z_k \} \rightarrow \text{Windowed Estimate}. } \]

Older information is discarded or marginalized.

This provides a middle ground between batch estimation and purely recursive filtering.

4.3.18.78 Moving-Horizon Estimation

MHE repeatedly solves a constrained optimization problem over a recent finite interval.

It can naturally incorporate:

• nonlinear dynamics,
• bounds,
• constraints,
• multi-sensor measurements,
• prior state information.

The main disadvantage is computational expense.

4.3.18.79 Constraints in Estimation

Physical constraints may include:

\[ m>0, \]

\[ C_D\ge0, \]

or:

• unit quaternion,
• bounded bias,
• geometric constraints,
• actuator limits.

A constrained estimator avoids physically impossible parameter values.

4.3.18.80 Hard versus Soft Constraints

A hard constraint must be exactly satisfied:

\[ \boxed{ g(\mathbf x)=0. } \]

A soft constraint allows uncertainty:

\[ \boxed{ \mathbf z_c = h_c(\mathbf x)+\mathbf v_c. } \]

Hard constraints define the feasible space.

Soft constraints behave like probabilistic knowledge.

4.3.18.81 Pseudo-Measurements

A known relationship may be written as

\[ \boxed{ z_p = h_p(\mathbf x)+v_p. } \]

The estimator then processes it similarly to an ordinary sensor measurement.

The assigned covariance determines how strongly it constrains the solution.

4.3.18.82 Real-Time versus Offline Estimation

Real-time estimation prioritizes

• predictable latency,
• bounded computation,
• robustness,
• memory efficiency,
• continuous output,
• fault response.

Offline estimation can use

• long iterations,
• complete measurement arcs,
• smoothing,
• expensive optimization,
• detailed diagnostics.

Timing requirements therefore strongly influence architecture.

4.3.18.83 Onboard versus Ground Estimation

Onboard estimation often prioritizes:

• deterministic computation,
• bounded memory,
• robustness,
• autonomy,
• numerical reliability.

Ground estimation can usually use:

• more complex force models,
• larger parameter sets,
• repeated processing,
• complete measurement arcs,
• more environmental data.

Many missions therefore use both onboard and ground estimators for different purposes.

4.3.18.84 Computational Complexity

Algorithm selection must consider:

• state dimension,
• measurement dimension,
• update frequency,
• number of sigma points,
• number of particles,
• factorization cost,
• memory,
• processor capability.

A theoretically excellent estimator is unsuitable if it cannot satisfy its real-time execution requirement.

4.3.18.85 Numerical Robustness

Estimator architecture must also account for:

• matrix conditioning,
• scaling,
• rank deficiency,
• covariance positivity,
• floating-point precision,
• roundoff accumulation,
• matrix factorization.

This connects directly with Section 4.3.17.

Square-root and factorized methods are useful when numerical robustness is especially important.

4.3.18.86 Square-Root Estimator Selection

Instead of propagating \mathbf P, propagate

\[ \boxed{ \mathbf P = \mathbf S \mathbf S^T. } \]

The factor \mathbf S can provide improved numerical stability and better preservation of positive definiteness.

This is particularly useful for:

• long-running filters,
• high-precision navigation,
• high-dynamic-range estimation.

4.3.18.87 Robust Estimation

When measurements occasionally contain large non-Gaussian errors, replace purely quadratic loss with

\[ \boxed{ J = \sum_i \rho(r_i). } \]

For example, Huber-type losses reduce the influence of large residuals.

But robust methods should not be used to suppress systematic modeling errors.

4.3.18.88 Data Association

If a sensor sees multiple possible targets or features, the estimator must determine which observation belongs to which object.

Possible methods include:

• nearest neighbor,
• statistical gating,
• multiple hypotheses,
• probabilistic association.

Wrong association can create catastrophic state-estimation errors even when the filter equations themselves are correct.

4.3.18.89 Time Synchronization

Measurements must be processed at the correct timestamp.

A measurement

\[ \mathbf z(t_k) \]

must be compared with the predicted state at t_k, not simply at its reception time.

Important timing effects include:

• timestamps,
• communication latency,
• sensor clock offset,
• processing delay,
• asynchronous sampling.

4.3.18.90 Delayed Measurements

If a measurement arrives late, possible strategies include:

• buffered-state processing,
• out-of-sequence update,
• fixed-lag smoothing,
• recent-window reprocessing.

Applying an old measurement directly to the current state without time correction can produce estimation error.

4.3.18.91 Estimator Reset and Reinitialization

A practical estimator needs recovery logic.

Triggers may include:

• invalid covariance,
• persistent innovation failures,
• measurement loss,
• numerical failure,
• unacceptable state uncertainty.

Responses may include:

• covariance inflation,
• state reset,
• reinitialization,
• sensor-mode switching,
• filter restart.

Recovery is part of estimator architecture, not merely an afterthought.

4.3.18.92 Measurement Dropout

During a sensor outage:

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

can continue propagating.

Covariance usually grows.

When measurements return, correction resumes.

The maximum tolerable dropout depends on:

• dynamic uncertainty,
• measurement accuracy,
• required navigation accuracy,
• elapsed time.

4.3.18.93 Fault Detection and Estimator Integrity

Residuals may indicate:

• sensor failure,
• bias change,
• timing problem,
• model error,
• unexpected maneuver.

However,

\[ \boxed{ \text{Large Residual} \neq \text{Automatically Sensor Failure}. } \]

FDIR logic should distinguish among plausible causes before isolating a sensor.

4.3.18.94 Validation with Truth Data

When truth is available:

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

Important performance metrics include:

• RMSE,
• mean error,
• bias,
• maximum error,
• convergence time,
• steady-state error,
• covariance consistency.

4.3.18.95 Monte Carlo Validation

A single successful trajectory is insufficient to demonstrate statistical estimator performance.

Monte Carlo should vary:

• initial-state error,
• sensor noise,
• biases,
• environmental uncertainty,
• timing,
• model parameters,
• geometry.

Then compare the resulting empirical error distribution with predicted covariance.

This is how estimator robustness and consistency are assessed statistically.

4.3.18.96 Real-Data Validation

Simulation cannot represent all real-world effects.

Real measurements may reveal:

• colored noise,
• unknown bias,
• clock/timing errors,
• calibration drift,
• environmental effects,
• unexpected sensor behavior.

A good development sequence is therefore:

\[ \boxed{ \text{Analytical Verification} \rightarrow \text{Simulation} \rightarrow \text{Monte Carlo} \rightarrow \text{Representative Real Data}. } \]

4.3.18.97 Common Estimator-Selection Mistakes

Important mistakes to avoid include:

1. Selecting EKF/UKF before defining the problem.
2. Adding every uncertainty to the state vector.
3. Ignoring observability.
4. Assuming greater algorithm complexity guarantees accuracy.
5. Treating Q and R purely as tuning parameters.
6. Using unrealistic \mathbf P_0.
7. Ignoring asynchronous timing.
8. Assuming sensor and estimator rates must match.
9. Ignoring parameter correlation.
10. Estimating parameters without excitation.
11. Using EKF far outside its linearization region.
12. Believing UKF fixes poor observability.
13. Using particle filters without computational justification.
14. Ignoring estimator consistency.
15. Confusing covariance with instantaneous error.
16. Calling every large residual an outlier.
17. Using robust weighting to hide model error.
18. Ignoring numerical conditioning.
19. Ignoring delayed measurements.
20. Omitting recovery/reinitialization logic.
21. Choosing algorithms only from nominal accuracy.
22. Ignoring latency or processor limits.
23. Separately estimating physically redundant parameters.
24. Skipping Monte Carlo or representative-data validation.
25. Reporting state estimates without uncertainty.

4.3.18.98 Practical Estimator Design Workflow

A disciplined design process is

\[ \boxed{ \begin{array}{c} \text{Define Estimation Objective}\\ \downarrow\\ \text{Define States and Parameters}\\ \downarrow\\ \text{Define Dynamics}\\ \downarrow\\ \text{Define Measurements}\\ \downarrow\\ \text{Define Noise, Bias, and Timing Models}\\ \downarrow\\ \text{Assess Observability and Identifiability}\\ \downarrow\\ \text{Assess Geometry and Correlation}\\ \downarrow\\ \text{Determine Real-Time / Offline Requirement}\\ \downarrow\\ \text{Choose Batch / Sequential / Smoothing Architecture}\\ \downarrow\\ \text{Assess Linear / Nonlinear Behavior}\\ \downarrow\\ \text{Select Candidate Algorithm}\\ \downarrow\\ \text{Scale States and Measurements}\\ \downarrow\\ \text{Choose Numerical Solver}\\ \downarrow\\ \text{Define }Q,R,P_0\\ \downarrow\\ \text{Implement Residual Monitoring}\\ \downarrow\\ \text{Test Nominal Convergence}\\ \downarrow\\ \text{Test Weak Geometry / Dropout / Outliers}\\ \downarrow\\ \text{Perform Statistical Validation}\\ \downarrow\\ \text{Validate Representative Data}\\ \downarrow\\ \text{Verify Computational Requirements} \end{array}} \]

The key point is that algorithm selection is intentionally in the middle, after the physical and information structure has been understood.

4.3.18.99 Practical Algorithm-Selection Guide

This is a starting guide, not an automatic selection rule.

4.3.18.100 Core Equations

Dynamic model

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

Measurement model

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

Linearized state model

\[ \boxed{ \delta\mathbf x_{k+1} = \mathbf F_k \delta\mathbf x_k + \mathbf w_k } \]

Linearized measurement

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

Innovation

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

Innovation covariance

\[ \boxed{ \mathbf S_k = \mathbf H_k \mathbf P_k^- \mathbf H_k^T + \mathbf R_k } \]

Kalman gain

\[ \boxed{ \mathbf K_k = \mathbf P_k^- \mathbf H_k^T \mathbf S_k^{-1} } \]

State correction

\[ \boxed{ \hat{\mathbf x}_k^+ = \hat{\mathbf x}_k^- + \mathbf K_k \mathbf y_k } \]

Weighted least squares

\[ \boxed{ J = \mathbf r^T \mathbf R^{-1} \mathbf r } \]

MAP objective

\[ \boxed{ J_{\rm MAP} = \mathbf r^T \mathbf R^{-1} \mathbf r + ( \mathbf x-\mathbf x_0 )^T \mathbf P_0^{-1} ( \mathbf x-\mathbf x_0 ) } \]

Augmented state

\[ \boxed{ \mathbf x_a = \begin{bmatrix} \mathbf x\\ \boldsymbol\theta \end{bmatrix} } \]

NIS

\[ \boxed{ {\rm NIS} = \mathbf y_k^T \mathbf S_k^{-1} \mathbf y_k } \]

NEES

\[ \boxed{ {\rm NEES} = ( \mathbf x_k- \hat{\mathbf x}_k )^T \mathbf P_k^{-1} ( \mathbf x_k- \hat{\mathbf x}_k ) } \]

4.3.18.101 Complete Estimator Architecture

The estimator sits inside a larger system:

\[ \boxed{ \begin{array}{c} \text{Physical System}\\ \downarrow\\ \text{Dynamics + Inputs + Disturbances}\\ \downarrow\\ \text{Sensors}\\ \downarrow\\ \text{Timing + Noise + Bias Models}\\ \downarrow\\ \text{Observability / Estimability Assessment}\\ \downarrow\\ \text{Estimator Architecture}\\ \downarrow\\ \text{Prediction}\\ \downarrow\\ \text{Measurement Update}\\ \downarrow\\ \text{State + Parameters + Covariance}\\ \downarrow\\ \text{Residual / Consistency Monitoring}\\ \downarrow\\ \text{Navigation / Guidance / Decision System} \end{array}} \]

The algorithm—EKF, UKF, least squares, etc.—is therefore only one part of a much larger estimation architecture.

4.3.18.102 Key Takeaways

Estimator design is fundamentally a system-design problem.

The correct sequence is:

\[ \boxed{ \text{Problem Definition} \rightarrow \text{Dynamics} \rightarrow \text{Measurements} \rightarrow \text{Observability} \rightarrow \text{Uncertainty} \rightarrow \text{Algorithm} \rightarrow \text{Validation}. } \]

Batch estimation is appropriate when a complete observation arc is processed together.

Sequential estimation is appropriate when the current state must remain continuously available.

Filtering uses measurements only through the current epoch:

\[ \boxed{ p( \mathbf x_k| \mathbf z_{1:k} ) } \]

whereas smoothing includes later measurements:

\[ \boxed{ p( \mathbf x_k| \mathbf z_{1:N} ). } \]

Augmented-state estimation can include unknown parameters,

\[ \boxed{ \mathbf x_a = \begin{bmatrix} \mathbf x\\ \boldsymbol\theta \end{bmatrix}, } \]

but only when those parameters are sufficiently observable.

One of the most important principles in estimator design is:

\[ \boxed{ \text{A More Sophisticated Estimator Cannot Create Information That the Measurements Do Not Contain.} } \]

Therefore observability, geometry, identifiability, and conditioning should be examined before increasing estimator complexity.

Uncertainty must also be modeled correctly using

\[ \boxed{ \mathbf Q,\qquad \mathbf R,\qquad \mathbf P_0. } \]

They are physical uncertainty descriptions—not simply tuning variables.

For multisensor systems,

\[ \boxed{ \text{Sensor Rate} \neq \text{Estimator Propagation Rate} \neq \text{Other Sensor Rates}. } \]

Measurements can arrive asynchronously and be processed at their actual epochs.

Estimator validation should assess not only trajectories but also:

• estimation error,
• residuals,
• innovations,
• covariance,
• NIS,
• NEES,
• parameter correlation,
• convergence,
• observability,
• conditioning,
• dropout robustness,
• model uncertainty.

The final estimator must balance

\[ \boxed{ \begin{array}{c} \text{Physics}\\ +\text{Measurement Information}\\ +\text{Uncertainty}\\ +\text{Geometry}\\ +\text{Numerical Robustness}\\ +\text{Computational Constraints}\\ +\text{Operational Requirements} \end{array}} \]

and the final design principle is:

\[ \boxed{ \text{Choose the Simplest Estimator That Correctly Represents the Physics, Information, Uncertainty, and Operational Need.} } \]

That principle connects the preceding topics—measurement modelling, covariance propagation, filtering, orbit determination, parameter estimation, geometry, identifiability, and numerical conditioning—into one coherent estimator-design methodology.

Continue Exploring Estimation and Navigation

This section completes the estimator-design workflow by connecting physics, measurements, observability, uncertainty, numerical conditioning, algorithm selection, timing, integrity monitoring, and validation. Use the Theory Notes index to continue to the next topic in the sequence.