Parameter and Augmented-State Estimation

4.3.10 Parameter and Augmented-State Estimation

State estimation often begins by estimating only the dynamic state of a spacecraft, such as position, velocity, attitude, and angular rate. In many practical navigation and GNC problems, however, important quantities that influence the dynamics or measurements are not known exactly.

These quantities may include:

• static physical parameters,
• slowly varying dynamic parameters,
• sensor biases,
• calibration coefficients,
• ballistic coefficient,
• drag-related parameters,
• scale factors,
• alignment errors,
• timing offsets,
• environmental parameters,
• nuisance parameters.

Instead of treating these quantities as perfectly known constants, they can be incorporated into an augmented state vector and estimated together with the spacecraft state.

If the original dynamic state is

\[ \mathbf x \]

and the uncertain parameter vector is

\[ \boldsymbol\theta, \]

the augmented state becomes

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

The estimator then solves simultaneously for the physical state and for the uncertain parameters that influence the dynamics or measurements.

The central concept is therefore

\[ \boxed{ \text{Dynamic State} + \text{Unknown Parameters} \rightarrow \text{Augmented-State Estimation} } \]

This is widely used in spacecraft navigation, orbit determination, attitude estimation, sensor calibration, atmospheric-drag estimation, autonomous navigation, relative navigation, system identification, and post-flight trajectory reconstruction.

4.3.10.1 Why Estimate Parameters?

Suppose the spacecraft dynamics are written as

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

and the measurements are

\[ \mathbf z_k = h(\mathbf x_k,\boldsymbol\theta,t_k) + \mathbf v_k. \]

If the parameter vector

\[ \boldsymbol\theta \]

is assumed known but is actually incorrect, the estimator may attribute the resulting mismatch to errors in the dynamic state.

For example, an incorrect drag coefficient may appear as a position or velocity error.

Similarly, a gyro bias may appear as a persistent attitude-rate error.

Therefore,

\[ \boxed{ \text{Incorrect Fixed Parameter} \rightarrow \text{Potential State-Estimation Bias} } \]

Estimating the uncertain parameter can reduce this systematic error.

4.3.10.2 State versus Parameter

A state is normally a quantity required to describe the instantaneous dynamical condition of the system.

Examples include

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

A parameter is usually a physical or calibration quantity that influences the system but may not have fast dynamics of its own.

Examples include

\[ C_D,\quad B,\quad b_g,\quad s_a,\quad \Delta t. \]

However, in estimation theory the distinction is flexible.

Once a parameter is included in the estimator, it becomes part of the mathematical state vector.

Thus,

\[ \boxed{ \text{Estimated Parameter} \rightarrow \text{Augmented State Component} } \]

4.3.10.3 General Augmented State

Suppose the navigation state is

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

and the uncertain parameters are

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

Then the augmented state is

\[ \boxed{ \mathbf x_a = \begin{bmatrix} \mathbf r\\ \mathbf v\\ \theta_1\\ \theta_2\\ \vdots\\ \theta_p \end{bmatrix} } \]

with dimension

\[ n_a=n+p. \]

The estimator must propagate not only the state but also the uncertainty and correlation associated with the parameters.

4.3.10.4 Static Parameter Model

A truly constant parameter can be modeled as

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

or in discrete time,

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

This states that the best dynamic model for the parameter is constant.

Examples may include approximately constant:

• geometric calibration constants,
• fixed scale factors,
• constant alignment angles,
• fixed mass parameters over a short arc,
• constant sensor offsets over a limited interval.

The parameter can still be estimated because measurements provide information about it even though its nominal time derivative is zero.

4.3.10.5 Slowly Varying Parameter Model

Many quantities are not perfectly constant.

A simple slowly varying model is

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

where

\[ w_{\theta,k}\sim\mathcal N(0,Q_\theta). \]

This is a random-walk model.

In continuous time,

\[ \boxed{ \dot{\theta} = w_\theta(t). } \]

The corresponding process noise allows the estimator to adapt gradually as the real parameter changes.

4.3.10.6 Why Process Noise Is Added to Parameters

If

\[ Q_\theta=0, \]

the estimator assumes that the parameter is perfectly constant.

If the real parameter changes, the estimator may no longer track it correctly.

Adding

\[ Q_\theta>0 \]

allows slow evolution.

However, too much process noise can make the estimated parameter fluctuate excessively.

Thus,

\[ \boxed{ Q_\theta \text{ controls how rapidly the estimator allows the parameter to change.} } \]

This is one of the most important tuning decisions in augmented-state estimation.

4.3.10.7 Random-Walk Parameter Model

For a scalar parameter,

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

with

\[ w_k\sim\mathcal N(0,q_\theta). \]

The variance grows over time when no measurements constrain the parameter.

This model is useful for quantities expected to drift slowly without a preferred nominal value.

Typical examples include:

• slowly drifting sensor bias,
• uncertain drag-related coefficient,
• calibration drift,
• environmental parameters.

4.3.10.8 First-Order Gauss-Markov Parameter Model

A parameter that tends to remain near a nominal value can be represented using a first-order Gauss-Markov process:

\[ \boxed{ \dot{\theta} = -\frac{1}{\tau} (\theta-\bar{\theta}) + w_\theta. } \]

Here,

\[ \tau \]

is the correlation time, and

\[ \bar{\theta} \]

is the long-term nominal value.

In discrete form,

\[ \boxed{ \theta_{k+1} = \bar{\theta} + e^{-\Delta t/\tau} (\theta_k-\bar{\theta}) + w_k. } \]

This model is often more physically realistic than a pure random walk when the parameter exhibits bounded temporal correlation.

4.3.10.9 Augmented Dynamics

For

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

the augmented dynamics become

\[ \boxed{ \dot{\mathbf x}_a = \begin{bmatrix} f(\mathbf x,\boldsymbol\theta,t)\\ g(\boldsymbol\theta,t) \end{bmatrix} } \]

where

\[ g(\boldsymbol\theta,t) \]

describes the chosen parameter dynamics.

For static parameters,

\[ g=0. \]

For random-walk parameters,

\[ g=w_\theta. \]

4.3.10.10 Augmented State Jacobian

The linearized augmented system matrix becomes

\[ \boxed{ \mathbf A_a = \frac{\partial f_a}{\partial \mathbf x_a} } \]

which often has block structure

\[ \boxed{ \mathbf A_a = \begin{bmatrix} \frac{\partial f}{\partial \mathbf x} & \frac{\partial f}{\partial\boldsymbol\theta} \\ \mathbf 0 & \frac{\partial g}{\partial\boldsymbol\theta} \end{bmatrix}. } \]

The block

\[ \frac{\partial f}{\partial\boldsymbol\theta} \]

is especially important because it describes how parameter errors affect state dynamics.

4.3.10.11 Augmented Measurement Model

The measurement model becomes

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

The augmented measurement Jacobian is

\[ \boxed{ \mathbf H_a = \begin{bmatrix} \frac{\partial h}{\partial\mathbf x} & \frac{\partial h}{\partial\boldsymbol\theta} \end{bmatrix}. } \]

A parameter can be estimated only if it influences the measurements directly or indirectly through the dynamics.

4.3.10.12 Direct and Indirect Parameter Observability

A parameter may enter the measurement model directly:

\[ z=h(\mathbf x,\theta). \]

Then

\[ \frac{\partial h}{\partial\theta}\neq0. \]

This provides direct measurement sensitivity.

Alternatively, a parameter may influence only the dynamics:

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

while

\[ h=h(\mathbf x). \]

Then the parameter becomes observable only through its effect on the propagated state.

Thus,

\[ \boxed{ \text{Parameter Observability} = \text{Direct Measurement Sensitivity} + \text{Dynamic Sensitivity} } \]

4.3.10.13 Parameter Sensitivity

The sensitivity of the state with respect to a parameter is

\[ \boxed{ \mathbf S_\theta = \frac{\partial\mathbf x} {\partial\theta}. } \]

If this sensitivity is nearly zero, changes in the parameter barely affect the state.

The parameter will then be difficult to estimate.

If

\[ \left\| \frac{\partial\mathbf x} {\partial\theta} \right\| \]

is significant, the trajectory carries information about that parameter.

4.3.10.14 Sensitivity Dynamics

Starting with

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

differentiate with respect to \theta:

\[ \frac{d}{dt} \left( \frac{\partial\mathbf x}{\partial\theta} \right) = \frac{\partial f}{\partial\mathbf x} \frac{\partial\mathbf x}{\partial\theta} + \frac{\partial f}{\partial\theta}. \]

Therefore,

\[ \boxed{ \dot{\mathbf S}_\theta = \mathbf A \mathbf S_\theta + \mathbf B_\theta } \]

where

\[ \mathbf B_\theta = \frac{\partial f}{\partial\theta}. \]

This equation is fundamental to parameter estimation and system identification.

4.3.10.15 Parameter Covariance

For the augmented state,

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

the covariance becomes

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

Here,

\[ \mathbf P_{xx} \]

represents state uncertainty,

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

represents parameter uncertainty, and

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

represents state-parameter correlation.

4.3.10.16 Why State-Parameter Correlation Matters

Suppose an error in a parameter produces a trajectory error similar to an error in the initial state.

Then the estimator may be unable to distinguish between them immediately.

The resulting covariance contains strong cross-correlation:

\[ \boxed{ \mathbf P_{x\theta}\neq0. } \]

This correlation is physically meaningful.

It indicates that uncertainty in the state and uncertainty in the parameter are not independent.

Ignoring this correlation can lead to inconsistent uncertainty estimates.

4.3.10.17 Example: Position versus Dynamic Parameter Correlation

Suppose an uncertain parameter changes acceleration.

A small acceleration error accumulated over time produces velocity error:

\[ \delta v \approx \delta a\,\Delta t \]

and then position error:

\[ \delta r \approx \frac12\delta a\,\Delta t^2. \]

Therefore the parameter becomes correlated with both velocity and position.

Thus,

\[ \boxed{ \text{Parameter Error} \rightarrow \text{Acceleration Error} \rightarrow \text{Velocity Error} \rightarrow \text{Position Error} } \]

This is why long observation arcs can sometimes improve dynamic-parameter estimation.

4.3.10.18 Sensor Bias as an Augmented State

A common application is sensor-bias estimation.

Suppose a gyro measurement is

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

Here,

\[ \mathbf b_g \]

is gyro bias and

\[ \mathbf n_g \]

is measurement noise.

Augment the state:

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

The estimator then attempts to separate true angular rate from sensor bias.

4.3.10.19 Gyro Bias Dynamics

A simple model is

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

This random-walk model allows the bias to drift.

Alternatively,

\[ \boxed{ \dot{\mathbf b}_g = -\frac{1}{\tau_b} \mathbf b_g + \mathbf w_b } \]

may be used for a Gauss-Markov bias model.

The correct model depends on the sensor behavior and time scale of interest.

4.3.10.20 Accelerometer Bias Estimation

An accelerometer measurement may be written

\[ \boxed{ \mathbf a_m = \mathbf a + \mathbf b_a + \mathbf n_a. } \]

The augmented state may include

\[ \mathbf b_a. \]

Because accelerometer bias integrates into velocity and position error,

\[ \boxed{ \text{Accelerometer Bias} \rightarrow \text{Velocity Drift} \rightarrow \text{Position Drift} } \]

its estimation can be very important in inertial-navigation systems.

4.3.10.21 Sensor Scale Factor

A measurement may contain scale-factor error:

\[ \boxed{ z_m = (1+s)z_{\rm true} + b + n. } \]

Here,

\[ s \]

is scale-factor error.

If s is uncertain, it may be added to the augmented state.

However, estimating scale factor requires sufficient excitation in the measured quantity.

If the true signal barely changes, scale factor may be difficult to distinguish from bias.

4.3.10.22 Misalignment Parameters

Sensor axes may not be perfectly aligned with the spacecraft body frame.

A small-angle misalignment can be represented as

\[ \boldsymbol\epsilon = \begin{bmatrix} \epsilon_x\\ \epsilon_y\\ \epsilon_z \end{bmatrix}. \]

For small angles,

\[ \boxed{ \mathbf C_{\text{actual}} \approx (\mathbf I-[\boldsymbol\epsilon\times]) \mathbf C_{\text{nominal}}. } \]

These alignment parameters may be calibrated on the ground or estimated in flight if the available motion and measurements provide sufficient information.

4.3.10.23 Calibration Parameters

Calibration parameters may include:

• bias,
• scale factor,
• non-orthogonality,
• alignment,
• timing offset,
• sensor lever arm,
• temperature coefficient,
• range offset,
• camera focal-length correction.

A general calibrated measurement may be expressed as

\[ \boxed{ \mathbf z_m = h( \mathbf x, \boldsymbol\theta_{\rm cal} ) + \mathbf v. } \]

The parameter vector

\[ \boldsymbol\theta_{\rm cal} \]

can be augmented when in-flight calibration is required.

4.3.10.24 Calibration versus Navigation State

Calibration and navigation states can become strongly correlated.

For example:

• camera alignment error may appear as attitude error,
• range bias may appear as relative-position error,
• clock error may appear as range error,
• accelerometer bias may appear as velocity error.

Therefore calibration estimation must be designed with observability in mind.

\[ \boxed{ \text{Successful Calibration} \neq \text{Simply Adding More States} } \]

The measurements must contain enough independent information to separate the calibration parameter from the physical state.

4.3.10.25 Ballistic Coefficient

A common orbit-determination parameter is the ballistic coefficient.

One convention is

\[ \boxed{ B = \frac{m}{C_DA} } \]

where

• m is spacecraft mass,
• C_D is drag coefficient,
• A is effective cross-sectional area.

Another frequently used parameter is its inverse:

\[ \boxed{ \beta = \frac{C_DA}{m}. } \]

The exact convention must always be stated because both definitions occur in the literature.

4.3.10.26 Drag Acceleration and Ballistic Coefficient

A simplified drag acceleration is

\[ \boxed{ \mathbf a_D = -\frac12 \rho \frac{C_DA}{m} v_{\rm rel}^2 \hat{\mathbf v}_{\rm rel}. } \]

Using

\[ B=\frac{m}{C_DA}, \]

this becomes

\[ \boxed{ \mathbf a_D = -\frac12 \rho \frac{1}{B} v_{\rm rel}^2 \hat{\mathbf v}_{\rm rel}. } \]

Thus uncertainty in ballistic coefficient directly affects predicted drag acceleration.

4.3.10.27 Why Estimate Ballistic Coefficient?

The effective ballistic coefficient may be uncertain because of:

• uncertainty in C_D,
• changing projected area,
• attitude changes,
• uncertain atmospheric density,
• imperfect spacecraft geometry,
• accommodation effects,
• mass uncertainty.

Therefore a fixed nominal ballistic coefficient may not represent the actual long-term drag behavior.

An estimator may augment

\[ B \]

or

\[ C_D \]

as a state.

4.3.10.28 Ballistic Coefficient as a Static State

Over a short arc, one may assume

\[ \boxed{ \dot B=0. } \]

Then

\[ B \]

is treated as constant but unknown.

The estimator uses accumulated trajectory information to determine which value best explains the observed orbital motion.

4.3.10.29 Ballistic Coefficient as a Slowly Varying State

For longer arcs,

\[ \boxed{ B_{k+1} = B_k+w_{B,k} } \]

may be more appropriate.

This allows the effective coefficient to adapt slowly.

However, increasing

\[ Q_B \]

too much can cause the filter to use ballistic coefficient as a generic correction term for model errors that actually originate elsewhere.

4.3.10.30 Ballistic Coefficient versus Atmospheric Density

Drag depends approximately on

\[ \rho\frac{C_DA}{m}. \]

Therefore errors in atmospheric density and errors in ballistic coefficient can produce similar trajectory effects.

This can create strong correlation:

\[ \boxed{ \text{Atmospheric Density Error} \leftrightarrow \text{Ballistic-Coefficient Error} } \]

Consequently, a fitted ballistic coefficient can sometimes absorb deficiencies in the atmospheric model.

This must be considered when interpreting the physical meaning of the estimate.

4.3.10.31 Effective Parameters

Sometimes the estimated parameter should be interpreted as an effective parameter rather than an exact physical constant.

For example, an estimated drag coefficient may compensate simultaneously for:

• atmospheric-model error,
• attitude-dependent projected area,
• true drag-coefficient uncertainty,
• density-model deficiencies.

Thus,

\[ \boxed{ \text{Estimated Effective Parameter} \neq \text{Necessarily a Pure Physical Constant} } \]

This distinction is important in orbit determination and model validation.

4.3.10.32 Mass as an Estimated Parameter

Spacecraft mass influences dynamics whenever force is converted to acceleration:

\[ \boxed{ \mathbf a = \frac{\mathbf F}{m}. } \]

Mass may be uncertain because of:

• propellant consumption,
• deployment events,
• payload changes,
• uncertain initial mass.

If necessary, mass can be included as a parameter state.

However, estimating mass requires motion that makes its effect distinguishable from force-model uncertainties.

4.3.10.33 Center-of-Mass Parameters

For rotational dynamics,

\[ \boldsymbol\tau = \mathbf r\times\mathbf F. \]

Uncertainty in center-of-mass position affects the lever arm

\[ \mathbf r. \]

Therefore uncertain center-of-mass offsets may influence attitude dynamics and actuator calibration.

Under suitable excitation, selected center-of-mass parameters may be estimated as augmented states.

4.3.10.34 Inertia Parameters

Rigid-body attitude dynamics depend on

\[ \boxed{ \mathbf I\dot{\boldsymbol\omega} + \boldsymbol\omega\times (\mathbf I\boldsymbol\omega) = \boldsymbol\tau. } \]

If inertia is poorly known, selected inertia coefficients may be estimated.

However, all inertia terms may not be independently observable from available maneuvers.

Parameterization should respect physical constraints such as symmetry:

\[ \boxed{ \mathbf I=\mathbf I^T. } \]

4.3.10.35 Actuator Calibration Parameters

Actuator models may contain uncertain parameters such as:

• thrust scale factor,
• torque scale factor,
• misalignment,
• deadband,
• delay,
• minimum impulse,
• wheel torque calibration.

A commanded actuator input

\[ u_c \]

may produce actual output

\[ \boxed{ u_{\rm actual} = k_u u_c+b_u. } \]

The parameters

\[ k_u,\quad b_u \]

may be estimated if the resulting spacecraft response is sufficiently measured.

4.3.10.36 Timing Offset Estimation

A measurement may be associated with an uncertain timestamp

\[ t+\Delta t. \]

Then

\[ \boxed{ z_k = h[ \mathbf x(t_k+\Delta t) ] + v_k. } \]

A small timing error can produce a significant apparent state error for rapidly changing measurements.

The timing offset

\[ \Delta t \]

can therefore be treated as an estimated parameter in precision navigation.

4.3.10.37 Clock Bias and Clock Drift

For ranging or timing-sensitive navigation, clock states may be augmented.

A simple clock model is

\[ \boxed{ \begin{aligned} b_{k+1} &= b_k+\dot b_k\Delta t+w_b\\ \dot b_{k+1} &= \dot b_k+w_{\dot b}. \end{aligned} } \]

where

\[ b \]

is clock bias and

\[ \dot b \]

is clock drift.

These states become correlated with position when measurement geometry depends on signal time of flight.

4.3.10.38 Nuisance Parameters

A nuisance parameter affects the measurements or dynamics but is not itself the main quantity of interest.

Examples include:

• measurement bias,
• scale factor,
• environmental coefficient,
• unknown background offset,
• calibration constant,
• timing delay,
• unknown noise parameter.

Suppose

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

where

\[ \mathbf x \]

is the desired state and

\[ \boldsymbol\eta \]

is a nuisance parameter.

Ignoring \boldsymbol\eta can bias the desired estimate.

4.3.10.39 Estimating Nuisance Parameters

One option is to augment them:

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

This allows uncertainty in the nuisance quantity to propagate properly into the state estimate.

However, adding nuisance states increases:

• estimator dimension,
• covariance size,
• computational cost,
• correlation,
• potential observability problems.

Therefore augmentation should be justified by estimation performance, not done automatically.

4.3.10.40 Profiling Nuisance Parameters

In batch maximum-likelihood estimation, nuisance parameters can sometimes be handled by profile likelihood:

\[ \boxed{ L_p(\mathbf x) = \max_{\boldsymbol\eta} L( \mathbf x, \boldsymbol\eta ). } \]

For every candidate state, the nuisance parameters are optimized.

This avoids reporting them as primary outputs while still accounting for their effect on the likelihood.

4.3.10.41 Marginalizing Nuisance Parameters

In Bayesian estimation, nuisance parameters can be integrated out:

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

This is marginalization.

Therefore,

\[ \boxed{ \text{Augmentation}, \quad \text{Profiling}, \quad \text{Marginalization} } \]

are different approaches for handling nuisance variables.

4.3.10.42 Parameter Identifiability

Adding a parameter to the estimator does not guarantee that it can be estimated.

A parameter is identifiable only if the observations provide enough independent information about it.

If

\[ \theta_1 \]

and

\[ \theta_2 \]

produce nearly identical measurement effects,

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

they may be difficult to separate.

This produces strong parameter correlation.

4.3.10.43 Observability of Augmented States

For a linearized augmented system,

\[ \delta\mathbf x_{a,k+1} = \mathbf F_{a,k} \delta\mathbf x_{a,k}, \]

\[ \delta\mathbf z_k = \mathbf H_{a,k} \delta\mathbf x_{a,k}, \]

the observability of the augmented system must be considered.

A parameter state may be theoretically included but practically unobservable.

Thus,

\[ \boxed{ \text{State Augmentation} \neq \text{Guaranteed Observability} } \]

4.3.10.44 Excitation and Parameter Estimation

Many parameters become observable only when the system is sufficiently excited.

For example:

• scale factor requires variation in the measured signal,
• inertia estimation requires rotational excitation,
• thrust calibration requires controlled actuation,
• sensor misalignment requires changing orientation,
• timing offset requires dynamic variation.

Therefore,

\[ \boxed{ \text{Parameter Estimation Quality} \propto \text{Informative System Excitation} } \]

in a qualitative sense.

4.3.10.45 Persistent Excitation

A parameter may remain weakly estimated if the spacecraft repeatedly experiences nearly identical conditions.

Rich motion produces greater variation in measurement sensitivities.

This is related to the concept of persistent excitation.

A practical parameter-estimation campaign should therefore consider not only estimator mathematics but also whether the trajectory or maneuver profile generates sufficient information.

4.3.10.46 Correlated Parameters

Suppose two estimated parameters have covariance block

\[ \mathbf P_{\theta\theta} = \begin{bmatrix} \sigma_{\theta_1}^2 & P_{12}\\ P_{21} & \sigma_{\theta_2}^2 \end{bmatrix}. \]

The correlation coefficient is

\[ \boxed{ \rho_{12} = \frac{P_{12}} {\sigma_{\theta_1}\sigma_{\theta_2}}. } \]

If

\[ |\rho_{12}|\approx1, \]

the parameters are strongly coupled.

This often indicates weak independent identifiability.

4.3.10.47 Parameter Correlation versus Parameter Error

Strong correlation does not necessarily mean that the estimates are incorrect.

It means the uncertainties are coupled.

For example, one parameter may increase while another decreases with little change in predicted measurements.

This corresponds geometrically to an elongated uncertainty region in parameter space.

4.3.10.48 Information Matrix for Parameters

For Gaussian measurements,

\[ \mathbf z=h(\boldsymbol\theta)+\mathbf v, \]

with

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

the local parameter information matrix is

\[ \boxed{ \mathbf I_\theta = \mathbf H_\theta^T \mathbf R^{-1} \mathbf H_\theta } \]

where

\[ \mathbf H_\theta = \frac{\partial h}{\partial\boldsymbol\theta}. \]

Poorly conditioned

\[ \mathbf I_\theta \]

indicates weak parameter identifiability.

4.3.10.49 Parameter Uncertainty from Information

Under local Gaussian assumptions,

\[ \boxed{ \mathbf P_\theta \approx \mathbf I_\theta^{-1}. } \]

Large eigenvalues of

\[ \mathbf P_\theta \]

correspond to poorly determined parameter combinations.

This provides a direct connection between sensitivity, information, and parameter covariance.

4.3.10.50 Prior Information for Parameters

Some parameters may already be known approximately from ground calibration or previous analysis.

Represent this by

\[ \boxed{ \boldsymbol\theta \sim \mathcal N( \bar{\boldsymbol\theta}, \mathbf P_{\theta,0} ). } \]

This prior can stabilize estimation of weakly observed parameters.

However,

\[ \boxed{ \text{Prior-Constrained Parameter} \neq \text{Strongly Observable Parameter} } \]

The posterior may be precise primarily because of the prior, not because of new measurement information.

4.3.10.51 Parameter Covariance Initialization

The initial covariance

\[ \mathbf P_{\theta,0} \]

should reflect actual prior knowledge.

If it is too small, the estimator may effectively freeze the parameter.

If it is too large, the parameter may initially absorb unrelated measurement residuals.

Thus,

\[ \boxed{ \mathbf P_{\theta,0} = \text{Quantitative Statement of Prior Parameter Uncertainty} } \]

4.3.10.52 Process Noise Tuning for Parameter States

For a random-walk parameter,

\[ \theta_{k+1} = \theta_k+w_k, \]

the tuning quantity is

\[ Q_\theta. \]

Too small:

\[ Q_\theta\downarrow \Rightarrow \text{parameter changes too slowly}. \]

Too large:

\[ Q_\theta\uparrow \Rightarrow \text{parameter becomes noisy and may absorb other errors}. \]

Therefore process noise should be connected to expected physical parameter variability.

4.3.10.53 Parameter Absorption

An augmented parameter can sometimes absorb errors from an unrelated model deficiency.

For example, an estimated coefficient may compensate for:

• wrong environmental model,
• incorrect sensor bias,
• timing error,
• unmodeled acceleration,
• geometry error.

Then the filter may fit the measurements well while the parameter loses direct physical meaning.

Thus,

\[ \boxed{ \text{Good Residual Fit} \neq \text{Correct Physical Parameter} } \]

4.3.10.54 Bias-State Absorption

Similarly, bias states can hide model errors.

Suppose a persistent residual is caused by incorrect sensor alignment.

If the filter contains only a bias state, it may partially absorb that residual into the bias estimate.

The measurement residual can improve even though the true cause has not been modeled correctly.

Therefore estimated biases should be checked for physical plausibility.

4.3.10.55 Parameter Constraints

Some parameters must satisfy physical bounds.

Examples include

\[ m>0, \]

\[ C_D>0, \]

\[ B>0. \]

A constrained estimator may impose

\[ \boxed{ \theta_{\min} \leq \theta \leq \theta_{\max}. } \]

Constraints prevent physically impossible solutions.

However, constraints and prior probabilities are conceptually different.

4.3.10.56 Parameter Transformation

Positive parameters may be reparameterized as

\[ \boxed{ \gamma = \ln\theta } \]

so that

\[ \boxed{ \theta=e^\gamma>0. } \]

This can make unconstrained numerical optimization easier.

However, covariance and prior distributions must be transformed consistently.

4.3.10.57 Covariance Transformation for Parameters

Suppose

\[ \theta=g(\gamma). \]

For small uncertainty,

\[ \boxed{ P_\theta \approx \left( \frac{\partial g}{\partial\gamma} \right)^2 P_\gamma. } \]

For a vector transformation,

\[ \boxed{ \mathbf P_\theta \approx \mathbf J \mathbf P_\gamma \mathbf J^T. } \]

This is particularly important when reporting parameter uncertainty in physical units after estimating transformed parameters.

4.3.10.58 Parameter Scaling

Augmented states often contain quantities with very different magnitudes.

For example:

\[ r\sim10^6\ \mathrm{m}, \]

\[ v\sim10^3\ \mathrm{m/s}, \]

while a calibration parameter may be

\[ 10^{-4}. \]

Poor scaling can make the estimator numerically ill-conditioned.

Introduce scaled parameters:

\[ \boxed{ \boldsymbol\theta = \mathbf S_\theta \tilde{\boldsymbol\theta}. } \]

Scaling improves numerical behavior without changing the underlying physical information when handled consistently.

4.3.10.59 Augmented-State Covariance Propagation

The augmented covariance propagates as

\[ \boxed{ \mathbf P_{a,k+1|k} = \mathbf F_{a,k} \mathbf P_{a,k|k} \mathbf F_{a,k}^T + \mathbf Q_{a,k}. } \]

Here,

\[ \mathbf Q_a \]

contains process noise for both dynamic states and parameter states.

A typical structure is

\[ \boxed{ \mathbf Q_a = \begin{bmatrix} \mathbf Q_x & \mathbf Q_{x\theta}\\ \mathbf Q_{\theta x} & \mathbf Q_\theta \end{bmatrix}. } \]

Often the cross-noise blocks are assumed zero, but this should reflect the actual model.

4.3.10.60 Augmented Measurement Update

Using the augmented Jacobian

\[ \mathbf H_a, \]

the innovation covariance becomes

\[ \boxed{ \mathbf S_k = \mathbf H_{a,k} \mathbf P_{a,k|k-1} \mathbf H_{a,k}^T + \mathbf R_k. } \]

The Kalman gain is

\[ \boxed{ \mathbf K_k = \mathbf P_{a,k|k-1} \mathbf H_{a,k}^T \mathbf S_k^{-1}. } \]

The update simultaneously corrects:

• physical states,
• parameter states,
• their correlations.

4.3.10.61 How Measurements Correct Parameters

Partition the gain as

\[ \mathbf K = \begin{bmatrix} \mathbf K_x\\ \mathbf K_\theta \end{bmatrix}. \]

Then

\[ \boxed{ \hat{\boldsymbol\theta}_{k|k} = \hat{\boldsymbol\theta}_{k|k-1} + \mathbf K_\theta\mathbf y_k. } \]

A nonzero parameter gain exists because the parameter is correlated with the measurement residual through direct or dynamic sensitivity.

4.3.10.62 Batch Parameter Estimation

Parameters can also be estimated using measurements across an entire observation arc.

Suppose

\[ \mathbf z_k = h_k( \mathbf x_0, \boldsymbol\theta ) + \mathbf v_k. \]

The batch objective may be

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

Then both reference state and parameters are estimated together.

4.3.10.63 Batch versus Recursive Parameter Estimation

Batch estimation uses a complete measurement arc at once.

Recursive estimation updates the parameter as measurements arrive.

Batch methods are useful for:

• post-processing,
• calibration,
• orbit determination,
• system identification.

Recursive methods are useful for:

• real-time navigation,
• adaptive estimation,
• onboard calibration,
• slowly varying parameters.

Neither is universally superior; the choice depends on mission architecture and computation constraints.

4.3.10.64 Static Parameter Estimation in a Kalman Filter

Even if

\[ \theta_{k+1}=\theta_k, \]

the parameter estimate can still change after measurements because the measurement update modifies it:

\[ \boxed{ \hat{\theta}_{k|k} = \hat{\theta}_{k|k-1} + K_\theta y_k. } \]

Thus,

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

does not mean

\[ \boxed{ \hat{\theta}\text{ cannot change}. } \]

It means only that the dynamic model predicts no physical change between measurements.

4.3.10.65 Augmented-State Estimation in Attitude Navigation

A general attitude-estimation state may include

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

More elaborate versions may include

• gyro scale factor,
• sensor alignment,
• accelerometer bias,
• torque-model coefficients.

The objective is to prevent persistent sensor errors from being mistaken for true spacecraft rotation.

4.3.10.66 Augmented-State Estimation in Orbit Determination

A general orbit-determination state may be written

\[ \boxed{ \mathbf x_a = \begin{bmatrix} \mathbf r\\ \mathbf v\\ B\\ \mathbf b_{\rm meas}\\ \boldsymbol\theta_{\rm dyn} \end{bmatrix}. } \]

Possible estimated quantities include:

• drag-related coefficients,
• radiation-pressure coefficient,
• measurement biases,
• timing offsets,
• empirical accelerations,
• maneuver corrections.

The selection depends on the measurement set and observation arc.

4.3.10.67 Augmented States in General Relative Navigation

A general relative-navigation state may be

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

Additional uncertain quantities may include

• sensor bias,
• target feature-location error,
• camera alignment,
• range bias,
• clock or timing offset,
• relative sensor calibration.

Then

\[ \boxed{ \mathbf x_{a,\rm rel} = \begin{bmatrix} \mathbf r_{\rm rel}\\ \mathbf v_{\rm rel}\\ \boldsymbol\theta_{\rm cal} \end{bmatrix}. } \]

This is a general relative-navigation principle and is not specific to any particular RPO mission.

4.3.10.68 Parameter Correlation in Relative Navigation

Suppose line-of-sight measurements are used.

A camera boresight error may produce an angular measurement effect similar to a relative-position error.

Then the estimator may produce significant correlation between:

\[ \boxed{ \text{Relative Position} \leftrightarrow \text{Camera Alignment} } \]

Similarly,

\[ \boxed{ \text{Range Bias} \leftrightarrow \text{Relative Range} } \]

may become strongly correlated.

Independent measurement geometry or maneuver excitation can help separate them.

4.3.10.69 Calibration before Flight versus In-Flight Estimation

Some parameters should preferably be determined on the ground.

Others may require in-flight estimation.

Ground calibration is attractive because it can provide controlled conditions.

In-flight estimation is useful when the parameter:

• changes after launch,
• depends on the space environment,
• changes with temperature,
• depends on deployment,
• cannot be fully calibrated on the ground.

A hybrid approach is common:

\[ \boxed{ \text{Ground Calibration} \rightarrow \text{Prior} \rightarrow \text{In-Flight Refinement} } \]

4.3.10.70 Parameter Convergence

A parameter estimate may converge more slowly than the primary state.

This occurs because the parameter may influence measurements only indirectly through accumulated dynamics.

Therefore the estimator should be evaluated over a sufficiently long observation period.

Convergence should be judged using:

• estimate history,
• covariance history,
• residuals,
• correlation,
• consistency,
• sensitivity to initialization.

4.3.10.71 False Parameter Convergence

A parameter may appear numerically stable even when it is not physically well determined.

For example,

\[ \hat{\theta}_k \]

may stop changing because:

• covariance became artificially small,
• process noise was set too low,
• measurements provide little new information,
• parameter is strongly constrained by prior.

Therefore,

\[ \boxed{ \text{Stable Estimate} \neq \text{Well-Identified Parameter} } \]

4.3.10.72 Parameter Covariance Collapse

If the estimator repeatedly receives apparently informative measurements, parameter covariance may become extremely small.

But if the underlying model neglects systematic errors, the true parameter error may remain larger than predicted.

This produces an overconfident estimator.

Covariance floors or process noise may help, but the underlying model should first be examined.

4.3.10.73 Parameter Consistency

If true parameter values are known in simulation, parameter consistency can be checked through

\[ \boxed{ \epsilon_\theta = \tilde{\boldsymbol\theta}^T \mathbf P_{\theta\theta}^{-1} \tilde{\boldsymbol\theta}. } \]

For a consistent Gaussian estimator,

\[ \epsilon_\theta \]

should follow the appropriate chi-square statistics.

This is the parameter analogue of NEES.

4.3.10.74 Innovation Diagnostics

Even when true parameters are unknown, innovation behavior can indicate parameter-model problems.

Persistent innovation bias may suggest:

• unmodeled sensor bias,
• wrong dynamic parameter,
• incorrect calibration,
• timing error,
• insufficient process noise.

Thus residual analysis remains essential in augmented-state estimation.

4.3.10.75 Parameter Estimate Validation

Estimated parameters should be checked against:

• physical bounds,
• manufacturer specifications,
• independent calibration,
• alternate observation arcs,
• alternate sensors,
• expected temporal behavior.

A parameter that produces excellent residuals but is physically implausible may indicate model compensation rather than true identification.

4.3.10.76 Over-Parameterization

Adding too many parameters may make the estimation problem weakly determined.

Possible symptoms include:

• large covariance,
• strong correlations,
• slow convergence,
• numerical instability,
• dependence on priors,
• unrealistic parameter values.

Therefore,

\[ \boxed{ \text{More Estimated Parameters} \not\Rightarrow \text{Better Estimation} } \]

4.3.10.77 Under-Parameterization

The opposite problem is estimating too few parameters.

If an important bias or physical coefficient is omitted, the estimator may force the dynamic state to compensate.

This may cause:

• biased state estimates,
• structured residuals,
• underestimated covariance,
• poor prediction.

Thus parameter selection must balance model completeness and observability.

4.3.10.78 Parameter Selection Strategy

A useful parameter should satisfy three conditions:

1. It has meaningful uncertainty.
2. It materially affects dynamics or measurements.
3. Available measurements provide enough information to estimate it.

Therefore,

\[ \boxed{ \text{Estimate Parameter} \quad\text{if}\quad \text{Uncertain} + \text{Influential} + \text{Observable}. } \]

Otherwise it may be better to keep it fixed or represent its effect as process noise.

4.3.10.79 Parameter versus Process Noise

Not every unknown effect should become an explicit parameter.

Sometimes an uncertain disturbance is better represented through

\[ \mathbf Q. \]

Parameter augmentation is appropriate when the uncertainty has persistent structure that can potentially be inferred.

Process noise is often more appropriate when the disturbance is rapidly varying or poorly parameterizable.

Thus,

\[ \boxed{ \text{Structured Persistent Uncertainty} \rightarrow \text{Candidate Parameter State} } \]

whereas

\[ \boxed{ \text{Unstructured Random Disturbance} \rightarrow \text{Candidate Process Noise} } \]

4.3.10.80 Parameter versus Bias

A bias is one type of parameter, but not every parameter is a bias.

Bias typically enters additively:

\[ z_m=z+b+n. \]

Scale factor enters multiplicatively:

\[ z_m=(1+s)z+n. \]

Other parameters may influence dynamics nonlinearly.

Correct parameterization matters because different error mechanisms produce different sensitivities.

4.3.10.81 Parameter versus Calibration Constant

A calibration constant may be:

• known from ground calibration,
• estimated once,
• estimated continuously,
• slowly varying.

Whether it should be part of the online state depends on its expected stability and mission requirement.

An estimator should not continuously estimate a perfectly known calibration quantity without a reason, because doing so unnecessarily increases state dimension and correlation.

4.3.10.82 Numerical Conditioning of Augmented Filters

Adding states increases covariance dimension.

For n original states and p parameters,

\[ \mathbf P_a \in \mathbb R^{(n+p)\times(n+p)}. \]

This increases computational burden and may worsen conditioning.

Good practice includes:

• sensible scaling,
• square-root filtering,
• robust matrix factorizations,
• avoiding explicit inverses,
• monitoring covariance eigenvalues,
• checking parameter correlations.

4.3.10.83 Reduced Parameter Sets

Instead of estimating many individual physical quantities, it may be preferable to estimate a smaller set of effective parameters.

For example, rather than separately estimating

\[ C_D,\quad A,\quad m, \]

one may estimate the combined quantity

\[ \boxed{ \frac{C_DA}{m}. } \]

This can improve identifiability because the measurements may respond only to the combined effect.

4.3.10.84 Composite Parameters

Suppose measurements depend on

\[ \gamma=\theta_1\theta_2. \]

If the data contain little independent information about \theta_1 and \theta_2, estimating them separately may be ill-conditioned.

Estimating

\[ \boxed{ \gamma } \]

directly may be more appropriate.

This is a general parameterization principle:

\[ \boxed{ \text{Estimate What the Measurements Actually Observe} } \]

4.3.10.85 Parameterization and Identifiability

The choice of parameterization can dramatically affect numerical conditioning.

Two mathematically equivalent physical descriptions may not be equally easy to estimate.

A useful parameterization should:

• respect physical constraints,
• avoid unnecessary redundancy,
• improve scaling,
• align with observable combinations,
• minimize excessive correlation.

4.3.10.86 Static versus Time-Varying Parameter Decision

Before choosing a model, ask:

Is the parameter physically constant over the estimation interval?

If yes,

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

If it drifts slowly,

\[ \dot{\theta}=w_\theta \]

or a Gauss-Markov model may be more appropriate.

If it changes rapidly and unpredictably, explicit estimation may no longer be the best representation.

The parameter dynamic model should reflect the physical time scale.

4.3.10.87 Parameter Time Scale

The estimator must separate three time scales:

• fast spacecraft dynamics,
• measurement update rate,
• slow parameter variation.

For a slowly varying parameter,

\[ T_\theta \gg T_{\rm measurement} \]

is usually desirable for clear separation.

If the parameter varies as rapidly as the state, a simple constant or random-walk model may be inadequate.

4.3.10.88 Correlation Time

For a first-order Gauss-Markov process,

\[ \dot{\theta} = -\frac{\theta}{\tau}+w, \]

the parameter

\[ \tau \]

defines the approximate memory of the process.

Large

\[ \tau \]

means slow variation.

Small

\[ \tau \]

means faster decorrelation.

The choice should be related to the expected physical behavior of the parameter.

4.3.10.89 State-Parameter Tradeoff

Augmented-state estimation inevitably introduces tradeoffs.

A larger parameter set can improve model fidelity but may reduce observability.

A smaller set improves robustness but may leave systematic errors unmodeled.

Therefore,

\[ \boxed{ \text{Parameter Estimation Design} = \text{Model Fidelity} \leftrightarrow \text{Observability and Robustness} } \]

4.3.10.90 Parameter Estimation and Prediction

A good parameter estimate improves not only current state reconstruction but also future prediction.

For example, an improved dynamic coefficient can reduce propagation error after measurements stop.

Thus parameter estimation can improve:

• trajectory prediction,
• maneuver planning,
• sensor scheduling,
• autonomy,
• future navigation performance.

4.3.10.91 Parameter Estimation and Smoothing

Parameters estimated over a complete arc can benefit strongly from smoothing or batch estimation.

Future measurements can provide information about parameters that affected earlier dynamics.

Thus,

\[ \boxed{ \text{Forward Measurements over an Arc} \rightarrow \text{Improved Parameter Reconstruction} } \]

This connects augmented-state estimation directly to the smoothing methods developed earlier.

4.3.10.92 Parameter Estimation and STM/Sensitivity Matrices

The state transition matrix maps state perturbations:

\[ \delta\mathbf x(t) = \boldsymbol\Phi(t,t_0) \delta\mathbf x_0. \]

Parameter sensitivity adds

\[ \mathbf S_\theta(t) \delta\boldsymbol\theta. \]

Therefore,

\[ \boxed{ \delta\mathbf x(t) = \boldsymbol\Phi(t,t_0) \delta\mathbf x_0 + \mathbf S_\theta(t) \delta\boldsymbol\theta. } \]

This directly links Section 4.3.8 to augmented-state estimation.

4.3.10.93 Combined State-Parameter Mapping

Define

\[ \delta\mathbf y = \begin{bmatrix} \delta\mathbf x_0\\ \delta\boldsymbol\theta \end{bmatrix}. \]

Then the propagated perturbation may be expressed as

\[ \boxed{ \delta\mathbf x(t) = \begin{bmatrix} \boldsymbol\Phi(t,t_0) & \mathbf S_\theta(t) \end{bmatrix} \delta\mathbf y. } \]

The same mapping propagates uncertainty from initial-state and parameter errors into trajectory uncertainty.

4.3.10.94 Parameter Contribution to Covariance

If initial-state and parameter errors are independent,

\[ \boxed{ \mathbf P_x(t) = \boldsymbol\Phi \mathbf P_{x0} \boldsymbol\Phi^T + \mathbf S_\theta \mathbf P_\theta \mathbf S_\theta^T. } \]

If they are correlated, cross terms must also be included.

This shows quantitatively how parameter uncertainty contributes to trajectory uncertainty.

4.3.10.95 Parameter Error Budget

Suppose output quantity

\[ y=g(\mathbf x,\boldsymbol\theta). \]

Then

\[ \delta y \approx \mathbf J_x\delta\mathbf x + \mathbf J_\theta\delta\boldsymbol\theta. \]

The output variance becomes

\[ \boxed{ \sigma_y^2 \approx \begin{bmatrix} \mathbf J_x & \mathbf J_\theta \end{bmatrix} \mathbf P_a \begin{bmatrix} \mathbf J_x & \mathbf J_\theta \end{bmatrix}^T. } \]

This provides a direct connection between augmented-state covariance and engineering error budgets.

4.3.10.96 Common Mistakes in Augmented-State Estimation

Common errors include:

1. Adding parameters without checking observability.
2. Estimating too many highly correlated parameters.
3. Assuming a parameter is constant when it actually varies.
4. Using excessive parameter process noise.
5. Using zero process noise for a drifting parameter.
6. Ignoring state-parameter cross-covariance.
7. Treating fitted effective parameters as exact physical constants.
8. Using unrealistic prior covariance.
9. Allowing parameters to absorb unrelated model errors.
10. Ignoring physical parameter bounds.
11. Using poorly scaled parameter states.
12. Confusing bias, scale factor, and alignment errors.
13. Declaring parameter convergence only because the numerical estimate stops changing.
14. Reporting posterior covariance without checking whether the information came primarily from the prior.
15. Failing to validate parameter estimates against independent data.

4.3.10.97 Practical Parameter-Estimation Workflow

A disciplined workflow is

\[ \boxed{ \begin{array}{c} \text{Identify Uncertain Physical/Calibration Quantity}\\ \downarrow\\ \text{Determine Whether It Influences Dynamics or Measurements}\\ \downarrow\\ \text{Choose Parameterization}\\ \downarrow\\ \text{Select Static / Random-Walk / Gauss-Markov Model}\\ \downarrow\\ \text{Define Initial Estimate and Covariance}\\ \downarrow\\ \text{Derive State and Measurement Sensitivities}\\ \downarrow\\ \text{Check Observability / Identifiability}\\ \downarrow\\ \text{Augment the State}\\ \downarrow\\ \text{Propagate State and Covariance}\\ \downarrow\\ \text{Process Measurements}\\ \downarrow\\ \text{Estimate State + Parameters}\\ \downarrow\\ \text{Examine Correlations}\\ \downarrow\\ \text{Check Residuals and Consistency}\\ \downarrow\\ \text{Validate Parameter Physical Plausibility}\\ \downarrow\\ \text{Validate Against Independent Data} \end{array} } \]

4.3.10.98 General Augmented-State Architecture

The overall architecture can be summarized as

\[ \boxed{ \begin{array}{c} \text{Physical State}\\ \mathbf x \\[2mm] + \\[2mm] \text{Unknown Parameters}\\ \boldsymbol\theta \\ \downarrow\\ \text{Augmented State}\\ \mathbf x_a= [\mathbf x^T\;\boldsymbol\theta^T]^T \\ \downarrow\\ \text{Augmented Dynamics}\\ \dot{\mathbf x}_a=f_a(\mathbf x_a) \\ \downarrow\\ \text{Measurement Model}\\ \mathbf z=h(\mathbf x_a)+\mathbf v \\ \downarrow\\ \text{Estimator} \\ \downarrow\\ \hat{\mathbf x},\hat{\boldsymbol\theta}, \mathbf P_{xx}, \mathbf P_{\theta\theta}, \mathbf P_{x\theta} \end{array} } \]

The result is not only an estimated trajectory but also estimates of the uncertain physical and calibration quantities influencing that trajectory.

4.3.10.99 Key Equations

Augmented state

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

Static parameter

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

Random-walk parameter

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

Gauss-Markov parameter

\[ \boxed{ \dot{\boldsymbol\theta} = -\mathbf T^{-1} ( \boldsymbol\theta-\bar{\boldsymbol\theta} ) + \mathbf w_\theta } \]

Augmented dynamics

\[ \boxed{ \dot{\mathbf x}_a = f_a(\mathbf x_a,t) } \]

Augmented measurement model

\[ \boxed{ \mathbf z_k = h( \mathbf x_{a,k} ) + \mathbf v_k } \]

Augmented Jacobian

\[ \boxed{ \mathbf A_a = \begin{bmatrix} \frac{\partial f}{\partial\mathbf x} & \frac{\partial f}{\partial\boldsymbol\theta}\\ \mathbf0 & \frac{\partial g}{\partial\boldsymbol\theta} \end{bmatrix} } \]

Measurement Jacobian

\[ \boxed{ \mathbf H_a = \begin{bmatrix} \frac{\partial h}{\partial\mathbf x} & \frac{\partial h}{\partial\boldsymbol\theta} \end{bmatrix} } \]

Augmented covariance

\[ \boxed{ \mathbf P_a = \begin{bmatrix} \mathbf P_{xx} & \mathbf P_{x\theta}\\ \mathbf P_{\theta x} & \mathbf P_{\theta\theta} \end{bmatrix} } \]

Parameter sensitivity

\[ \boxed{ \mathbf S_\theta = \frac{\partial\mathbf x} {\partial\boldsymbol\theta} } \]

Sensitivity dynamics

\[ \boxed{ \dot{\mathbf S}_\theta = \mathbf A \mathbf S_\theta + \frac{\partial f} {\partial\boldsymbol\theta} } \]

Augmented covariance propagation

\[ \boxed{ \mathbf P_{a,k+1|k} = \mathbf F_{a,k} \mathbf P_{a,k|k} \mathbf F_{a,k}^T + \mathbf Q_{a,k} } \]

Parameter correction

\[ \boxed{ \hat{\boldsymbol\theta}_{k|k} = \hat{\boldsymbol\theta}_{k|k-1} + \mathbf K_\theta \mathbf y_k } \]

Ballistic coefficient

\[ \boxed{ B=\frac{m}{C_DA} } \]

Relative parameter information

\[ \boxed{ \mathbf I_\theta = \mathbf H_\theta^T \mathbf R^{-1} \mathbf H_\theta } \]

4.3.10.100 Key Takeaways

Parameter and augmented-state estimation extend ordinary state estimation by recognizing that many quantities influencing spacecraft dynamics and measurements are not perfectly known.

The essential construction is

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

Parameters may be modeled as static,

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

slowly varying,

\[ \boxed{ \dot{\boldsymbol\theta} = \mathbf w_\theta, } \]

or correlated in time through models such as Gauss-Markov processes.

Sensor biases, scale factors, alignment errors, calibration coefficients, ballistic coefficient, timing offsets, actuator coefficients, and environmental parameters are all possible augmented-state candidates.

However,

\[ \boxed{ \text{Adding a Parameter to the State} \neq \text{Making the Parameter Observable}. } \]

Successful estimation requires sufficient sensitivity and measurement information.

State and parameter uncertainties are generally correlated:

\[ \boxed{ \mathbf P_{x\theta}\neq0. } \]

These cross-correlations must be retained because they describe how uncertainty in parameters maps into uncertainty in the physical state.

For drag estimation, an important example is

\[ \boxed{ B=\frac{m}{C_DA}. } \]

Because atmospheric density and ballistic coefficient can produce similar trajectory effects, their estimates may become strongly correlated.

Sensor biases and calibration parameters behave similarly: a calibration error can often resemble a physical state error unless additional measurements or system excitation separate them.

Nuisance parameters should not simply be ignored. They may be:

• estimated as augmented states,
• profiled in maximum likelihood,
• marginalized in Bayesian estimation.

The most important engineering principle is therefore

\[ \boxed{ \text{Estimate Only Parameters That Are} \quad \text{Uncertain, Influential, and Observable}. } \]

A well-designed augmented-state estimator can improve state accuracy, calibration, dynamic-model fidelity, and future trajectory prediction. A poorly designed augmented estimator can instead create strong correlations, weak observability, numerical instability, and physically misleading parameter estimates.

Next: 4.3.11 Orbit Determination Fundamentals

Augmented-state estimation provides the machinery for estimating uncertain dynamic and calibration quantities together with the navigation state. The next section applies the broader state-estimation framework to orbit determination, including reference-epoch states, force models, trajectory propagation, observation arcs, measurement prediction, and the overall orbit-determination architecture.