Recursive Estimation from Least-Squares and Bayesian Viewpoints

Sequential information accumulation through recursive least squares, Bayesian prior-to-posterior updating, information and covariance forms, innovations, recursive gains, dynamic prediction, process and measurement uncertainty, multisensor estimation, observability, consistency, and the mathematical bridge to Kalman filtering.

4.3.4 Recursive Estimation from Least-Squares and Bayesian Viewpoints

Batch estimation processes a collection of observations together. In many spacecraft applications, however, measurements arrive continuously or asynchronously, and the navigation system must update its estimate without repeatedly solving the complete historical data set from the beginning.

This motivates recursive estimation.

The central idea is

\[ \boxed{ \text{Previous Estimate} + \text{New Measurement Information} \rightarrow \text{Updated Estimate} } \]

Rather than storing and reprocessing every previous observation, the information accumulated from past measurements is summarized by quantities such as

\[ \hat{\mathbf x}_{k-1}, \qquad \mathbf P_{k-1}, \]

or, equivalently, by an information matrix and information vector.

When a new measurement arrives, the previous estimate becomes the prior, the measurement provides new information, and the estimator produces a posterior estimate.

Conceptually,

\[ \boxed{ \text{Prior} + \text{Likelihood} \rightarrow \text{Posterior} } \]

This is the Bayesian interpretation of recursive estimation.

From a least-squares viewpoint,

\[ \boxed{ \text{Previously Accumulated Normal Equations} + \text{New Measurement Contribution} \rightarrow \text{Updated Normal Equations} } \]

The two viewpoints are closely related.

For linear systems with Gaussian uncertainty,

\[ \boxed{ \text{Recursive Weighted Least Squares} \Longleftrightarrow \text{Bayesian Gaussian Updating} \Longleftrightarrow \text{Kalman Measurement Update} } \]

under the corresponding assumptions.

This section develops that connection without repeating the detailed Kalman-filter implementation already covered in Section 4.2.

4.3.4.1 Why Recursive Estimation Is Needed

Suppose measurements arrive sequentially:

\[ \mathbf z_1, \mathbf z_2, \mathbf z_3, \ldots \]

A batch estimator could, in principle, solve

\[ \{\mathbf z_1,\ldots,\mathbf z_k\} \]

every time a new measurement arrives.

However, this would repeatedly reprocess old information.

For a real-time spacecraft system, this can be inefficient because

Recursive estimation instead carries forward a compact representation of previously accumulated information.

Thus,

\[ \boxed{ \text{Historical Measurements} \rightarrow \text{Compressed Into Current Estimate and Uncertainty}. } \]

When the next observation arrives, only the new information needs to be incorporated.

4.3.4.2 Batch versus Recursive Processing

Consider a static linear model:

\[ \mathbf z_i = \mathbf H_i\mathbf x+\mathbf v_i. \]

In batch estimation, all measurements are stacked:

\[ \mathbf z = \begin{bmatrix} \mathbf z_1\\ \mathbf z_2\\ \vdots\\ \mathbf z_N \end{bmatrix} \]

with

\[ \mathbf H = \begin{bmatrix} \mathbf H_1\\ \mathbf H_2\\ \vdots\\ \mathbf H_N \end{bmatrix}. \]

The complete solution is found in one operation.

Recursive estimation instead proceeds as

\[ \boxed{ \mathbf z_1 \rightarrow \hat{\mathbf x}_1 \rightarrow \mathbf z_2 \rightarrow \hat{\mathbf x}_2 \rightarrow \cdots \rightarrow \hat{\mathbf x}_N. } \]

The final estimate can contain the same information as the batch solution under appropriate assumptions.

The difference is how that information is accumulated and represented.

4.3.4.3 The Fundamental Recursive Question

At measurement step k-1, suppose we have already processed all observations up to that point.

The available information is summarized by

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

and uncertainty

\[ \mathbf P_{k-1}. \]

A new measurement arrives:

\[ \mathbf z_k. \]

The recursive-estimation problem is

\[ \boxed{ ( \hat{\mathbf x}_{k-1}, \mathbf P_{k-1} ) + \mathbf z_k \rightarrow ( \hat{\mathbf x}_{k}, \mathbf P_{k} ). } \]

The estimator must determine

  1. how strongly the previous estimate should be trusted,
  2. how strongly the new measurement should be trusted,
  3. how the new measurement changes the state estimate,
  4. how the state uncertainty changes.

4.3.4.4 Prior and Posterior Terminology

Recursive estimation commonly distinguishes between a prior and a posterior.

Before processing measurement \(k\),

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

denotes the prior estimate.

After processing measurement \(k\),

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

denotes the posterior estimate.

Similarly,

\[ \boxed{ \mathbf P_k^{-} } \]

is prior covariance and

\[ \boxed{ \mathbf P_k^{+} } \]

is posterior covariance.

Conceptually,

\[ \boxed{ \text{Prior} \xrightarrow{\text{Measurement Update}} \text{Posterior}. } \]

For a dynamic estimator, the posterior from one epoch later becomes the basis for the next prediction.

4.3.4.5 Measurement Model

Consider a linear measurement model:

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

Assume

\[ E[\mathbf v_k]=\mathbf0 \]

and

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

The matrix

\[ \mathbf H_k \]

describes measurement sensitivity, while

\[ \mathbf R_k \]

describes measurement uncertainty.

A new measurement therefore contributes information through both

\[ \boxed{ \mathbf H_k } \]

and

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

4.3.4.6 Prior Information as a Least-Squares Term

Suppose the previous estimate is

\[ \hat{\mathbf x}^{-} \]

with covariance

\[ \mathbf P^{-}. \]

This prior can be included in a least-squares objective:

\[ \boxed{ J(\mathbf x) = (\mathbf x-\hat{\mathbf x}^{-})^T (\mathbf P^{-})^{-1} (\mathbf x-\hat{\mathbf x}^{-}) + (\mathbf z-\mathbf H\mathbf x)^T \mathbf R^{-1} (\mathbf z-\mathbf H\mathbf x). } \]

The first term represents prior information.

The second term represents the new measurement.

Therefore,

\[ \boxed{ \text{Recursive Update} = \text{Prior-Constrained Weighted Least Squares}. } \]

This provides the first major connection between least squares and recursive estimation.

4.3.4.7 Information Matrix

Define the prior information matrix

\[ \boxed{ \boldsymbol{\Lambda}^{-} = (\mathbf P^{-})^{-1}. } \]

The measurement contributes information

\[ \boxed{ \Delta\boldsymbol{\Lambda} = \mathbf H^T \mathbf R^{-1} \mathbf H. } \]

The posterior information matrix is therefore

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

This is one of the most important equations in recursive estimation.

It shows that, for independent Gaussian information,

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

Engineering interpretation: The information matrix is more than an algebraic convenience. Its eigenvectors identify combinations of states that are strongly or weakly constrained, while its eigenvalues indicate the strength of those constraints. A small eigenvalue corresponds to a weak-information direction.

4.3.4.8 Information Vector

The information matrix alone is not enough to recover the state estimate.

Define the information vector

\[ \boxed{ \boldsymbol{\eta} = \boldsymbol{\Lambda}\hat{\mathbf x}. } \]

For the prior,

\[ \boldsymbol{\eta}^{-} = \boldsymbol{\Lambda}^{-} \hat{\mathbf x}^{-}. \]

The new measurement contributes

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

Therefore,

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

The posterior estimate is then

\[ \boxed{ \hat{\mathbf x}^{+} = (\boldsymbol{\Lambda}^{+})^{-1} \boldsymbol{\eta}^{+}. } \]

This is the information-form recursive update.

4.3.4.9 Recursive Least Squares from Information Accumulation

Suppose independent measurements

\[ \mathbf z_1,\ldots,\mathbf z_k \]

have been processed.

The information matrix is

\[ \boxed{ \boldsymbol{\Lambda}_k = \boldsymbol{\Lambda}_0 + \sum_{i=1}^{k} \mathbf H_i^T \mathbf R_i^{-1} \mathbf H_i. } \]

The information vector is

\[ \boxed{ \boldsymbol{\eta}_k = \boldsymbol{\eta}_0 + \sum_{i=1}^{k} \mathbf H_i^T \mathbf R_i^{-1} \mathbf z_i. } \]

Hence,

\[ \boxed{ \hat{\mathbf x}_k = \boldsymbol{\Lambda}_k^{-1} \boldsymbol{\eta}_k. } \]

Each new measurement simply adds another information contribution.

4.3.4.10 Batch and Recursive Least Squares Are Closely Related

For batch WLS,

\[ \boxed{ \hat{\mathbf x} = ( \mathbf H^T \mathbf R^{-1} \mathbf H )^{-1} \mathbf H^T \mathbf R^{-1} \mathbf z. } \]

If the measurements are independent and stacked, then

\[ \mathbf H^T \mathbf R^{-1} \mathbf H = \sum_i \mathbf H_i^T \mathbf R_i^{-1} \mathbf H_i. \]

Similarly,

\[ \mathbf H^T \mathbf R^{-1} \mathbf z = \sum_i \mathbf H_i^T \mathbf R_i^{-1} \mathbf z_i. \]

Thus recursive estimation is not introducing different information.

It is accumulating the same least-squares information progressively.

Therefore,

\[ \boxed{ \text{Batch Solution} \leftrightarrow \text{Sequential Accumulation of the Same Information}. } \]

Engineering interpretation: This equivalence depends on using the same models, covariance assumptions, and independent measurement information. If data are rejected, reweighted, correlated, or relinearized differently, the recursive and batch results can diverge.

4.3.4.11 Measurement Order in the Static Linear Case

If

then the final information matrix is

\[ \boldsymbol{\Lambda}_N = \boldsymbol{\Lambda}_0 + \sum_{k=1}^{N} \Delta\boldsymbol{\Lambda}_k. \]

Because matrix addition is commutative,

\[ \boxed{ \text{Measurement Ordering Does Not Change the Final Static Linear WLS Information}. } \]

This is an important theoretical result.

However, measurement ordering can matter in nonlinear estimation because the system may be repeatedly linearized around changing state estimates.

4.3.4.12 Scalar Recursive Example

Suppose the unknown quantity is a scalar \(x\).

The prior estimate is

\[ \hat x^{-} \]

with variance

\[ P^{-}. \]

A new scalar measurement is

\[ z=x+v \]

with variance

\[ R. \]

The prior information is

\[ \frac{1}{P^{-}}. \]

The measurement information is

\[ \frac{1}{R}. \]

Therefore,

\[ \boxed{ \frac{1}{P^{+}} = \frac{1}{P^{-}} + \frac{1}{R}. } \]

The posterior estimate is

\[ \boxed{ \hat x^{+} = P^{+} \left( \frac{\hat x^{-}}{P^{-}} + \frac{z}{R} \right). } \]

Thus the posterior is a statistically weighted combination of the prior and measurement.

4.3.4.13 Interpretation of the Scalar Update

If

\[ R\ll P^{-}, \]

the measurement is much more precise than the prior.

Then the posterior estimate moves strongly toward the measurement.

If

\[ P^{-}\ll R, \]

the prior is much more precise.

Then the new measurement has relatively little effect.

Thus,

\[ \boxed{ \text{Relative Uncertainty Determines Relative Influence}. } \]

This is the same principle encountered in weighted least squares.

4.3.4.14 From Information Form to Correction Form

Recursive estimation is often written not in information form but in a correction form.

Define the measurement residual or innovation

\[ \boxed{ \mathbf y = \mathbf z-\mathbf H\hat{\mathbf x}^{-}. } \]

The posterior estimate can be written as

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

where

\[ \mathbf K \]

determines how strongly the residual corrects the prior estimate.

This matrix is the familiar gain appearing in Kalman-style estimators.

4.3.4.15 Innovation

The innovation is

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

It represents the new information that was not predicted by the prior estimate.

Conceptually,

\[ \boxed{ \text{Measurement} - \text{Prior Prediction} = \text{Innovation}. } \]

If the measurement is exactly predicted,

\[ \mathbf y_k=\mathbf0. \]

Then there is no measurement-driven state correction.

4.3.4.16 Innovation Covariance

The innovation itself has uncertainty.

For a linear measurement,

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

The first term represents uncertainty in the predicted measurement caused by prior state uncertainty.

The second term represents sensor measurement uncertainty.

Thus,

\[ \boxed{ \text{Innovation Uncertainty} = \text{Predicted-State Contribution} + \text{Measurement-Noise Contribution}. } \]

Engineering interpretation: The innovation covariance provides the statistically correct scale for judging an innovation. It combines uncertainty in the predicted state with uncertainty in the measurement, which is why normalized innovation tests use \(\mathbf S^{-1}\) rather than raw residual magnitude alone.

4.3.4.17 Recursive Gain

The correction gain is

\[ \boxed{ \mathbf K = \mathbf P^{-} \mathbf H^T \mathbf S^{-1}. } \]

Substituting

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

gives

\[ \boxed{ \mathbf K = \mathbf P^{-} \mathbf H^T ( \mathbf H\mathbf P^{-}\mathbf H^T + \mathbf R )^{-1}. } \]

The state update becomes

\[ \boxed{ \hat{\mathbf x}^{+} = \hat{\mathbf x}^{-} + \mathbf K ( \mathbf z-\mathbf H\hat{\mathbf x}^{-} ). } \]

This is the covariance-form equivalent of the recursive least-squares update.

4.3.4.18 Physical Interpretation of the Gain

The gain balances prior uncertainty against measurement uncertainty.

If the prior covariance is large,

\[ \mathbf P^{-}\uparrow, \]

the estimator is less confident in its prior prediction.

The measurement tends to influence the posterior more strongly.

If measurement covariance is large,

\[ \mathbf R\uparrow, \]

the new measurement is considered less reliable.

The gain generally decreases in the measurement-sensitive directions.

Thus,

\[ \boxed{ \mathbf K \rightarrow \text{How Measurement Innovation Is Mapped Into State Correction}. } \]

4.3.4.19 Posterior Covariance

A common covariance update is

\[ \boxed{ \mathbf P^{+} = ( \mathbf I-\mathbf K\mathbf H ) \mathbf P^{-}. } \]

The measurement normally reduces uncertainty in the state directions to which it is informative.

Conceptually,

\[ \boxed{ \text{Useful Measurement} \rightarrow \text{Additional Information} \rightarrow \text{Reduced Posterior Uncertainty}. } \]

However, uncertainty may remain largely unchanged in state directions that the measurement does not constrain.

4.3.4.20 Joseph Covariance Form

For numerical robustness, the posterior covariance can also be written as

\[ \boxed{ \mathbf P^{+} = ( \mathbf I-\mathbf K\mathbf H ) \mathbf P^{-} ( \mathbf I-\mathbf K\mathbf H )^T + \mathbf K\mathbf R\mathbf K^T. } \]

This is commonly called the Joseph form.

It is algebraically equivalent under exact arithmetic but often provides better numerical preservation of symmetry and positive semidefiniteness.

Thus,

\[ \boxed{ \text{Equivalent Mathematics} \neq \text{Equivalent Numerical Robustness}. } \]

Engineering interpretation: The Joseph form is especially useful in long-running onboard estimators because repeated finite-precision arithmetic can otherwise slowly destroy covariance symmetry or positive semidefiniteness.

4.3.4.21 Recursive Least Squares for Static Parameters

Suppose a constant parameter vector

\[ \boldsymbol{\theta} \]

is observed through

\[ \mathbf z_k = \mathbf H_k\boldsymbol{\theta} + \mathbf v_k. \]

A recursive least-squares estimator can update

\[ \hat{\boldsymbol{\theta}}_k \]

as new observations arrive.

The update has the same form:

\[ \boxed{ \hat{\boldsymbol{\theta}}_k = \hat{\boldsymbol{\theta}}_{k-1} + \mathbf K_k [ \mathbf z_k - \mathbf H_k \hat{\boldsymbol{\theta}}_{k-1} ]. } \]

This is particularly useful for online calibration and slowly varying parameter estimation.

4.3.4.22 Recursive Least-Squares Covariance Update

For static RLS,

\[ \boxed{ \mathbf K_k = \mathbf P_{k-1} \mathbf H_k^T ( \mathbf H_k \mathbf P_{k-1} \mathbf H_k^T + \mathbf R_k )^{-1}. } \]

The covariance becomes

\[ \boxed{ \mathbf P_k = ( \mathbf I-\mathbf K_k\mathbf H_k ) \mathbf P_{k-1}. } \]

Then

\[ \hat{\boldsymbol{\theta}}_k \]

contains the information from all previously processed measurements.

4.3.4.23 Forgetting Factor

A conventional static RLS estimator assumes the unknown parameter remains constant.

If the parameter changes slowly, old measurements may become less relevant.

A forgetting factor

\[ \lambda \]

with

\[ 0<\lambda\le1 \]

can reduce the influence of older data.

A common information interpretation is

\[ \boxed{ \boldsymbol{\Lambda}_{k}^{-} = \lambda \boldsymbol{\Lambda}_{k-1}^{+}. } \]

For

\[ \lambda=1, \]

no information is intentionally forgotten.

For

\[ \lambda<1, \]

older information gradually loses influence.

4.3.4.24 Meaning of the Forgetting Factor

A forgetting factor can allow an estimator to track slowly changing parameters.

However,

\[ \boxed{ \text{Forgetting} \neq \text{Free Information}. } \]

Reducing old information increases estimator responsiveness but generally increases estimate variance.

Thus,

\[ \boxed{ \lambda\downarrow \rightarrow \text{Faster Adaptation} + \text{Less Memory} + \text{Greater Uncertainty}. } \]

The forgetting factor should therefore have a physical or statistical justification.

4.3.4.25 Bayesian Estimation Viewpoint

The same recursive process can be expressed using probability.

Let

\[ Z_{k-1} = \{ \mathbf z_1, \ldots, \mathbf z_{k-1} \} \]

represent all observations available before measurement \(k\).

The prior distribution is

\[ \boxed{ p(\mathbf x_k|Z_{k-1}). } \]

When measurement

\[ \mathbf z_k \]

arrives, Bayes' theorem gives

\[ \boxed{ p(\mathbf x_k|Z_k) = \frac{ p(\mathbf z_k|\mathbf x_k) p(\mathbf x_k|Z_{k-1}) }{ p(\mathbf z_k|Z_{k-1}) }. } \]

This equation is the conceptual foundation of Bayesian recursive estimation.

4.3.4.26 Prior, Likelihood, and Posterior

The three central Bayesian quantities are:

Prior

\[ \boxed{ p(\mathbf x|\text{previous information}) } \]

which describes what is believed before processing the new measurement.

Likelihood

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

which describes how compatible a candidate state is with the new measurement.

Posterior

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

which describes the updated state uncertainty after combining prior and measurement information.

Thus,

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

4.3.4.27 Evidence or Normalizing Constant

The denominator in Bayes' theorem is

\[ \boxed{ p(\mathbf z) = \int p(\mathbf z|\mathbf x) p(\mathbf x) \,d\mathbf x. } \]

This ensures that the posterior distribution integrates to one.

For state estimation, it is often sufficient to write

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

when only the shape or maximizer of the posterior is required.

4.3.4.28 Gaussian Prior

Suppose the prior is Gaussian:

\[ \boxed{ p(\mathbf x) = \mathcal N ( \hat{\mathbf x}^{-}, \mathbf P^{-} ). } \]

Its negative log-probability, ignoring constants, is proportional to

\[ \boxed{ ( \mathbf x-\hat{\mathbf x}^{-} )^T ( \mathbf P^{-} )^{-1} ( \mathbf x-\hat{\mathbf x}^{-} ). } \]

This is exactly the prior least-squares term introduced earlier.

4.3.4.29 Gaussian Likelihood

Suppose

\[ \mathbf z = \mathbf H\mathbf x+\mathbf v \]

with

\[ \mathbf v \sim \mathcal N(\mathbf0,\mathbf R). \]

Then

\[ \boxed{ p(\mathbf z|\mathbf x) \propto \exp \left[ -\frac12 ( \mathbf z-\mathbf H\mathbf x )^T \mathbf R^{-1} ( \mathbf z-\mathbf H\mathbf x ) \right]. } \]

The negative log-likelihood is therefore the weighted residual cost.

4.3.4.30 Gaussian Posterior

Multiplying Gaussian prior and Gaussian likelihood produces another Gaussian distribution.

Thus,

\[ \boxed{ p(\mathbf x|\mathbf z) = \mathcal N ( \hat{\mathbf x}^{+}, \mathbf P^{+} ). } \]

The posterior information matrix is

\[ \boxed{ (\mathbf P^{+})^{-1} = (\mathbf P^{-})^{-1} + \mathbf H^T \mathbf R^{-1} \mathbf H. } \]

The posterior information vector is

\[ \boxed{ (\mathbf P^{+})^{-1} \hat{\mathbf x}^{+} = (\mathbf P^{-})^{-1} \hat{\mathbf x}^{-} + \mathbf H^T \mathbf R^{-1} \mathbf z. } \]

These are exactly the recursive information equations derived from least squares.

4.3.4.31 Least-Squares and Bayesian Equivalence

We have now arrived at the same posterior from two viewpoints.

Least-squares viewpoint

Minimize

\[ J = \text{prior penalty} + \text{measurement residual penalty}. \]

Bayesian viewpoint

Compute

\[ p(\mathbf x|\mathbf z) \propto p(\mathbf z|\mathbf x) p(\mathbf x). \]

With Gaussian distributions,

\[ \boxed{ \text{Prior-Constrained WLS} \Longleftrightarrow \text{Gaussian MAP Estimation}. } \]

Because Gaussian mean and MAP coincide,

\[ \boxed{ \text{Gaussian Recursive WLS} \Longleftrightarrow \text{Gaussian Bayesian Update}. } \]

Engineering interpretation: The equivalence is exact for the stated Gaussian-linear assumptions. Outside that setting, least-squares, MAP, and posterior-mean estimates need not coincide.

4.3.4.32 Maximum A Posteriori Interpretation

The posterior estimate may be written as

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

Equivalently,

\[ \hat{\mathbf x}_{MAP} = \arg\min_{\mathbf x} \left[ -\ln p(\mathbf x|\mathbf z) \right]. \]

Under Gaussian assumptions this gives

\[ \boxed{ J = ( \mathbf x-\hat{\mathbf x}^{-} )^T (\mathbf P^{-})^{-1} ( \mathbf x-\hat{\mathbf x}^{-} ) + ( \mathbf z-\mathbf H\mathbf x )^T \mathbf R^{-1} ( \mathbf z-\mathbf H\mathbf x ). } \]

This is the same objective used for recursive WLS.

4.3.4.33 Dynamic Recursive Estimation

So far, the unknown state has effectively been treated as static during the measurement update.

A spacecraft state evolves with time:

\[ \boxed{ \mathbf x_k = f_{k-1} ( \mathbf x_{k-1}, \mathbf u_{k-1} ) + \mathbf w_{k-1}. } \]

Recursive dynamic estimation therefore requires two stages:

\[ \boxed{ \text{Prediction} \rightarrow \text{Measurement Update}. } \]

The prediction uses the dynamic model.

The measurement update uses the new observation.

4.3.4.34 Prediction Step

Given the previous posterior estimate

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

the dynamic model produces the prior for the next epoch:

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

The covariance is also propagated.

For a linear system

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

the covariance prediction is

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

4.3.4.35 Why Process Noise Appears in Prediction

The matrix

\[ \mathbf Q_{k-1} \]

represents uncertainty introduced by imperfect dynamics.

Possible sources include

Without process noise,

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

With process uncertainty,

\[ \boxed{ \text{Prediction Uncertainty} = \text{Propagated Previous Uncertainty} + \text{New Dynamic Uncertainty}. } \]

4.3.4.36 Measurement Update after Prediction

When measurement \(\mathbf z_k\) arrives,

\[ \mathbf y_k = \mathbf z_k - \mathbf H_k \hat{\mathbf x}_k^{-}. \]

Then

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

The gain is

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

The posterior estimate becomes

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

This is the standard linear Gaussian recursive update.

4.3.4.37 Relationship to the Kalman Filter

The equations just obtained are mathematically the core of the linear Kalman filter.

However, there are two useful ways to interpret them.

Control/navigation interpretation

The Kalman filter is a recursive state estimator.

Estimation-theory interpretation

The Kalman filter is a sequential Gaussian Bayesian estimator and recursive least-squares solution for a dynamic linear system.

Thus,

\[ \boxed{ \text{Kalman Filter} = \text{Dynamic Model} + \text{Recursive Bayesian Update} + \text{Gaussian Linear Estimation}. } \]

Section 4.2 develops the Kalman filter as a sensor-fusion architecture.

Here, the emphasis is on why those equations arise mathematically.

Engineering interpretation: This viewpoint is useful because it shows that the Kalman filter is not an isolated algorithm: it is the recursive realization of a broader information-estimation framework.

4.3.4.38 Why This Section Does Not Repeat Section 4.2

Section 4.2 addresses questions such as

Section 4.3.4 instead asks:

\[ \boxed{ \text{How does recursive filtering arise from least-squares information accumulation and Bayesian inference?} } \]

Therefore the focus here is mathematical connection rather than repeating individual filter architectures.

4.3.4.39 Recursive Information Form

Instead of propagating covariance,

\[ \mathbf P, \]

an estimator may work with information

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

The measurement update is particularly simple:

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

Similarly,

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

This makes independent information accumulation explicit.

4.3.4.40 Covariance Form versus Information Form

The covariance form naturally represents uncertainty:

\[ \mathbf P. \]

The information form naturally represents certainty:

\[ \boldsymbol{\Lambda} = \mathbf P^{-1}. \]

Conceptually,

\[ \boxed{ \mathbf P\uparrow \Rightarrow \text{Uncertainty}\uparrow } \]

while

\[ \boxed{ \boldsymbol{\Lambda}\uparrow \Rightarrow \text{Information}\uparrow. } \]

The two contain equivalent information when the matrices are nonsingular.

4.3.4.41 Why Measurement Updates Are Simple in Information Form

Independent measurement information adds:

\[ \boldsymbol{\Lambda}^{+} = \boldsymbol{\Lambda}^{-} + \Delta\boldsymbol{\Lambda}. \]

For multiple independent sensors,

\[ \boxed{ \boldsymbol{\Lambda}^{+} = \boldsymbol{\Lambda}^{-} + \sum_j \mathbf H_j^T \mathbf R_j^{-1} \mathbf H_j. } \]

This makes information form especially attractive conceptually for distributed sensing and multisensor estimation.

Each sensor contributes its own information block.

4.3.4.42 Prediction Is Less Simple in Information Form

Although the measurement update becomes simple in information form, dynamic prediction is generally less convenient.

For covariance form,

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

The corresponding information prediction involves inversion of this expression.

Therefore,

\[ \boxed{ \text{Information Form Simplifies Measurement Fusion but Can Complicate Dynamic Prediction}. } \]

The preferred form depends on the estimation architecture and computational structure.

4.3.4.43 Independent Sensor Information

Suppose sensors \(A\) and \(B\) provide independent measurements.

Then

\[ \Delta\boldsymbol{\Lambda}_A = \mathbf H_A^T \mathbf R_A^{-1} \mathbf H_A \]

and

\[ \Delta\boldsymbol{\Lambda}_B = \mathbf H_B^T \mathbf R_B^{-1} \mathbf H_B. \]

The combined information is

\[ \boxed{ \boldsymbol{\Lambda}^{+} = \boldsymbol{\Lambda}^{-} + \Delta\boldsymbol{\Lambda}_A + \Delta\boldsymbol{\Lambda}_B. } \]

Thus,

\[ \boxed{ \text{Independent Sensor Information Adds}. } \]

This is the recursive equivalent of stacking independent measurements in batch WLS.

4.3.4.44 Correlated Measurements

The simple additive form assumes that measurement errors are appropriately represented and independent of previously incorporated measurement errors.

If observations are correlated, treating them as independent can cause information to be counted more than once.

For jointly correlated measurements,

\[ \mathbf R = \begin{bmatrix} \mathbf R_A & \mathbf R_{AB}\\ \mathbf R_{BA} & \mathbf R_B \end{bmatrix}. \]

The complete covariance structure should be used.

Therefore,

\[ \boxed{ \text{Correlated Measurements} \neq \text{Independent Information Contributions}. } \]

4.3.4.45 Double Counting Information

Recursive systems must be especially careful about information reuse.

Suppose measurement \(B\) has already been computed using information from measurement \(A\).

If both are later treated as independent raw measurements, the estimator may count some information twice.

This problem can occur in distributed navigation systems where multiple subsystems exchange already-fused estimates.

Therefore,

\[ \boxed{ \text{Fusing Estimates Without Tracking Cross-Correlation Can Produce Overconfidence}. } \]

This is one motivation for conservative fusion methods and federated architectures.

Engineering interpretation: Double counting is particularly dangerous because the estimator may appear numerically stable while its covariance becomes too small. The state estimate can then be overconfident even when the mean appears reasonable.

4.3.4.46 Asynchronous Measurements

Recursive estimation naturally supports sensors operating at different rates.

For example,

\[ \text{IMU}: \text{high rate}, \]

\[ \text{GNSS}: \text{lower rate}, \]

\[ \text{camera}: \text{event dependent}. \]

The state can propagate continuously or at a fixed estimator rate, while measurement updates occur whenever measurements become available.

Thus,

\[ \boxed{ \text{Prediction at System Rate} + \text{Updates at Measurement Times}. } \]

This is one of the major practical advantages of recursive estimation.

4.3.4.47 Missing Measurements

If no measurement is available at epoch \(k\), the estimator can simply perform prediction:

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

and

\[ \boxed{ \mathbf P_k^{+} = \mathbf P_k^{-} } \]

if no update occurs.

The dynamic model continues to propagate the state.

As measurement gaps increase, uncertainty may grow.

Thus,

\[ \boxed{ \text{No Measurement} \rightarrow \text{Prediction Continues but Measurement Information Does Not Increase}. } \]

4.3.4.48 Delayed or Out-of-Sequence Measurements

A measurement may arrive after the estimator has already advanced beyond its measurement epoch.

Such observations are called delayed or out-of-sequence measurements.

Naively applying them at the current time can be incorrect because

\[ \mathbf z_k \]

corresponds to an earlier state.

Possible strategies include

The timestamp therefore belongs to the measurement information itself.

Engineering interpretation: Correct handling requires preserving the measurement timestamp and the estimator's historical state relationship. An out-of-sequence observation should not simply be treated as though it were measured at the current epoch.

4.3.4.49 Initialization

A recursive estimator requires an initial estimate:

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

and initial covariance

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

The covariance should describe how uncertain the initialization actually is.

If

\[ \mathbf P_0 \]

is unrealistically small, the estimator may resist valid measurements.

If it is extremely large, measurements may produce very aggressive corrections.

Thus,

\[ \boxed{ \text{Initial Covariance} \rightarrow \text{Initial Trust in the State Estimate}. } \]

4.3.4.50 Noninformative or Weak Prior

When little is known about the initial state, a weak prior may be represented by relatively large covariance.

Conceptually,

\[ \mathbf P_0\rightarrow\text{large} \]

means

\[ \boldsymbol{\Lambda}_0 = \mathbf P_0^{-1} \rightarrow \text{small}. \]

Thus early measurements dominate the estimate.

A completely uninformative prior is mathematically delicate because a singular information matrix may result before sufficient measurements have been accumulated.

4.3.4.51 Strong Prior

If the initial state is well known,

\[ \mathbf P_0 \]

may be small.

Then the estimator initially places substantial confidence in the prior.

A new measurement must contain sufficiently strong information to move the estimate significantly.

Thus,

\[ \boxed{ \text{Strong Prior} \rightarrow \text{Smaller Initial Measurement-Driven Corrections}. } \]

An incorrect but overconfident prior can therefore delay recovery.

4.3.4.52 Recursive Parameter Estimation

Unknown physical parameters can also be estimated recursively.

Define

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

The estimator can update both state and parameter estimates as new measurements arrive.

Examples include

However, each parameter must have sufficient measurement sensitivity.

4.3.4.53 Constant Parameter Model

A constant parameter may be modeled as

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

Its process-noise variance may be set very small or zero if the parameter is truly constant.

Then information about \(\theta\) accumulates over time.

The corresponding uncertainty generally decreases as more informative measurements are processed.

4.3.4.54 Slowly Varying Parameter Model

If a parameter changes slowly,

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

The parameter process noise

\[ Q_\theta \]

prevents the estimator from becoming permanently overconfident.

Conceptually,

\[ \boxed{ \text{Parameter Process Noise} \rightarrow \text{Allows Controlled Parameter Adaptation}. } \]

Too little process noise may prevent tracking real parameter changes.

Too much can make the estimate noisy.

4.3.4.55 Recursive Estimation and Identifiability

A recursive estimator cannot estimate a parameter merely because it has been added to the state vector.

If

\[ \frac{\partial h}{\partial\theta} \approx0, \]

then measurements contain little information about \(\theta\).

If two parameters produce almost identical signatures,

\[ \frac{\partial h}{\partial\theta_1} \approx c \frac{\partial h}{\partial\theta_2}, \]

their estimates may remain strongly correlated.

Therefore,

\[ \boxed{ \text{State Augmentation} \neq \text{Parameter Identifiability}. } \]

Engineering interpretation: Adding a parameter state without sensitivity can increase state dimension and correlation without adding information. Identifiability should therefore be assessed before or alongside augmentation.

4.3.4.56 Persistent Excitation

For online parameter estimation, measurements must contain sufficient independent variation over time.

This is often described through persistent excitation.

If the system remains in a condition where a parameter has almost no observable effect, repeated measurements may add little useful information.

Thus,

\[ \boxed{ \text{Repeated Measurements} + \text{No Parameter Excitation} \rightarrow \text{Little Parameter Information}. } \]

System motion and changing geometry can therefore be essential for recursive parameter estimation.

4.3.4.57 General Relative-Navigation Example

Consider a general rendezvous or proximity-navigation problem with relative state

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

The previous navigation solution provides

\[ \hat{\mathbf x}_{\text{rel},k}^{-} \]

with covariance

\[ \mathbf P_k^{-}. \]

Suppose a ranging sensor provides

\[ z_{\rho,k}. \]

The predicted range is

\[ \hat\rho_k = \| \hat{\mathbf r}_{\text{rel},k}^{-} \|. \]

The innovation is

\[ \boxed{ y_{\rho,k} = z_{\rho,k}-\hat\rho_k. } \]

The estimator uses this innovation, measurement sensitivity, and uncertainty to update the relative state.

Later, an optical observation may provide line-of-sight information and contribute a different sensitivity direction.

Thus,

\[ \boxed{ \text{Sequential Range Information} + \text{Sequential Angular Information} + \text{Relative Dynamics} \rightarrow \text{Continuously Updated Relative Navigation State}. } \]

This is a general RPO estimation concept rather than a particular mission architecture.

4.3.4.58 Why Different Measurements Correct Different State Directions

Suppose a range measurement is primarily sensitive to one combination of relative-position components.

Its measurement Jacobian may constrain that direction strongly.

A bearing measurement may constrain another combination.

The gain

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

maps each measurement innovation into the state directions correlated with that observation.

Thus,

\[ \boxed{ \text{Sensor Geometry} + \text{Prior State Correlation} \rightarrow \text{Direction of Recursive Correction}. } \]

4.3.4.59 Measurement Rejection and Gating

Not every measurement should necessarily be incorporated.

A large innovation may indicate

A normalized innovation statistic is often based on

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

This measures innovation magnitude relative to expected uncertainty.

Measurements with unusually large normalized innovations may be investigated or rejected according to defined validation logic.

Engineering interpretation: A gating threshold should be selected from the assumed statistical model and the measurement dimension, then validated in simulation. A large innovation is evidence of inconsistency, not proof of a particular failure cause.

4.3.4.60 Why Raw Residual Magnitude Is Not Enough

Suppose one sensor has

\[ \sigma=0.01 \]

while another has

\[ \sigma=10. \]

A residual of magnitude 1 means very different things for these two sensors.

Therefore measurement consistency should be judged relative to

\[ \mathbf S \]

rather than only raw residual magnitude.

Thus,

\[ \boxed{ \text{Innovation Significance} \neq \text{Innovation Magnitude Alone}. } \]

4.3.4.61 Innovation Consistency

For a properly modeled linear-Gaussian estimator, innovations should be broadly consistent with the predicted innovation covariance.

Persistent behaviour such as

may indicate

Therefore innovations are valuable diagnostic signals.

4.3.4.62 Innovation Should Be Unpredictable

A well-modeled innovation sequence should not contain strong predictable structure.

If future innovations can be predicted from past innovations, some systematic information may not have been captured by the estimator model.

Conceptually,

\[ \boxed{ \text{Structured Innovation} \rightarrow \text{Possible Model Deficiency}. } \]

This is related to innovation whiteness testing used in filter verification.

4.3.4.63 Covariance Consistency

An estimator can produce small errors while reporting unrealistic uncertainty, or large errors while appearing overly confident.

Therefore state accuracy and covariance consistency are different questions.

A consistent estimator should have actual errors broadly compatible with its predicted covariance.

This can be evaluated in simulation using truth information and metrics such as normalized estimation error.

The detailed Monte Carlo consistency framework belongs to Section 4.2 verification topics.

Engineering interpretation: In simulation, covariance consistency can be assessed against known truth; in flight, direct truth is unavailable, so innovation statistics, cross-comparisons, and external references become especially important.

4.3.4.64 Process Noise and Responsiveness

Process noise controls how rapidly prior certainty is lost during prediction.

Larger

\[ \mathbf Q \]

generally produces larger

\[ \mathbf P^{-}, \]

which often increases measurement influence.

Therefore,

\[ \boxed{ \mathbf Q\uparrow \rightarrow \text{Less Confidence in Dynamics} \rightarrow \text{Greater Measurement Responsiveness}. } \]

But excessive process noise can create unnecessarily noisy estimates.

4.3.4.65 Measurement Noise and Responsiveness

Increasing

\[ \mathbf R \]

indicates lower trust in the measurement.

The gain tends to decrease.

Thus,

\[ \boxed{ \mathbf R\uparrow \rightarrow \text{Lower Measurement Influence}. } \]

Reducing \(R\) does the opposite.

However, \(Q\) and \(R\) should represent actual uncertainty rather than being treated only as arbitrary response-tuning parameters.

4.3.4.66 Recursive Estimation Is Not Simple Averaging

The update

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

may look similar to weighted averaging.

However, in a multidimensional system,

Therefore recursive estimation is much richer than simply averaging sensor outputs.

4.3.4.67 Dynamic Coupling Transfers Measurement Information

Suppose a measurement directly observes position but not velocity.

If position and velocity are dynamically coupled, a position measurement can still reduce velocity uncertainty.

This occurs because covariance contains cross-correlation terms:

\[ \mathbf P = \begin{bmatrix} \mathbf P_{rr} & \mathbf P_{rv} \\ \mathbf P_{vr} & \mathbf P_{vv} \end{bmatrix}. \]

The gain can therefore map a position innovation into both position and velocity corrections.

Thus,

\[ \boxed{ \text{A Measurement Can Improve an Unmeasured State Through Dynamic Correlation}. } \]

4.3.4.68 Covariance Carries More Than Variances

The covariance matrix contains both

These cross-correlations are essential.

For example,

\[ P_{rv}\neq0 \]

indicates that position and velocity errors are statistically coupled.

Ignoring off-diagonal covariance can prevent correct information transfer between state components.

Thus,

\[ \boxed{ \text{Covariance Structure} \rightarrow \text{How Information Propagates Between States}. } \]

4.3.4.69 Sequential Processing of Multiple Measurements at One Epoch

Suppose two independent measurements are available at the same epoch.

They may be processed together:

\[ \mathbf z = \begin{bmatrix} \mathbf z_A\\ \mathbf z_B \end{bmatrix}, \]

or sequentially:

\[ \mathbf z_A \rightarrow \text{update} \rightarrow \mathbf z_B \rightarrow \text{update}. \]

For linear independent Gaussian measurements, these approaches are mathematically equivalent under exact arithmetic.

Thus,

\[ \boxed{ \text{Joint Update} \Longleftrightarrow \text{Sequential Independent Updates} } \]

under the appropriate assumptions.

4.3.4.70 Why Sequential Scalar Updates Can Be Useful

Processing scalar or small measurement blocks sequentially can reduce the size of matrix inversions.

Instead of inverting a large innovation covariance, the estimator may perform several small updates.

This may be computationally attractive for onboard applications.

However, correlations between measurements must still be handled correctly.

4.3.4.71 Numerical Stability

Recursive covariance updates occur repeatedly, sometimes for millions of cycles.

Numerical roundoff can gradually produce

Therefore implementation details matter.

Useful strategies include

Engineering interpretation: Numerical stability is an estimator-design requirement, not merely a software detail. A mathematically correct recursion can still fail over long operation if finite-precision effects are ignored.

4.3.4.72 Square-Root Recursive Estimation

Instead of propagating

\[ \mathbf P \]

directly, square-root methods propagate a factor such as

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

This can improve numerical stability and preserve positive semidefiniteness more reliably.

Square-root forms are especially valuable when

4.3.4.73 Information Square-Root Form

Similarly, the information matrix may be factored:

\[ \boldsymbol{\Lambda} = \mathbf R_I^T \mathbf R_I. \]

Measurements can then update the square-root information representation.

This avoids explicitly forming some poorly conditioned matrices and provides numerically stable information accumulation.

4.3.4.74 Computational Complexity

For a state dimension \(n\), covariance propagation and matrix operations can become expensive as \(n\) grows.

Recursive estimation avoids continuously increasing the number of historical measurements in the active optimization problem.

The size of the core estimator therefore remains primarily tied to the state dimension rather than the entire observation history.

Thus,

\[ \boxed{ \text{Recursive Estimation} \rightarrow \text{Bounded Online State Representation}. } \]

This makes it attractive for onboard use.

4.3.4.75 Memory Advantage

A full batch estimator may need access to

\[ \mathbf z_1,\ldots,\mathbf z_N. \]

A recursive estimator generally needs only

Therefore memory requirements can be substantially lower.

This is particularly relevant for long-duration autonomous operation.

4.3.4.76 Recursive Estimation Loses Easy Reprocessing

The computational efficiency of recursive estimation has a tradeoff.

Once information is compressed into

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

individual historical measurements are no longer explicitly represented in the current estimator state.

If an old calibration error is later discovered, reprocessing may require access to stored raw data.

Thus,

\[ \boxed{ \text{Recursive Efficiency} \leftrightarrow \text{Less Direct Access to Historical Measurement Contributions}. } \]

Batch reprocessing is often easier for offline analysis.

4.3.4.77 Recursive Estimation and Smoothing

Filtering estimates the state using measurements available up to the current epoch:

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

Smoothing additionally uses future measurements:

\[ \boxed{ \hat{\mathbf x}_{k|N}, \qquad N>k. } \]

Therefore,

\[ \boxed{ \text{Filter} = \text{Past + Present Information} } \]

while

\[ \boxed{ \text{Smoother} = \text{Past + Present + Future Information}. } \]

Smoothing is developed in Section 4.3.7.

4.3.4.78 Recursive Estimation and Nonlinear Systems

Most spacecraft dynamics and measurements are nonlinear:

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

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

The linear recursive equations cannot simply be applied globally.

Possible approaches include

These lead to estimator families such as EKF, UKF, and particle filters, already treated in Section 4.2 from the sensor-fusion perspective.

4.3.4.79 Local Linearization in Recursive Estimation

For a nonlinear measurement model,

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

linearize around the prior estimate:

\[ \boxed{ h(\mathbf x_k) \approx h(\hat{\mathbf x}_k^{-}) + \mathbf H_k ( \mathbf x_k-\hat{\mathbf x}_k^{-} ). } \]

where

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

The recursive correction is then based on this local approximation.

This connects recursive estimation to nonlinear least-squares linearization.

4.3.4.80 Recursive versus Iterative Measurement Update

A conventional recursive update processes the measurement once.

For a strongly nonlinear measurement model, one may instead relinearize around an updated state and process the same measurement iteratively.

Conceptually,

\[ \boxed{ \text{Predict} \rightarrow \text{Linearize} \rightarrow \text{Update} \rightarrow \text{Relinearize} \rightarrow \text{Repeat}. } \]

This creates a closer connection between nonlinear recursive estimation and Gauss-Newton-style differential correction.

4.3.4.81 Constraints in Recursive Estimation

Physical constraints may exist, such as

\[ \|\mathbf q\|=1 \]

or bounds on parameters.

Possible approaches include

Constraints should represent genuine physical information.

4.3.4.82 Pseudo-Measurements

A known relationship may sometimes be represented as an artificial observation.

Suppose approximately

\[ \mathbf C\mathbf x = \mathbf d. \]

Then define

\[ \boxed{ \mathbf z_c = \mathbf d, \qquad \mathbf H_c=\mathbf C } \]

with covariance

\[ \mathbf R_c. \]

This allows the constraint to be incorporated through the same recursive measurement-update machinery.

A very small covariance makes the constraint strong.

A larger covariance makes it soft.

4.3.4.83 Sensor Bias in Recursive Estimation

Suppose

\[ \mathbf z_k = h(\mathbf x_k) + \mathbf b_k + \boldsymbol{\eta}_k. \]

If bias is ignored, the state may absorb it.

Instead, augment the state:

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

A bias model may be

\[ \mathbf b_{k+1} = \mathbf b_k+\mathbf w_b. \]

This allows the estimator to separate, where observable, physical state changes from sensor-bias evolution.

4.3.4.84 State-Parameter Correlation

When a bias or parameter is added to the state, it may become strongly correlated with physical states.

For example,

\[ \mathbf P = \begin{bmatrix} \mathbf P_{xx} & \mathbf P_{x\theta}\\ \mathbf P_{\theta x} & \mathbf P_{\theta\theta} \end{bmatrix}. \]

Large

\[ \mathbf P_{x\theta} \]

indicates that uncertainty in the state and parameter is coupled.

This can be physically meaningful, but strong correlation may also signal weak distinguishability.

4.3.4.85 Recursive Information Growth

For a truly static parameter with repeated independent informative measurements,

\[ \boldsymbol{\Lambda}_k = \boldsymbol{\Lambda}_{k-1} + \Delta\boldsymbol{\Lambda}_k. \]

Therefore information tends to increase monotonically.

Correspondingly, uncertainty tends to decrease.

However, in a dynamic estimator, process noise can reduce confidence between measurements.

Therefore dynamic estimation contains competing effects:

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

while

\[ \boxed{ \text{Process Uncertainty Reduces Predictive Confidence}. } \]

4.3.4.86 Steady-State Behaviour

For some time-invariant linear systems, the covariance and gain may converge toward approximately constant values.

Then

\[ \mathbf K_k \rightarrow \mathbf K_\infty. \]

This is called steady-state filtering behaviour.

It can reduce onboard computation in some applications because a nearly constant gain may eventually be used.

However, this is only appropriate when the assumptions leading to steady state are valid.

Engineering interpretation: Steady-state gain is most meaningful when system dynamics, measurement geometry, sampling, and noise statistics remain approximately stationary. Changing geometry or sensor availability can invalidate a fixed-gain assumption.

4.3.4.87 Covariance Collapse

If an estimator repeatedly processes measurements while assuming too little process noise or too-small measurement uncertainty, the covariance may become unrealistically small.

Then

\[ \mathbf P\rightarrow\text{very small} \]

and subsequent valid measurements may have insufficient influence.

This is sometimes called overconfidence or covariance collapse.

Thus,

\[ \boxed{ \text{Small Covariance} \neq \text{Automatically Accurate Estimate}. } \]

Covariance must remain statistically credible.

4.3.4.88 Divergence

An estimator is said to diverge when the estimate moves increasingly away from the actual state or loses statistical consistency.

Possible causes include

Divergence is therefore often a symptom of a model or consistency problem rather than simply an equation failure.

4.3.4.89 Overweighting Measurements

If

\[ \mathbf R \]

is made unrealistically small, measurements are treated as more accurate than they really are.

This produces large gain and potentially aggressive corrections.

Noise can then be injected into the state estimate.

Therefore,

\[ \boxed{ \text{Underestimated Measurement Noise} \rightarrow \text{Overconfident Measurement Updates}. } \]

4.3.4.90 Underweighting Measurements

If

\[ \mathbf R \]

is too large, useful measurements are treated as uninformative.

The estimator may depend excessively on the model.

This can allow model errors to accumulate.

Thus,

\[ \boxed{ \text{Overestimated Measurement Noise} \rightarrow \text{Weak Measurement Correction}. } \]

4.3.4.91 Underestimated Process Noise

If

\[ \mathbf Q \]

is too small, the estimator assumes the dynamics are more accurate than they actually are.

The covariance can become too small during prediction.

Then the estimator may reject or underweight valid measurement corrections.

This is a common source of inconsistent estimation.

4.3.4.92 Overestimated Process Noise

If

\[ \mathbf Q \]

is too large, predicted covariance grows excessively.

Measurements then dominate updates.

The state estimate may become unnecessarily noisy and sensitive to individual observations.

Thus,

\[ \boxed{ Q \text{ and } R \text{ should represent uncertainty, not arbitrary gains}. } \]

4.3.4.93 Model Error versus Process Noise

Process noise can represent uncertainty not explicitly modeled in the dynamics.

However, increasing \(Q\) is not a substitute for correcting known modelling errors.

If a deterministic bias or missing force is known to exist, explicitly modeling it is generally more meaningful than indefinitely inflating process noise.

Thus,

\[ \boxed{ \text{Known Physics} \rightarrow \text{Model It}; \qquad \text{Residual Uncertainty} \rightarrow \text{Represent Statistically}. } \]

Engineering interpretation: Process noise is best interpreted as a model for unresolved uncertainty. When a repeatable deterministic effect is known, explicitly representing that physics or estimating the associated parameter usually gives a more interpretable solution.

4.3.4.94 Relationship to Moving-Window Estimation

Recursive estimation uses a compact summary of all past information.

A moving-window estimator instead keeps only a finite recent measurement history.

Conceptually,

\[ \boxed{ \{\mathbf z_{k-L+1},\ldots,\mathbf z_k\} } \]

is processed repeatedly.

This can provide a compromise between

It is useful when old information should not dominate or when recent states and parameters require re-estimation.

4.3.4.95 Recursive Estimation and Real-Time Navigation

Recursive estimation is naturally suited to onboard navigation because the cycle is

\[ \boxed{ \text{Propagate} \rightarrow \text{Receive Measurement} \rightarrow \text{Validate} \rightarrow \text{Update} \rightarrow \text{Output Navigation State}. } \]

This can be repeated whenever new sensor data arrive.

The estimator therefore maintains a continuously updated navigation solution.

4.3.4.96 Real-Time Timing Requirement

A recursive algorithm must usually complete its update before the next required processing deadline.

Thus computation time must satisfy approximately

\[ \boxed{ t_{\text{compute}} < t_{\text{available}}. } \]

If the estimator cannot meet this requirement,

Computational architecture is therefore part of recursive-estimator design.

4.3.4.97 Estimation Latency

A mathematically accurate estimate can still be operationally poor if it is delivered too late.

If a state estimate corresponds to

\[ t_k \]

but becomes available at

\[ t_k+\Delta t, \]

guidance and control are acting on delayed information.

Thus,

\[ \boxed{ \text{Navigation Performance} = \text{Accuracy} + \text{Uncertainty Quality} + \text{Timeliness}. } \]

4.3.4.98 Sequential Sensor Scheduling

If several measurements are available but computational resources are limited, the estimator may need to decide which observations to process first or whether all are necessary.

Useful considerations include

This connects recursive estimation to sensor-management and information-based scheduling problems.

4.3.4.99 Information Contribution of One Measurement

For measurement \(k\),

\[ \boxed{ \Delta\boldsymbol{\Lambda}_k = \mathbf H_k^T \mathbf R_k^{-1} \mathbf H_k. } \]

This allows the contribution of a measurement to be interpreted before or after processing.

A measurement with

can provide significant information.

A highly precise measurement may still contribute little to a particular state direction if its sensitivity to that direction is weak.

4.3.4.100 Recursive Observability Development

Observability can develop over time.

At one instant,

\[ \mathbf H_k \]

may not constrain every state component.

But dynamics change the state and measurement geometry.

Across multiple recursive updates, previously weak state directions may become informative.

Thus,

\[ \boxed{ \text{Instantaneous Weak Observability} \not\Rightarrow \text{Long-Term Unobservability}. } \]

Dynamic evolution must be considered.

Engineering interpretation: This is why recursive observability should be assessed over time rather than from one measurement Jacobian alone. Dynamic coupling and changing sensor geometry can gradually reveal states that are weakly constrained instantaneously.

4.3.4.101 Covariance as an Observability Indicator

Persistent large covariance in one state direction may indicate weak available information.

Similarly, strong state correlations may reveal that certain state combinations are difficult to separate.

However, covariance depends on the assumed model and noise statistics.

Therefore,

\[ \boxed{ \text{Large Covariance} \rightarrow \text{Possible Weak Estimation Direction}, } \]

but formal observability analysis may still be required.

4.3.4.102 Recursive Estimation with Maneuvers

If a maneuver is known and commanded, its effect can be included in state propagation:

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

If maneuver magnitude is uncertain, its uncertainty should influence prediction covariance.

An unknown maneuver can also be treated as

The appropriate treatment depends on what information is known about the maneuver.

4.3.4.103 Maneuver Detection from Innovations

An unexpected maneuver may cause innovation sequences to depart strongly from predicted behaviour.

For example,

\[ \mathbf y_k \]

may suddenly become large and remain structured.

This can indicate

Innovation monitoring can therefore help identify dynamic events, but the cause should not be inferred from residual magnitude alone.

4.3.4.104 Recursive Multi-Sensor Architecture

A general multisensor recursive architecture can be written as

\[ \boxed{ \begin{array}{c} \text{Previous Posterior State}\\ \downarrow\\ \text{Dynamic Prediction}\\ \downarrow\\ \text{Prior State and Covariance}\\ \downarrow\\ \text{Sensor A Update}\\ \downarrow\\ \text{Sensor B Update}\\ \downarrow\\ \cdots\\ \downarrow\\ \text{Current Posterior Navigation State} \end{array} } \]

Sensors may update at different times and observe different combinations of states.

The common state and covariance carry information between these updates.

4.3.4.105 Centralized versus Distributed Recursive Estimation

In a centralized estimator, raw or minimally processed measurements are combined in one state estimator.

In a distributed architecture, local estimators may generate partial estimates that are later fused.

Centralized fusion simplifies correlation accounting when all information is available.

Distributed fusion can improve modularity and fault containment but requires care with cross-correlation.

Thus,

\[ \boxed{ \text{Estimator Architecture} \rightarrow \text{How Information and Correlation Are Managed}. } \]

4.3.4.106 Recursive Estimation and Sensor Failure

If a sensor becomes unavailable, recursive estimation can continue using

However, uncertainty may grow in directions that depended strongly on the failed sensor.

This may be visible as increasing covariance.

Thus covariance can support navigation-health monitoring and sensor-failure assessment.

4.3.4.107 Why Covariance Should Grow During Information Loss

Suppose no measurements constrain a particular state for some interval.

Process noise and propagation uncertainty continue to act.

Therefore uncertainty should usually increase.

If the estimator covariance remains unrealistically small despite long measurement outages, the uncertainty model may be inconsistent.

Conceptually,

\[ \boxed{ \text{Information Loss} + \text{Dynamic Uncertainty} \rightarrow \text{Covariance Growth}. } \]

4.3.4.108 Measurement Update as Error Correction

The recursive state equation

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

can be interpreted as

\[ \boxed{ \text{Posterior} = \text{Prediction} + \text{Estimated Prediction Error}. } \]

The term

\[ \mathbf K\mathbf y \]

is the estimated state correction inferred from the measurement innovation.

This gives a useful physical interpretation of recursive estimation.

4.3.4.109 Bayesian Prediction Step

The dynamic prediction also has a Bayesian interpretation.

The prior distribution at time \(k\) is

\[ \boxed{ p(\mathbf x_k|Z_{k-1}) = \int p(\mathbf x_k|\mathbf x_{k-1}) p(\mathbf x_{k-1}|Z_{k-1}) \,d\mathbf x_{k-1}. } \]

This is the Chapman-Kolmogorov prediction equation.

It propagates the entire state probability distribution through the transition model.

4.3.4.110 Bayesian Measurement Update

Once measurement \(\mathbf z_k\) arrives,

\[ \boxed{ p(\mathbf x_k|Z_k) \propto p(\mathbf z_k|\mathbf x_k) p(\mathbf x_k|Z_{k-1}). } \]

Thus the complete Bayesian recursion consists of

\[ \boxed{ \text{Prediction Integral} \rightarrow \text{Bayesian Measurement Update}. } \]

For linear Gaussian systems, these probability-distribution equations reduce to the familiar mean and covariance recursions.

4.3.4.111 Why Gaussian Estimation Is Special

The Gaussian distribution is completely characterized by

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

For linear dynamics and linear measurements with Gaussian uncertainties, Gaussian distributions remain Gaussian under both prediction and update.

Therefore the estimator only needs to propagate

\[ \hat{\mathbf x} \]

and

\[ \mathbf P. \]

This is why the linear Kalman filter has such a compact recursive form.

4.3.4.112 Non-Gaussian Recursive Estimation

If the posterior distribution is strongly non-Gaussian, mean and covariance may no longer describe all relevant information.

The state distribution may be

Then more general methods may be required.

Examples include

These methods are covered conceptually elsewhere in the sensor-fusion section.

4.3.4.113 Batch versus Recursive Information Storage

Batch estimation retains explicit measurement equations:

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

Recursive estimation compresses previously processed observations into quantities such as

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

or

\[ \boldsymbol{\eta}, \boldsymbol{\Lambda}. \]

Therefore,

\[ \boxed{ \text{Recursive Estimation} = \text{Information Compression Through Sufficient Statistics} } \]

for cases where those statistics completely describe the relevant posterior information.

For Gaussian estimation, mean and covariance serve this role.

4.3.4.114 Advantages of Recursive Estimation

Important advantages include

These features explain why recursive estimation is central to spacecraft navigation systems.

4.3.4.115 Limitations of Recursive Estimation

Important limitations include

Therefore recursive implementation efficiency does not eliminate the need for estimation-system verification.

4.3.4.116 Common Failure Modes

Common recursive-estimation failures include

These should be diagnosed systematically.

4.3.4.117 Recursive Estimation Development Workflow

A disciplined workflow is:

  1. Define the state vector.
  2. Define any estimated parameters.
  3. Define the dynamic model.
  4. Define process-noise assumptions.
  5. Define all measurement models.
  6. Define measurement covariance.
  7. Define initial state.
  8. Define initial covariance.
  9. Establish prediction timing.
  10. Associate measurements with correct epochs.
  11. Predict the state.
  12. Predict covariance.
  13. Predict measurement.
  14. Compute innovation.
  15. Compute innovation covariance.
  16. Perform measurement validation.
  17. Compute recursive gain.
  18. Update state.
  19. Update covariance.
  20. Preserve covariance symmetry and numerical stability.
  21. Process additional measurements.
  22. Monitor innovation statistics.
  23. Monitor covariance growth or collapse.
  24. Check observability and parameter correlations.
  25. Validate against batch solutions where possible.
  26. Perform Monte Carlo consistency analysis.
  27. Test measurement outages.
  28. Test sensor biases.
  29. Test model mismatch.
  30. Test delayed and asynchronous measurements.
  31. Verify computational timing.
  32. Verify physical plausibility.

Conceptually,

\[ \boxed{ \begin{array}{c} \text{Initialize}\\ \downarrow\\ \text{Predict State}\\ \downarrow\\ \text{Predict Uncertainty}\\ \downarrow\\ \text{Receive Measurement}\\ \downarrow\\ \text{Compute Innovation}\\ \downarrow\\ \text{Evaluate Measurement Information}\\ \downarrow\\ \text{Update State}\\ \downarrow\\ \text{Update Uncertainty}\\ \downarrow\\ \text{Validate}\\ \downarrow\\ \text{Repeat} \end{array} } \]

Engineering interpretation: The workflow should be treated as an engineering loop rather than a one-time implementation checklist. Innovation statistics, covariance behaviour, observability, timing, and failure tests should feed back into model and covariance refinement.

4.3.4.118 Complete Recursive Estimation Architecture

The complete dynamic recursive-estimation architecture may be summarized as

\[ \boxed{ \begin{array}{c} \text{Previous Posterior}\\ \hat{\mathbf x}_{k-1}^{+}, \mathbf P_{k-1}^{+} \\ \downarrow\\ \text{Dynamic Prediction}\\ \hat{\mathbf x}_{k}^{-} = f( \hat{\mathbf x}_{k-1}^{+} ) \\ \downarrow\\ \text{Covariance Prediction}\\ \mathbf P_k^{-} \\ \downarrow\\ \text{Measurement Prediction}\\ \hat{\mathbf z}_k = h( \hat{\mathbf x}_k^{-} ) \\ \downarrow\\ \text{Innovation}\\ \mathbf y_k = \mathbf z_k-\hat{\mathbf z}_k \\ \downarrow\\ \text{Innovation Covariance}\\ \mathbf S_k \\ \downarrow\\ \text{Information / Gain Calculation}\\ \mathbf K_k \\ \downarrow\\ \text{State Update}\\ \hat{\mathbf x}_k^{+} = \hat{\mathbf x}_k^{-} + \mathbf K_k\mathbf y_k \\ \downarrow\\ \text{Covariance Update}\\ \mathbf P_k^{+} \\ \downarrow\\ \text{Posterior Becomes Basis for Next Prediction} \end{array} } \]

4.3.4.119 Key Equations

Linear measurement model

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

Measurement covariance

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

Prior-constrained least-squares cost

\[ \boxed{ \begin{aligned} J(\mathbf x) ={}& ( \mathbf x-\hat{\mathbf x}^{-} )^T ( \mathbf P^{-} )^{-1} ( \mathbf x-\hat{\mathbf x}^{-} ) \\ &+ ( \mathbf z-\mathbf H\mathbf x )^T \mathbf R^{-1} ( \mathbf z-\mathbf H\mathbf x ) \end{aligned} } \]

Information matrix

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

Information vector

\[ \boxed{ \boldsymbol{\eta} = \boldsymbol{\Lambda} \hat{\mathbf x} } \]

Information update

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

Information-vector update

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

Innovation

\[ \boxed{ \mathbf y = \mathbf z-\mathbf H\hat{\mathbf x}^{-} } \]

Innovation covariance

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

Recursive gain

\[ \boxed{ \mathbf K = \mathbf P^{-} \mathbf H^T \mathbf S^{-1} } \]

State update

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

Covariance update

\[ \boxed{ \mathbf P^{+} = ( \mathbf I-\mathbf K\mathbf H ) \mathbf P^{-} } \]

Joseph covariance form

\[ \boxed{ \mathbf P^{+} = ( \mathbf I-\mathbf K\mathbf H ) \mathbf P^{-} ( \mathbf I-\mathbf K\mathbf H )^T + \mathbf K\mathbf R\mathbf K^T } \]

Dynamic prediction

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

Linear covariance prediction

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

Bayesian prediction

\[ \boxed{ p(\mathbf x_k|Z_{k-1}) = \int p(\mathbf x_k|\mathbf x_{k-1}) p(\mathbf x_{k-1}|Z_{k-1}) \,d\mathbf x_{k-1} } \]

Bayesian measurement update

\[ \boxed{ p(\mathbf x_k|Z_k) \propto p(\mathbf z_k|\mathbf x_k) p(\mathbf x_k|Z_{k-1}) } \]

Measurement information increment

\[ \boxed{ \Delta\boldsymbol{\Lambda}_k = \mathbf H_k^T \mathbf R_k^{-1} \mathbf H_k } \]

Accumulated information

\[ \boxed{ \boldsymbol{\Lambda}_k = \boldsymbol{\Lambda}_0 + \sum_{i=1}^{k} \mathbf H_i^T \mathbf R_i^{-1} \mathbf H_i } \]

4.3.4.120 Key Takeaways

Recursive estimation processes information incrementally rather than repeatedly solving the complete measurement history.

The basic concept is

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

From the least-squares viewpoint, recursive estimation accumulates information:

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

Thus,

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

From the Bayesian viewpoint,

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

For Gaussian linear estimation, these two viewpoints are mathematically equivalent.

The measurement innovation is

\[ \boxed{ \mathbf y = \mathbf z-\mathbf H\hat{\mathbf x}^{-}, } \]

and the state update is

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

The gain determines how measurement-space disagreement becomes a state correction.

The estimator must balance

\[ \boxed{ \text{Dynamic Confidence} \leftrightarrow \text{Measurement Confidence}. } \]

Process covariance

\[ \mathbf Q \]

represents uncertainty introduced during state propagation, while measurement covariance

\[ \mathbf R \]

represents observation uncertainty.

Neither should be treated merely as an arbitrary tuning gain.

Recursive estimation is especially suitable for real-time spacecraft navigation because it maintains a fixed-size representation of previously accumulated information and naturally handles measurements as they arrive.

However,

\[ \boxed{ \text{Recursive Processing} \neq \text{Automatic Consistency}. } \]

The estimator can still fail because of

A central engineering principle is

\[ \boxed{ \text{Recursive Estimation Does Not Generate New Information; It Efficiently Carries Forward and Updates the Information Already Available From Dynamics, Measurements, and Prior Knowledge}. } \]

Finally,

\[ \boxed{ \text{Batch Least Squares} \leftrightarrow \text{Recursive Least Squares} \leftrightarrow \text{Bayesian Gaussian Estimation} } \]

are not isolated subjects. They are different mathematical views of how information about an unknown state is accumulated.

4.3.4.121 What Comes Next

The recursive relationships developed in this section are especially elegant when the system is linear:

\[ \mathbf z = \mathbf H\mathbf x+\mathbf v. \]

Real spacecraft-estimation problems are rarely perfectly linear.

Instead,

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

and the trajectory may also evolve through nonlinear dynamics:

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

The estimator must therefore solve a more difficult problem:

\[ \boxed{ \text{How do we estimate unknown quantities when the residuals depend nonlinearly on the state?} } \]

A common approach is to linearize the nonlinear model around the current estimate and repeatedly solve for a correction.

This produces methods such as

That leads directly to

\[ \boxed{ \textbf{4.3.5 Nonlinear Least Squares and Differential Corrections} } \]

where the next section should develop how

\[ \boxed{ \text{Nonlinear Residual} \rightarrow \text{Jacobian} \rightarrow \text{Linearized Correction} \rightarrow \text{State Update} \rightarrow \text{Relinearization} } \]

turns a nonlinear spacecraft estimation problem into a sequence of locally linear estimation problems.

Next: 4.3.5 Nonlinear Least Squares and Differential Corrections

Recursive estimation provides a compact sequential framework for carrying forward prior information and incorporating new measurements. The linear-Gaussian case leads naturally to information-form, covariance-form, and Kalman-style updates.

Real spacecraft estimation problems are frequently nonlinear. The next section therefore develops nonlinear residuals, Jacobians, local linearization, Gauss–Newton iteration, differential corrections, convergence behaviour, damping, Levenberg–Marquardt, trust-region ideas, initial-guess sensitivity, and practical diagnostics for nonlinear estimation problems.

4.3.5 Nonlinear Least Squares and Differential Corrections

Nonlinear residuals, Jacobians, Gauss–Newton iteration, differential correction, weighted nonlinear least squares, convergence criteria, damping, Levenberg–Marquardt, trust-region interpretation, initial-guess sensitivity, covariance approximation, and practical nonlinear-estimation diagnostics.

Continue to Nonlinear Least Squares and Differential Corrections →