Estimation Problem Formulation

4.3.1 Estimation Problem Formulation

Before selecting a particular estimation algorithm, the estimation problem itself must first be formulated correctly.

This is a fundamental step in spacecraft guidance, navigation, control, orbit determination, sensor calibration, and relative navigation because an estimator can only infer quantities for which sufficient information exists in the measurements, dynamics, constraints, and prior knowledge.

A sophisticated estimator cannot compensate automatically for an incorrectly defined state, incorrect measurement model, inconsistent reference frames, missing parameters, poor geometry, or information that simply does not exist in the observations.

The central estimation problem is

\[ \boxed{ \text{Imperfect Observations} + \text{Mathematical Model} + \text{Uncertainty Information} \rightarrow \text{Unknown States and Parameters} } \]

A general measurement model is

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

where

• \mathbf z_k is the observation vector at measurement epoch t_k,
• \mathbf x_k is the unknown dynamic state,
• \boldsymbol{\theta} represents unknown model or calibration parameters,
• h_k(\cdot) is the measurement or observation function,
• \mathbf v_k represents measurement uncertainty.

If the physical system evolves dynamically, the estimator must also consider a state model such as

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

or in discrete time,

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

where \mathbf w_k represents uncertainty in the dynamic model.

A dynamic estimation problem therefore normally contains two fundamental mathematical relationships:

\[ \boxed{ \text{Dynamic Model} \rightarrow \text{How the State Evolves} } \]

and

\[ \boxed{ \text{Measurement Model} \rightarrow \text{How the State Produces Observations}. } \]

The estimator connects these two pieces of information.

4.3.1.1 What Must Be Defined Before Estimation Begins

An estimation algorithm should not be selected before the estimation objective is clearly defined.

A complete formulation should answer:

• What quantities are unknown?
• Which of those quantities are dynamic states?
• Which are constant or slowly varying parameters?
• What quantities are actually measured?
• In what reference frames are states and measurements expressed?
• At what epochs are observations obtained?
• How do measurements depend on the unknown quantities?
• How does the system evolve between measurements?
• What uncertainty exists in the measurements?
• What uncertainty exists in the dynamics?
• Is any prior state or parameter information available?
• Are there physical constraints?
• Are the unknown quantities observable or identifiable?
• Is the measurement geometry sufficient?
• Will the observations be processed in batch or sequentially?
• What numerical method will eventually be needed?

Conceptually,

\[ \boxed{ \begin{array}{c} \text{Define Unknowns}\\ \downarrow\\ \text{Define Dynamics}\\ \downarrow\\ \text{Define Measurements}\\ \downarrow\\ \text{Define Uncertainty}\\ \downarrow\\ \text{Check Information Content}\\ \downarrow\\ \text{Choose Estimation Method} \end{array} } \]

The estimation algorithm should therefore be regarded as the last major decision in the formulation stage, not the first.

4.3.1.2 Unknown State Vector

The first quantity to define is the state vector.

A state is a set of variables that describes the instantaneous condition of a dynamic system sufficiently to propagate that system forward in time when the necessary inputs and models are known.

For translational spacecraft dynamics,

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

where

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

and

\[ \mathbf v = \begin{bmatrix} v_x\\v_y\\v_z \end{bmatrix}. \]

Therefore,

\[ \mathbf x = \begin{bmatrix} x& y& z& v_x& v_y& v_z \end{bmatrix}^{T}. \]

For attitude dynamics, the state may contain

\[ \boxed{ \mathbf x = \begin{bmatrix} \mathbf q\\ \boldsymbol{\omega} \end{bmatrix} } \]

where

• \mathbf q represents spacecraft attitude,
• \boldsymbol{\omega} represents angular velocity.

A more complete navigation state may be

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

where

\[ \mathbf b_g \]

and

\[ \mathbf b_a \]

represent gyroscope and accelerometer biases.

The state should contain only quantities that are necessary for the estimation problem and that can be supported by the available information.

Thus,

\[ \boxed{ \text{More Estimated States} \neq \text{Automatically Better Estimation}. } \]

Adding unnecessary or weakly observable states can actually make an estimation problem more difficult.

4.3.1.3 State Definition Must Include Frame, Units, and Epoch

A state vector is not fully defined merely by listing its numerical components.

For example,

\[ \mathbf r = \begin{bmatrix} 100\\ 20\\ 5 \end{bmatrix} \]

is meaningless unless the following are also known:

• coordinate frame,
• unit,
• origin,
• reference epoch,
• axis convention.

Position may be expressed in

\[ \text{ECI}, \qquad \text{ECEF}, \qquad \text{LVLH/Hill}, \qquad \text{body frame}, \]

depending on the problem.

Similarly, relative position in a proximity-navigation application may be expressed in a target-centered local orbital frame or another defined relative-navigation frame.

Therefore,

\[ \boxed{ \text{State Definition} = \text{Variables} + \text{Frame} + \text{Units} + \text{Epoch} + \text{Convention}. } \]

Many apparent estimator failures are actually frame-definition or convention errors.

4.3.1.4 Unknown Parameters

Not every unknown quantity is a conventional dynamic state.

A model may contain uncertain physical, calibration, environmental, or maneuver parameters.

Collect these into

\[ \boxed{ \boldsymbol{\theta} = \begin{bmatrix} \theta_1\\ \theta_2\\ \vdots\\ \theta_p \end{bmatrix}. } \]

Typical spacecraft parameters may include

\[ \boldsymbol{\theta} = \begin{bmatrix} C_D\\ C_R\\ BC\\ \mathbf b_g\\ \mathbf b_a\\ \mathbf s_g\\ \mathbf s_a\\ \Delta\mathbf v\\ \cdots \end{bmatrix}. \]

Examples include

• drag coefficient C_D,
• solar-radiation-pressure coefficient C_R,
• ballistic coefficient BC,
• gyroscope bias,
• accelerometer bias,
• scale-factor errors,
• alignment parameters,
• receiver clock bias,
• receiver clock drift,
• thrust scale factor,
• maneuver magnitude,
• maneuver direction,
• empirical acceleration terms.

These parameters may influence either the spacecraft dynamics, the measurement model, or both.

4.3.1.5 State versus Parameter

The distinction between a state and a parameter is not always absolute.

A quantity may begin as a model parameter but later be incorporated into the estimated state.

For example, gyroscope bias may be modelled initially as

\[ \mathbf b_g=\text{constant}. \]

If the estimator must continuously estimate it, an augmented state may be formed:

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

More generally,

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

A constant parameter might satisfy

\[ \dot{\boldsymbol{\theta}}=\mathbf 0, \]

while a slowly varying parameter may be represented using

\[ \dot{\boldsymbol{\theta}} = \mathbf w_\theta. \]

This idea becomes important later in 4.3.10 Parameter and Augmented-State Estimation.

4.3.1.6 State Estimation

State estimation seeks the current or historical physical condition of the system.

Conceptually,

\[ \boxed{ \text{Measurements} \rightarrow \hat{\mathbf x}. } \]

Typical estimated states include

\[ \mathbf r, \qquad \mathbf v, \qquad \mathbf q, \qquad \boldsymbol{\omega}. \]

For a general relative-navigation problem,

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

may describe the state of one spacecraft relative to another.

State estimation therefore answers:

\[ \boxed{ \text{What is the current physical state of the system?} } \]

4.3.1.7 Parameter Estimation

Parameter estimation instead seeks unknown quantities that influence the model.

Conceptually,

\[ \boxed{ \text{Measurements} \rightarrow \hat{\boldsymbol{\theta}}. } \]

For example, a ballistic coefficient influences aerodynamic drag.

A simplified chain is

\[ BC \rightarrow \mathbf a_D \rightarrow \text{Trajectory} \rightarrow \text{Measurements}. \]

The estimation process operates in the opposite direction:

\[ \boxed{ \text{Measurements} \rightarrow \text{Observed Trajectory Behaviour} \rightarrow \widehat{BC}. } \]

This illustrates an important principle:

\[ \boxed{ \text{A quantity does not need to be measured directly in order to be estimated.} } \]

It only needs to influence the available observations in a distinguishable way.

4.3.1.8 Observation Equation

The next fundamental element is the observation or measurement equation.

A measurement generally does not equal the complete state.

Instead,

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

The function

\[ h_k(\cdot) \]

maps the unknown physical state into the quantity actually observed by the sensor.

Thus,

\[ \boxed{ \text{State} \xrightarrow{h(\cdot)} \text{Predicted Measurement}. } \]

This distinction is fundamental because estimators compare measured quantities with predicted measured quantities, rather than directly comparing a measurement with the entire state vector.

4.3.1.9 Example: Range Measurement

Suppose a tracking system observes spacecraft range.

Let spacecraft position be

\[ \mathbf r \]

and station position be

\[ \mathbf r_s. \]

Define

\[ \boldsymbol{\rho} = \mathbf r-\mathbf r_s. \]

The ideal range is

\[ \boxed{ \rho = \|\boldsymbol{\rho}\| = \|\mathbf r-\mathbf r_s\|. } \]

A real observation may be

\[ \boxed{ z_\rho = \|\mathbf r-\mathbf r_s\| + v_\rho. } \]

The measurement is one scalar quantity.

The spacecraft position contains three components.

Therefore,

\[ \boxed{ \text{One range measurement} \neq \text{complete three-dimensional position}. } \]

Additional geometry, measurements, dynamics, or prior information are required.

4.3.1.10 Example: Range-Rate Measurement

Relative velocity is

\[ \dot{\boldsymbol{\rho}} = \mathbf v-\mathbf v_s. \]

Range rate is

\[ \boxed{ \dot\rho = \frac{ \boldsymbol{\rho}^{T} \dot{\boldsymbol{\rho}} }{ \|\boldsymbol{\rho}\| }. } \]

A measurement model may therefore be

\[ z_{\dot\rho} = \dot\rho + v_{\dot\rho}. \]

Range rate primarily contains information about motion along the line of sight.

It does not directly provide the complete three-dimensional velocity vector.

Thus,

\[ \boxed{ \text{Measurement Type} \rightarrow \text{Particular Information About the State}. } \]

4.3.1.11 Example: General Relative Navigation

Consider a general proximity-navigation problem.

Suppose the relative position is

\[ \mathbf r_{\text{rel}}. \]

An optical sensor may measure only line of sight:

\[ \boxed{ \mathbf u_{\text{LOS}} = \frac{\mathbf r_{\text{rel}}} {\|\mathbf r_{\text{rel}}\|}. } \]

The measurement becomes

\[ \mathbf z_{\text{LOS}} = \frac{\mathbf r_{\text{rel}}} {\|\mathbf r_{\text{rel}}\|} + \mathbf v_{\text{LOS}}. \]

This observation provides directional information but does not independently provide range.

A separate range measurement may provide

\[ \rho = \|\mathbf r_{\text{rel}}\| + v_\rho. \]

Together,

\[ \boxed{ \text{Direction} + \text{Range} \rightarrow \text{Stronger Relative-Position Information}. } \]

This is a general estimation principle and does not depend on any particular RPO implementation.

4.3.1.12 Measurement Vector

Several observations can be collected into one vector:

\[ \boxed{ \mathbf z = \begin{bmatrix} z_1\\ z_2\\ \vdots\\ z_m \end{bmatrix}. } \]

For example,

\[ \mathbf z = \begin{bmatrix} \rho\\ \dot\rho\\ \alpha\\ \delta \end{bmatrix} \]

might contain range, range rate, and two angular observations.

The corresponding measurement function is

\[ h(\mathbf x) = \begin{bmatrix} h_\rho(\mathbf x)\\ h_{\dot\rho}(\mathbf x)\\ h_\alpha(\mathbf x)\\ h_\delta(\mathbf x) \end{bmatrix}. \]

Thus,

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

Different measurement components may have completely different units, noise levels, update rates, and sensitivities.

4.3.1.13 Predicted Measurement

Given a candidate estimate

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

the estimator predicts what the sensor should observe:

\[ \boxed{ \hat{\mathbf z} = h(\hat{\mathbf x},\hat{\boldsymbol{\theta}}). } \]

This may be called the

• predicted observation,
• computed observation,
• modelled measurement.

The fundamental comparison is therefore

\[ \boxed{ \text{Actual Observation} - \text{Predicted Observation}. } \]

This quantity is the residual.

4.3.1.14 Residual Vector

The residual is

\[ \boxed{ \mathbf r = \mathbf z - h(\hat{\mathbf x},\hat{\boldsymbol{\theta}}). } \]

For multiple observations,

\[ \mathbf r = \begin{bmatrix} r_1\\ r_2\\ \vdots\\ r_m \end{bmatrix}, \]

where

\[ r_i = z_i-\hat z_i. \]

Conceptually,

\[ \boxed{ \text{Observation} - \text{Prediction} = \text{Residual}. } \]

Residuals provide the information used by many estimators to correct the current state or parameter estimate.

4.3.1.15 Residual Is Not State Error

The residual must not be confused with estimation error.

The state estimation error is

\[ \boxed{ \mathbf e_x = \mathbf x_{\text{true}} - \hat{\mathbf x}. } \]

The residual is

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

These quantities live in different spaces.

\[ \mathbf e_x \]

belongs to state space.

\[ \mathbf r \]

belongs to measurement space.

During simulation, the truth state may be known and \mathbf e_x can therefore be calculated.

In real operation, truth is generally unavailable, so the estimator works primarily with residuals, covariance, dynamics, and measurement information.

Thus,

\[ \boxed{ \text{Small Residual} \not\Rightarrow \text{Perfect State Estimate}. } \]

Two parameters can compensate for each other while still producing small residuals.

Engineering interpretation: A small measurement residual does not prove that every state or model parameter is correct. Several uncertain quantities can compensate for one another while reproducing the observations, so residual analysis should be considered together with covariance, sensitivity, observability, and parameter-correlation analysis.

4.3.1.16 Linear Observation Model

For a linear estimation problem,

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

If

\[ \mathbf x\in\mathbb R^n \]

and

\[ \mathbf z\in\mathbb R^m, \]

then

\[ \mathbf H \in \mathbb R^{m\times n}. \]

The matrix \mathbf H is the observation, measurement, sensitivity, or design matrix depending on context.

The residual becomes

\[ \boxed{ \mathbf r = \mathbf z-\mathbf H\mathbf x. } \]

Each row of \mathbf H describes how one measurement depends on the unknown variables.

Each column describes how the observation set responds to one state component.

4.3.1.17 Measurement Sensitivity

The sensitivity of a measurement to state component x_j is represented by

\[ \boxed{ \frac{\partial h}{\partial x_j}. } \]

For a multidimensional problem,

\[ \boxed{ \mathbf H = \frac{\partial h}{\partial\mathbf x}. } \]

If

\[ \frac{\partial h}{\partial x_j} = 0, \]

the measurement has no local sensitivity to x_j.

If

\[ \left| \frac{\partial h}{\partial x_j} \right| \]

is extremely small compared with measurement uncertainty, then x_j may be only weakly estimable.

Thus,

\[ \boxed{ \text{Measurement Sensitivity} \rightarrow \text{Information About the Unknown State}. } \]

4.3.1.18 Nonlinear Observation Model

Most practical spacecraft measurement equations are nonlinear.

In general,

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

Near a nominal point \mathbf x_0,

\[ h(\mathbf x_0+\delta\mathbf x) \]

can be approximated using a first-order Taylor expansion:

\[ \boxed{ h(\mathbf x_0+\delta\mathbf x) \approx h(\mathbf x_0) + \mathbf H\delta\mathbf x } \]

where

\[ \boxed{ \mathbf H = \left. \frac{\partial h}{\partial\mathbf x} \right|_{\mathbf x_0}. } \]

The residual becomes approximately

\[ \mathbf r \approx \mathbf H\delta\mathbf x+\mathbf v. \]

This provides the foundation for iterative nonlinear estimation.

4.3.1.19 Dynamic Model

If observations are distributed over time, measurement equations alone are not sufficient.

The state must be propagated using a dynamic model:

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

In discrete form,

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

The dynamics connect states at different epochs:

\[ \boxed{ \mathbf x_0 \rightarrow \mathbf x_1 \rightarrow \mathbf x_2 \rightarrow \cdots \rightarrow \mathbf x_N. } \]

Therefore a measurement taken later in time may provide information about an earlier state.

This is fundamental to dynamic state estimation and orbit determination.

4.3.1.20 Dynamic Constraints Add Information

The dynamic model itself contributes information.

Suppose

\[ \mathbf x(t_0) \]

is known approximately.

The equations of motion determine a predicted future state:

\[ \boxed{ \mathbf x(t) = \phi(t,t_0,\mathbf x(t_0)). } \]

Thus observations at

\[ t_1,t_2,\ldots,t_N \]

are connected through the same dynamic trajectory.

This means

\[ \boxed{ \text{Measurements at Different Times} + \text{Dynamics} \rightarrow \text{Information About a Common State}. } \]

This relationship is central to Batch Dynamic State Estimation, developed in Section 4.3.3.

4.3.1.21 Observation Arc

Measurements used for dynamic estimation are often collected over an interval called the observation arc.

If observations span

\[ t_0 \]

to

\[ t_N, \]

then

\[ \boxed{ [t_0,t_N] } \]

is the observation arc.

The arc length can strongly affect estimation quality.

A short arc may provide insufficient dynamic evolution to separate

\[ \mathbf r_0, \qquad \mathbf v_0, \qquad \boldsymbol{\theta}. \]

A longer arc can improve sensitivity because the effects of different states or parameters accumulate over time.

However, longer arcs may also increase sensitivity to imperfect dynamics.

Therefore,

\[ \boxed{ \text{Longer Observation Arc} \neq \text{Automatically Better Estimation}. } \]

Model fidelity must also be considered.

4.3.1.22 Deterministic Estimation Formulation

One way to formulate estimation is deterministically.

Suppose

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

The estimator may seek the value of \mathbf x that minimizes the mismatch:

\[ \boxed{ \hat{\mathbf x} = \arg\min_{\mathbf x} \|\mathbf z-\mathbf H\mathbf x\|^2. } \]

The deterministic viewpoint focuses on

• equations,
• residuals,
• objective functions,
• optimization,
• numerical solution.

The uncertainty need not initially be interpreted probabilistically.

4.3.1.23 Stochastic Estimation Formulation

A stochastic formulation explicitly represents uncertainty using probability distributions.

For example,

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

and

\[ \boxed{ \mathbf w \sim \mathcal N(\mathbf 0,\mathbf Q). } \]

The estimator may then seek a maximum-likelihood solution:

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

or a maximum-a-posteriori solution:

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

This leads to the probabilistic interpretation of estimation.

Under Gaussian assumptions,

\[ \boxed{ \text{Weighted Least Squares} \Longleftrightarrow \text{Gaussian Maximum Likelihood}. } \]

This relationship is developed later in Section 4.3.6.

4.3.1.24 Cost Function

A candidate estimate must be evaluated according to some mathematical criterion.

This is represented using a cost function

\[ J. \]

For ordinary least squares,

\[ \boxed{ J(\mathbf x) = \mathbf r^T\mathbf r. } \]

Since

\[ \mathbf r = \mathbf z-h(\mathbf x), \]

we obtain

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

The estimate is

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

Thus estimation can be understood as a search for the state that best explains the measurements according to the chosen objective.

4.3.1.25 Why Residuals Are Squared

Suppose

\[ r_1,r_2,\ldots,r_m \]

are measurement residuals.

Simply computing

\[ \sum_{i=1}^{m}r_i \]

can allow positive and negative residuals to cancel.

Instead,

\[ \boxed{ J = \sum_{i=1}^{m}r_i^2. } \]

This ensures

\[ J\ge0 \]

and penalizes larger residual magnitudes.

In vector form,

\[ \boxed{ J = \mathbf r^T\mathbf r. } \]

This is the foundation of ordinary least-squares estimation.

4.3.1.26 Measurement Uncertainty

Different measurements generally have different accuracy.

Their uncertainty is represented statistically using

\[ \boxed{ \mathbf R = E[\mathbf v\mathbf v^T]. } \]

For independent measurements,

\[ \mathbf R = \begin{bmatrix} \sigma_1^2&0&\cdots\\ 0&\sigma_2^2&\cdots\\ \vdots&\vdots&\ddots \end{bmatrix}. \]

A highly accurate measurement has smaller variance.

A less accurate measurement has larger variance.

Therefore a statistically meaningful estimator should normally account for measurement uncertainty rather than treating all observations equally.

4.3.1.27 Measurement Weighting

A common weighting matrix is

\[ \boxed{ \mathbf W = \mathbf R^{-1}. } \]

The weighted objective becomes

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

For a scalar measurement,

\[ W_i = \frac{1}{\sigma_i^2}. \]

Therefore,

\[ \sigma_i\downarrow \quad\Rightarrow\quad W_i\uparrow. \]

Conceptually,

\[ \boxed{ \text{Lower Measurement Uncertainty} \Rightarrow \text{Greater Influence on the Estimate}. } \]

The detailed derivation belongs in Section 4.3.2.

4.3.1.28 Correlated Measurement Errors

Measurement errors are not always independent.

If two measurement errors are correlated,

\[ E[v_1v_2]\neq0. \]

Then

\[ \mathbf R = \begin{bmatrix} \sigma_1^2&R_{12}\\ R_{21}&\sigma_2^2 \end{bmatrix}. \]

The off-diagonal terms represent correlation.

The weighting must then use the complete covariance:

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

Ignoring significant measurement correlation can cause the estimator to overestimate the amount of independent information available.

4.3.1.29 Prior Information

Measurements are not always the only source of information.

Suppose an approximate state

\[ \mathbf x_0 \]

is already available with uncertainty

\[ \mathbf P_0. \]

Then the estimator may combine the prior and the new observations.

A common objective is

\[ \boxed{ \begin{aligned} J(\mathbf x) ={}& [\mathbf z-h(\mathbf x)]^T \mathbf R^{-1} [\mathbf z-h(\mathbf x)] \\ &+ (\mathbf x-\mathbf x_0)^T \mathbf P_0^{-1} (\mathbf x-\mathbf x_0). \end{aligned} } \]

The first term measures agreement with new observations.

The second measures agreement with prior knowledge.

Therefore,

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

4.3.1.30 Strength of Prior Information

Prior covariance determines how strongly the prior influences the solution.

If

\[ \mathbf P_0 \]

is small, the prior is considered precise.

Then

\[ \mathbf P_0^{-1} \]

is large and deviations from the prior are penalized strongly.

Thus,

\[ \boxed{ \text{Small Prior Covariance} \Rightarrow \text{Strong Prior}. } \]

Conversely,

\[ \boxed{ \text{Large Prior Covariance} \Rightarrow \text{Weak Prior}. } \]

Prior information should reflect genuine knowledge rather than being artificially strengthened merely to force a desired solution.

4.3.1.31 Overdetermined Problems

Suppose there are

\[ n \]

unknown quantities and

\[ m \]

measurements.

If

\[ \boxed{ m>n, } \]

the problem is overdetermined.

For example,

\[ \mathbf H \in \mathbb R^{100\times6} \]

might represent 100 observations used to estimate six unknown state components.

Because measurements contain noise, there will generally be no state that satisfies every observation exactly.

The estimator therefore seeks the best fit:

\[ \boxed{ \hat{\mathbf x} = \arg\min_{\mathbf x} \|\mathbf z-\mathbf H\mathbf x\|^2. } \]

This is the classical least-squares situation.

4.3.1.32 Exactly Determined Problems

If

\[ \boxed{ m=n, } \]

the number of equations equals the number of unknowns.

For the noise-free linear system

\[ \mathbf z=\mathbf H\mathbf x, \]

if \mathbf H is nonsingular,

\[ \boxed{ \mathbf x = \mathbf H^{-1}\mathbf z. } \]

However, equal numbers of equations and unknowns do not guarantee a reliable real-world estimate.

Noise, modelling error, weak geometry, and poor conditioning may still create substantial uncertainty.

4.3.1.33 Underdetermined Problems

If

\[ \boxed{ m

the problem is underdetermined.

Consider

\[ z=x+y. \]

If

\[ z=10, \]

then all of the following satisfy the equation:

\[ (x,y) = (1,9), (2,8), (5,5), \ldots \]

The observations are insufficient to determine a unique solution.

Additional information may come from

• more measurements,
• additional sensor types,
• dynamic relationships,
• longer observation arcs,
• prior information,
• physical constraints,
• regularization.

The fundamental principle is

\[ \boxed{ \text{An Estimator Cannot Create Information That Is Absent From the Problem}. } \]

4.3.1.34 Rank of the Estimation Problem

The number of measurements alone does not determine solvability.

For the linear problem,

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

estimating all n independent components generally requires

\[ \boxed{ \operatorname{rank}(\mathbf H)=n. } \]

If

\[ \operatorname{rank}(\mathbf H)

then at least one combination of unknowns is not independently constrained.

Thus,

\[ \boxed{ \text{Many Measurements} \neq \text{Many Independent Pieces of Information}. } \]

Repeated nearly identical observations may provide little additional information.

Engineering interpretation: Increasing the number of measurements does not necessarily increase useful information. Repeated measurements with nearly identical sensitivity or geometry may add much less information than a smaller set of independent observations.

4.3.1.35 Measurement Geometry

Measurement geometry strongly affects estimation accuracy.

Suppose several line-of-sight measurements originate from almost the same direction.

They may constrain one direction strongly while providing little information in another.

The resulting uncertainty may satisfy

\[ \sigma_x \ll \sigma_y. \]

Changing the measurement geometry can improve estimation substantially.

Geometry is important in

• GNSS,
• ground tracking,
• optical navigation,
• inter-satellite ranging,
• formation flying,
• relative navigation,
• proximity operations.

Conceptually,

\[ \boxed{ \text{Diverse Geometry} \rightarrow \text{Independent Sensitivity} \rightarrow \text{Stronger Estimation}. } \]

4.3.1.36 Observability

Observability concerns whether a dynamic state can be reconstructed from measurements over time.

Conceptually,

\[ \boxed{ \text{Dynamics} + \text{Measurements} + \text{Geometry} + \text{System Motion} \rightarrow \text{State Observability}. } \]

A state may not be measured directly but may still be observable because it influences future measurements through the dynamics.

Observability is therefore not simply a property of a sensor.

It is a property of the complete dynamic and measurement system.

4.3.1.37 Identifiability

Identifiability primarily concerns unknown parameters.

Suppose

\[ z = \theta_1+\theta_2. \]

A single observation cannot uniquely determine both parameters.

For example,

\[ 10=4+6 \]

and

\[ 10=3+7. \]

The total may be known, but the individual contributions are not.

Thus,

\[ \boxed{ \text{Parameter Identifiability} \rightarrow \text{Can Different Parameter Effects Be Distinguished?} } \]

This issue is important when estimating

• sensor bias,
• scale factor,
• drag parameters,
• ballistic coefficient,
• maneuver parameters,
• environmental parameters.

4.3.1.38 Observability versus Identifiability

The two terms are related but should not be used interchangeably.

\[ \boxed{ \text{Observability} \rightarrow \text{Dynamic State Reconstruction} } \]

while

\[ \boxed{ \text{Identifiability} \rightarrow \text{Unique Parameter Determination}. } \]

A spacecraft position and velocity may be observable while two model parameters remain highly correlated and difficult to identify individually.

When states and parameters are estimated together, both questions matter.

4.3.1.39 Parameter Correlation

Two parameters may influence observations in very similar ways.

Suppose

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

Then a change in \theta_1 may be compensated by a corresponding change in \theta_2.

The estimator may accurately determine some combination of the parameters while individual estimates remain uncertain.

Therefore,

\[ \boxed{ \text{Small Residuals} \neq \text{Individually Well-Determined Parameters}. } \]

Parameter covariance and correlation must therefore be examined.

4.3.1.40 Estimability

A quantity can be theoretically observable yet still be estimated poorly.

Suppose

\[ \frac{\partial h}{\partial x_i} \neq0 \]

but is extremely small.

Then the state technically influences the measurements, but the effect may be smaller than the measurement noise.

This motivates the engineering concept of estimability:

\[ \boxed{ \text{Can the Quantity Be Estimated With Useful Accuracy From the Available Information?} } \]

Estimability depends on

• sensitivity,
• measurement noise,
• geometry,
• observation duration,
• system excitation,
• parameter correlation,
• model fidelity.

Engineering interpretation: A state or parameter may be theoretically observable or identifiable yet still be practically difficult to estimate if its signature is weak compared with measurement noise, modelling error, or other correlated unknowns.

4.3.1.41 Batch Estimation

Batch estimation processes a collection of observations together.

Suppose measurements are collected at

\[ t_1,t_2,\ldots,t_N. \]

Define

\[ Z_N = \{ \mathbf z_1, \mathbf z_2, \ldots, \mathbf z_N \}. \]

A typical batch objective is

\[ \boxed{ \hat{\mathbf x} = \arg\min_{\mathbf x} \sum_{k=1}^{N} \mathbf r_k^T \mathbf W_k \mathbf r_k. } \]

Conceptually,

\[ \boxed{ \begin{array}{c} \mathbf z_1\\ \mathbf z_2\\ \vdots\\ \mathbf z_N \end{array} \rightarrow \text{Batch Estimator} \rightarrow \hat{\mathbf x}. } \]

Batch methods are useful for

• orbit determination,
• post-processing,
• calibration,
• trajectory reconstruction,
• parameter estimation,
• maneuver reconstruction.

4.3.1.42 Sequential Estimation

Sequential estimation processes observations as they arrive.

Suppose the estimator currently has

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

A new measurement

\[ \mathbf z_k \]

arrives.

The estimator produces

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

Conceptually,

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

Sequential estimation is particularly suitable for

• onboard navigation,
• real-time orbit determination,
• attitude estimation,
• autonomous navigation,
• relative navigation.

The detailed recursive filters belong primarily to Section 4.2.

Section 4.3.4 will instead develop the mathematical connection between sequential estimation, Bayesian updating, recursive least squares, and accumulated information.

4.3.1.43 Batch versus Sequential Processing

A useful comparison is:

Neither method is universally superior.

The appropriate choice depends on

\[ \boxed{ \text{Real-Time Need} + \text{Data Volume} + \text{Computational Resources} + \text{Estimation Objective} + \text{Required Accuracy}. } \]

4.3.1.44 Initial Guess

Nonlinear estimation frequently requires an initial guess:

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

The estimator evaluates

\[ \mathbf r^{(0)} = \mathbf z - h(\mathbf x^{(0)}) \]

and computes a correction

\[ \Delta\mathbf x^{(0)}. \]

The estimate is updated:

\[ \mathbf x^{(1)} = \mathbf x^{(0)} + \Delta\mathbf x^{(0)}. \]

In general,

\[ \boxed{ \mathbf x^{(i+1)} = \mathbf x^{(i)} + \Delta\mathbf x^{(i)}. } \]

A poor initial guess may lead to

• slow convergence,
• invalid linearization,
• convergence to an undesirable local solution,
• divergence.

Initial-state quality is therefore part of the estimation problem formulation.

4.3.1.45 Physical and Mathematical Constraints

Some estimation problems contain known constraints.

Equality constraints may be written as

\[ g(\mathbf x)=0. \]

Inequality constraints may be written as

\[ c(\mathbf x)\le0. \]

A common attitude constraint is

\[ \boxed{ \|\mathbf q\|=1. } \]

A physical parameter may also satisfy

\[ C_D>0. \]

Other examples include

• parameter bounds,
• known geometry,
• sensor mounting restrictions,
• physical conservation relationships.

Constraints can reduce the feasible solution space.

However,

\[ \boxed{ \text{A Constraint Should Represent Genuine Physical Knowledge, Not Artificially Force the Desired Solution}. } \]

4.3.1.46 Measurement Bias

A common simplified noise assumption is

\[ E[\mathbf v]=\mathbf0. \]

However, sensors may contain systematic bias.

Then

\[ \boxed{ \mathbf z = h(\mathbf x) + \mathbf b + \boldsymbol{\eta} } \]

where

• \mathbf b is bias,
• \boldsymbol{\eta} is zero-mean random noise.

If bias is ignored, the estimator may compensate by shifting other state components.

Bias may therefore be included in an augmented state:

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

However, adding the bias to the state is useful only if it can be distinguished from other errors.

4.3.1.47 Dynamic-Model Error

The estimator's dynamics are never perfectly identical to reality.

Suppose the true dynamics are

\[ \dot{\mathbf x}_{true} = f_{true}(\mathbf x,\mathbf u,\mathbf p_{true}), \]

while the estimator uses

\[ \dot{\hat{\mathbf x}} = f_{est} ( \hat{\mathbf x}, \mathbf u, \mathbf p_{nom} ). \]

Possible modelling errors include

• imperfect gravity modelling,
• drag uncertainty,
• solar radiation pressure uncertainty,
• unmodelled disturbances,
• actuator uncertainty,
• neglected flexible dynamics,
• uncertain environmental parameters.

Therefore a residual may contain contributions from several sources:

\[ \boxed{ \text{Residual} = \text{State Error Effect} + \text{Parameter Error Effect} + \text{Measurement Error} + \text{Model Error}. } \]

An estimator should therefore not interpret every residual purely as state error.

Engineering interpretation: Residual structure should not automatically be attributed to state error. Persistent residual bias can originate from a missing parameter, an incorrect force model, a sensor bias, timing error, or another mismatch between the physical system and estimator model.

4.3.1.48 The Forward Problem

The physical model naturally defines a forward problem.

Given

\[ \mathbf x \]

and

\[ \boldsymbol{\theta}, \]

we can predict observations:

\[ \boxed{ \mathbf x,\boldsymbol{\theta} \rightarrow \text{Dynamics} \rightarrow h(\mathbf x,\boldsymbol{\theta}) \rightarrow \hat{\mathbf z}. } \]

This direction is usually straightforward:

\[ \boxed{ \text{Known State} \rightarrow \text{Predicted Measurement}. } \]

4.3.1.49 The Inverse Problem

Estimation reverses the forward problem.

Measurements are known, while the state or parameters are unknown.

Thus,

\[ \boxed{ \mathbf z \rightarrow \hat{\mathbf x}, \hat{\boldsymbol{\theta}}. } \]

This is an inverse problem.

Inverse problems are more difficult because

• measurements contain noise,
• information may be incomplete,
• states may be indirectly observed,
• multiple solutions may fit the data,
• parameters may be correlated,
• models may be nonlinear,
• numerical conditioning may be poor.

Therefore,

\[ \boxed{ \text{Estimation} = \text{Solving an Inverse Problem From Imperfect Information}. } \]

4.3.1.50 General Optimization Formulation

Many estimation methods can be represented through the generic optimization problem

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

For measurements collected over N epochs,

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

where

\[ \mathbf r_k = \mathbf z_k - h_k(\mathbf x_k,\boldsymbol{\theta}). \]

With prior information,

\[ \boxed{ \begin{aligned} J ={}& (\mathbf x_0-\bar{\mathbf x}_0)^T \mathbf P_0^{-1} (\mathbf x_0-\bar{\mathbf x}_0) \\ &+ \sum_{k=1}^{N} \mathbf r_k^T \mathbf R_k^{-1} \mathbf r_k. \end{aligned} } \]

This formulation creates the mathematical bridge to least squares, MLE, MAP, and dynamic estimation.

4.3.1.51 Simple Scalar Estimation Example

Suppose an unknown scalar position x is measured three times:

\[ z_1=10.1, \qquad z_2=9.9, \qquad z_3=10.2. \]

Assume

\[ z_i=x+v_i. \]

For a candidate value of x,

\[ r_1=10.1-x, \]

\[ r_2=9.9-x, \]

\[ r_3=10.2-x. \]

The least-squares objective is

\[ \boxed{ J(x) = (10.1-x)^2 + (9.9-x)^2 + (10.2-x)^2. } \]

The estimate is

\[ \hat{x} = \arg\min_xJ(x). \]

For equal measurement uncertainty,

\[ \boxed{ \hat{x} = \frac{10.1+9.9+10.2}{3} \approx 10.067. } \]

Even this simple example already contains the fundamental estimation structure:

\[ \boxed{ \text{Unknown} + \text{Measurements} + \text{Measurement Model} + \text{Residuals} + \text{Cost Function} \rightarrow \text{Estimate}. } \]

4.3.1.52 General RPO Estimation Example

Consider a general rendezvous and proximity-operations problem.

Suppose the relative translational state is

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

Relative dynamics may be written generally as

\[ \dot{\mathbf x}_{\text{rel}} = f_{\text{rel}} ( \mathbf x_{\text{rel}}, \mathbf u,t ). \]

Suppose one sensor provides range:

\[ z_\rho = \|\mathbf r_{\text{rel}}\| + v_\rho, \]

while another provides line of sight:

\[ \mathbf z_{\text{LOS}} = \frac{ \mathbf r_{\text{rel}} }{ \|\mathbf r_{\text{rel}}\| } + \mathbf v_{\text{LOS}}. \]

Neither measurement necessarily gives complete relative position and velocity instantaneously.

The estimator combines

\[ \boxed{ \text{Relative Dynamics} + \text{Range} + \text{Line of Sight} + \text{Measurement Uncertainty} } \]

to estimate

\[ \boxed{ \hat{\mathbf r}_{\text{rel}}, \qquad \hat{\mathbf v}_{\text{rel}}. } \]

If range bias is also estimated,

\[ \mathbf x_a = \begin{bmatrix} \mathbf r_{\text{rel}}\\ \mathbf v_{\text{rel}}\\ b_\rho \end{bmatrix}, \]

the measurements must contain enough information to distinguish

\[ b_\rho \]

from actual relative-position error.

This illustrates why state definition, geometry, dynamics, measurement models, and identifiability must be considered together.

4.3.1.53 Common Problem-Formulation Errors

Estimation failures often originate before the estimator begins operating.

Common formulation problems include

• estimating too many quantities,
• omitting necessary parameters,
• incorrect observation equations,
• wrong coordinate-frame transformations,
• inconsistent units,
• wrong timestamps,
• poor initial conditions,
• insufficient observation geometry,
• ignored measurement correlation,
• unmodelled sensor bias,
• unrealistic covariance assumptions,
• rank-deficient design matrices,
• poor observation-arc selection,
• incorrect dynamic models,
• weakly identifiable parameters.

Thus,

\[ \boxed{ \text{Poor Problem Formulation} + \text{Advanced Estimator} \neq \text{Reliable Navigation Solution}. } \]

Changing from one filter to another does not fix missing physical information.

Engineering interpretation: Changing the estimator algorithm does not repair missing information, inconsistent frames, or an incorrect observation model. Problem formulation should therefore be verified before filter tuning or algorithm comparison.

4.3.1.54 Information Content of a Measurement

The deeper interpretation of estimation is information extraction.

A measurement provides information about x_i only if changing x_i changes the expected measurement.

Mathematically,

\[ \boxed{ \frac{\partial h}{\partial x_i} } \]

describes that local sensitivity.

If

\[ \frac{\partial h}{\partial x_i}=0, \]

then the measurement carries no local information about x_i.

If

\[ \left| \frac{\partial h}{\partial x_i} \right| \]

is very small compared with measurement noise, the information is weak.

If two state variables produce almost identical measurement responses, they are difficult to distinguish.

Therefore,

\[ \boxed{ \text{Estimation Quality} \text{ depends on } \text{Measurement Sensitivity} + \text{Geometry} + \text{Uncertainty} + \text{Dynamics}. } \]

4.3.1.55 Estimation Problem Formulation Workflow

A disciplined estimation workflow is:

1. define the estimation objective,
2. identify unknown states,
3. identify unknown parameters,
4. define units, frames, and reference epochs,
5. define the dynamic model,
6. define known inputs,
7. define each observation model,
8. define sensor and measurement uncertainty,
9. define process/model uncertainty where required,
10. define prior information,
11. identify physical constraints,
12. generate predicted observations,
13. define residuals,
14. determine state sensitivities,
15. determine parameter sensitivities,
16. examine measurement geometry,
17. examine rank,
18. assess observability,
19. assess parameter identifiability,
20. assess parameter correlation,
21. determine whether batch or sequential processing is appropriate,
22. define the estimation objective or probability model,
23. choose the numerical estimation algorithm,
24. define convergence criteria,
25. assess residuals, covariance, numerical conditioning, and physical plausibility.

Conceptually,

\[ \boxed{ \begin{array}{c} \text{Define Unknowns}\\ \downarrow\\ \text{Define Dynamics}\\ \downarrow\\ \text{Define Measurement Models}\\ \downarrow\\ \text{Define Uncertainties and Priors}\\ \downarrow\\ \text{Form Predicted Measurements}\\ \downarrow\\ \text{Compute Residuals}\\ \downarrow\\ \text{Evaluate Sensitivity and Information Content}\\ \downarrow\\ \text{Check Observability / Identifiability}\\ \downarrow\\ \text{Select Estimator}\\ \downarrow\\ \text{Solve and Assess} \end{array} } \]

This makes the estimator design traceable rather than simply selecting an algorithm by convention.

4.3.1.56 Complete Estimation Architecture

The complete architecture can be summarized as

\[ \boxed{ \begin{array}{c} \text{Define Unknown State } \mathbf x \\ + \\ \text{Define Unknown Parameters } \boldsymbol{\theta} \\ \downarrow\\ \text{Dynamic Model} \\ \dot{\mathbf x}=f(\mathbf x,\mathbf u,\boldsymbol{\theta},t) \\ \downarrow\\ \text{Predicted State} \\ \downarrow\\ \text{Measurement Model} \\ \hat{\mathbf z}=h(\hat{\mathbf x},\hat{\boldsymbol{\theta}}) \\ \downarrow\\ \text{Compare with Actual Observation } \mathbf z \\ \downarrow\\ \text{Residual} \\ \mathbf r=\mathbf z-\hat{\mathbf z} \\ \downarrow\\ \text{Weight Using Measurement / Prior Uncertainty} \\ \downarrow\\ \text{Evaluate Sensitivities and Information Content} \\ \downarrow\\ \text{Solve Estimation Problem} \\ \downarrow\\ \hat{\mathbf x},\hat{\boldsymbol{\theta}},\mathbf P \end{array} } \]

This is the general structure from which the later estimation algorithms are developed.

4.3.1.57 Relationship to the Remaining Estimation Chapters

The problem formulation developed here leads directly to the remaining sections.

For a linear measurement model,

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

the next question is:

How do we solve for the unknown state when many noisy measurements are available?

This leads to

\[ \boxed{ \text{4.3.2 Linear and Weighted Least Squares}. } \]

When observations occur across a dynamic trajectory,

\[ \boxed{ \text{4.3.3 Batch Dynamic State Estimation}. } \]

When measurements arrive sequentially,

\[ \boxed{ \text{4.3.4 Recursive Estimation}. } \]

For nonlinear measurement models,

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

For a probabilistic interpretation,

\[ \boxed{ \text{4.3.6 Maximum Likelihood and MAP Estimation}. } \]

Using future observations to improve earlier estimates leads to

\[ \boxed{ \text{4.3.7 Smoothing and Retrodiction}. } \]

Sensitivity propagation leads to

\[ \boxed{ \text{4.3.8 State Transition and Sensitivity Matrices}. } \]

Uncertainty propagation leads to

\[ \boxed{ \text{4.3.9 Covariance Analysis and Error Propagation}. } \]

Estimating physical parameters leads to

\[ \boxed{ \text{4.3.10 Parameter and Augmented-State Estimation}. } \]

Finally, applying these ideas specifically to spacecraft orbital states leads to

\[ \boxed{ \text{4.3.11 onward: Orbit Determination}. } \]

4.3.1.58 Key Equations

Dynamic model

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

or

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

General observation model

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

Predicted observation

\[ \boxed{ \hat{\mathbf z}_k = h_k ( \hat{\mathbf x}_k, \hat{\boldsymbol{\theta}} ) } \]

Residual

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

Linear observation model

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

Linearized nonlinear model

\[ \boxed{ h(\mathbf x+\delta\mathbf x) \approx h(\mathbf x) + \mathbf H\delta\mathbf x } \]

with

\[ \boxed{ \mathbf H = \frac{\partial h}{\partial\mathbf x} } \]

Measurement covariance

\[ \boxed{ \mathbf R = E[\mathbf v\mathbf v^T] } \]

Ordinary residual cost

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

Weighted residual cost

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

Cost with prior information

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

General estimation problem

\[ \boxed{ ( \hat{\mathbf x}, \hat{\boldsymbol{\theta}} ) = \arg\min_{\mathbf x,\boldsymbol{\theta}} J(\mathbf x,\boldsymbol{\theta}) } \]

4.3.1.59 Key Takeaways

Estimation begins before an estimator algorithm is selected.

The first requirement is to define

\[ \boxed{ \text{Unknown States} + \text{Unknown Parameters} + \text{Dynamics} + \text{Measurements} + \text{Uncertainty} + \text{Prior Information}. } \]

The fundamental measurement relationship is

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

The estimator does not normally compare measurements directly with the complete state.

Instead, it predicts the expected observations

\[ \hat{\mathbf z} = h(\hat{\mathbf x},\hat{\boldsymbol{\theta}}) \]

and forms residuals

\[ \boxed{ \mathbf r = \mathbf z-\hat{\mathbf z}. } \]

The residual is not the same as state estimation error.

\[ \boxed{ \mathbf r \neq \mathbf x_{\text{true}}-\hat{\mathbf x}. } \]

Measurement uncertainty is represented through quantities such as

\[ \mathbf R, \]

while prior uncertainty may be represented by

\[ \mathbf P_0. \]

A meaningful estimation problem must also contain sufficient independent information.

Therefore,

\[ \boxed{ \text{Number of Measurements} \neq \text{Amount of Independent Information}. } \]

Rank, sensitivity, geometry, observability, identifiability, parameter correlation, and estimability determine whether the unknown quantities can actually be recovered.

The distinction between batch and sequential processing is also fundamental:

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

whereas

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

The central engineering principle is

\[ \boxed{ \text{A Good Estimator Cannot Rescue a Poorly Formulated Estimation Problem}. } \]

A reliable estimation solution begins by defining the physics, unknowns, observations, frames, uncertainty, and information content correctly.

4.3.1.60 What Comes Next

Once the estimation problem has been formulated, the next question is:

\[ \boxed{ \text{Given a set of noisy observations, how do we mathematically solve for the unknown state?} } \]

For the linear model

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

the natural starting point is least squares.

The next section therefore develops 4.3.2 Linear and Weighted Least Squares, including

• linear observation equations,
• design matrices,
• residual minimization,
• normal equations,
• ordinary least squares,
• weighted least squares,
• measurement covariance,
• a priori constraints,
• covariance of the estimated solution,
• geometric interpretation,
• rank deficiency,
• and numerically robust solution methods.

The progression is therefore

\[ \boxed{ \text{4.3.1 Formulate the Estimation Problem} \rightarrow \text{4.3.2 Solve the Linear Estimation Problem}. } \]

That gives 4.3.1 the same structural style as your 4.2.30 chapter: numbered subsections, short conceptual blocks, boxed engineering principles, equations after the physical explanation, a general RPO example rather than project-specific material, a complete workflow, key equations, Key Takeaways, and an explicit transition to the next section.

Next: 4.3.2 Linear and Weighted Least Squares

The estimation problem is now defined in terms of unknown states and parameters, dynamics, observation models, residuals, uncertainty, prior information, geometry, observability, identifiability, and processing architecture.

The next section develops the first major solution method: linear and weighted least squares. It begins with the linear observation model and develops residual minimization, the normal equations, measurement weighting, a priori information, estimate covariance, rank deficiency, geometric interpretation, and numerically robust solution methods.