Filter Initialization and Convergence

4.2.18 Filter Initialization and Convergence

A spacecraft navigation filter does not begin with perfect knowledge of the vehicle state. Before the estimator can recursively combine sensor measurements with the dynamic model, it must be given an initial estimate of the state and an initial estimate of the uncertainty associated with that state.

For a general spacecraft navigation problem, the estimated state may contain

\[ \mathbf{x} = \begin{bmatrix} \mathbf r & \mathbf v & \mathbf q & \boldsymbol\omega & \mathbf b_g & \mathbf b_a & \mathbf s_g & \mathbf s_a & \Delta t & \cdots \end{bmatrix}^{T} \]

where the components may represent position, velocity, attitude quaternion, angular velocity, gyro bias, accelerometer bias, scale factors, clock errors, sensor misalignments, or other calibration parameters.

At initialization,

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

is the estimator's best initial state estimate, while

\[ \mathbf P_0 = E\left[ (\mathbf x_0-\hat{\mathbf x}_0) (\mathbf x_0-\hat{\mathbf x}_0)^T \right] \]

describes how uncertain that estimate is believed to be.

These two quantities are tightly connected:

\[ \boxed{ \text{Filter initialization} = \hat{\mathbf x}_0+\mathbf P_0 } \]

A filter can have excellent equations, correct process models, and high-quality sensors and still perform poorly if it is initialized inconsistently.

Initialization is therefore not simply a software startup operation. It is an important part of the navigation-system design.

4.2.18.1 Why Filter Initialization Matters

Consider a Kalman-type estimator.

The innovation is

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

with innovation covariance

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

The Kalman gain is

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

Therefore, the initial covariance directly influences how strongly the filter responds to the first measurements.

If \mathbf P_0 is unrealistically small, the estimator behaves as though its initial state is already known very accurately. Measurements may then receive insufficient weight.

If \mathbf P_0 is excessively large, the estimator can react very strongly to the first few measurements, producing large corrections or unstable-looking transients.

Thus,

\[ \boxed{ \mathbf P_0 \rightarrow \mathbf K_k \rightarrow \text{initial filter behaviour} } \]

The initialization problem is therefore both a state-estimation problem and an uncertainty-modelling problem.

4.2.18.2 What Must Be Initialized?

Initialization should be considered for every state included in the estimator.

For a representative spacecraft navigation filter,

\[ \mathbf x = \begin{bmatrix} \mathbf r\\ \mathbf v\\ \delta\boldsymbol\theta\\ \mathbf b_g\\ \mathbf b_a \end{bmatrix}, \]

the estimator requires initial values for

\[ \hat{\mathbf r}_0, \qquad \hat{\mathbf v}_0, \qquad \hat{\mathbf q}_0, \qquad \hat{\mathbf b}_{g,0}, \qquad \hat{\mathbf b}_{a,0}, \]

together with their uncertainties.

If additional calibration states are estimated, they must also be initialized:

\[ \hat{\mathbf s}_{g,0}, \qquad \hat{\mathbf s}_{a,0}, \qquad \hat{\boldsymbol\epsilon}_{mis,0}, \qquad \widehat{\Delta t}_0, \qquad \ldots \]

The correct initialization strategy is therefore strongly dependent on the state vector definition.

4.2.18.3 Sources of Initial-State Information

The initial state may come from several sources.

Typical spacecraft sources include:

• GNSS position and velocity,
• propagated orbital ephemeris,
• ground-based orbit determination,
• star tracker attitude,
• Sun sensors,
• Earth-horizon sensors,
• magnetometers,
• gyroscopes,
• accelerometers,
• camera-based relative navigation,
• LiDAR or laser range measurements,
• radar,
• previous navigation solutions,
• docking-target information,
• mission-planning data.

The initial estimate may therefore be written conceptually as

\[ \hat{\mathbf x}_0 = f( \mathbf z_{\text{GNSS}}, \mathbf z_{\text{ST}}, \mathbf z_{\text{IMU}}, \mathbf z_{\text{camera}}, \mathbf z_{\text{LiDAR}}, \ldots). \]

Not every sensor must be available at startup. The initialization architecture should explicitly define which sensors establish which states.

4.2.18.4 Position Initialization

For a spacecraft with GNSS availability, the simplest position initialization may be

\[ \hat{\mathbf r}_0 = \mathbf r_{\text{GNSS}}. \]

The corresponding covariance can be initialized from the expected GNSS uncertainty:

\[ \mathbf P_{r,0} = \begin{bmatrix} \sigma_x^2 & 0 & 0\\ 0 & \sigma_y^2 & 0\\ 0 & 0 & \sigma_z^2 \end{bmatrix}. \]

If correlations exist, the covariance should not automatically be assumed diagonal.

For example,

\[ \mathbf P_{r,0} = \begin{bmatrix} \sigma_x^2 & \rho_{xy}\sigma_x\sigma_y & \rho_{xz}\sigma_x\sigma_z\\ \rho_{xy}\sigma_x\sigma_y & \sigma_y^2 & \rho_{yz}\sigma_y\sigma_z\\ \rho_{xz}\sigma_x\sigma_z & \rho_{yz}\sigma_y\sigma_z & \sigma_z^2 \end{bmatrix}. \]

The important principle is that

\[ \boxed{ \mathbf P_{r,0} \text{ should represent actual position knowledge} } \]

rather than simply being assigned an arbitrary convenient value.

4.2.18.5 Velocity Initialization

Velocity can be initialized from GNSS velocity, orbit determination, Doppler measurements, or multiple position measurements separated in time.

A simple finite-difference estimate is

\[ \hat{\mathbf v}_0 \approx \frac{\mathbf r(t_2)-\mathbf r(t_1)} {t_2-t_1}. \]

However, differentiation amplifies measurement noise. Consequently, the uncertainty of such a velocity estimate may be substantially larger than that obtained from a dedicated velocity solution.

For spacecraft relative navigation,

\[ \mathbf v_{rel} = \mathbf v_c-\mathbf v_t \]

must be expressed consistently in the chosen relative-navigation frame.

Initialization therefore requires not only a numerical value but also correct reference-frame handling.

4.2.18.6 Attitude Initialization

Attitude initialization is particularly important because attitude is nonlinear and normally represented using a quaternion or rotation matrix.

The initial attitude may come from a star tracker:

\[ \hat{\mathbf q}_0 = \mathbf q_{\text{ST}}. \]

Alternatively, attitude may be determined using two or more reference vectors.

Suppose known inertial vectors are

\[ \mathbf v_1^I,\qquad \mathbf v_2^I \]

and corresponding body-frame measurements are

\[ \mathbf v_1^B,\qquad \mathbf v_2^B. \]

The attitude must satisfy approximately

\[ \mathbf v_i^B = \mathbf C_{BI}\mathbf v_i^I. \]

Algorithms such as TRIAD, QUEST, Davenport's q-method, or Wahba-problem solutions can then provide an initial attitude.

4.2.18.7 Quaternion Initialization

A quaternion must satisfy

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

so an initial quaternion should be normalized:

\[ \hat{\mathbf q}_0 \leftarrow \frac{\hat{\mathbf q}_0} {\|\hat{\mathbf q}_0\|}. \]

A zero quaternion is not a valid attitude initialization.

For an error-state filter, the nominal quaternion and attitude-error covariance must also be distinguished carefully.

The nominal attitude may be

\[ \hat{\mathbf q}_0, \]

while the small attitude error is represented as

\[ \delta\boldsymbol\theta_0. \]

The covariance then describes uncertainty in the three-dimensional attitude-error coordinates rather than directly treating the four quaternion components as independent states.

4.2.18.8 Coarse Attitude Alignment

The initial attitude does not necessarily need to satisfy the accuracy required during normal navigation.

A coarse alignment stage can provide an initial attitude estimate sufficiently accurate for the nonlinear estimator to begin operating.

The sequence may be

\[ \boxed{ \text{Coarse Attitude} \rightarrow \text{Filter Initialization} \rightarrow \text{Fine Estimation} } \]

For example, Sun-sensor and magnetometer measurements may provide a coarse orientation, after which a star tracker or fused attitude filter improves the solution.

This is particularly useful when the high-accuracy sensor has acquisition constraints.

4.2.18.9 Angular-Rate Initialization

Angular rate is often initialized directly from gyroscope measurements:

\[ \hat{\boldsymbol\omega}_0 \approx \boldsymbol\omega_m. \]

But a gyro measurement contains bias and noise:

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

Therefore,

\[ \hat{\boldsymbol\omega}_0 = \boldsymbol\omega_m-\hat{\mathbf b}_{g,0}. \]

The quality of angular-rate initialization is consequently connected to gyro-bias initialization.

4.2.18.10 Gyro-Bias Initialization

Gyro bias is frequently included as an estimated state:

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

A simple initialization may be

\[ \hat{\mathbf b}_{g,0} = \mathbf 0. \]

But this does not mean the filter should assume that the true bias is zero.

The uncertainty should reflect the expected bias range:

\[ \mathbf P_{b_g,0} = \sigma_{b_g,0}^2\mathbf I. \]

If the spacecraft is known to be stationary or its angular rate is independently known during initialization, several gyro samples can be averaged:

\[ \hat{\mathbf b}_{g,0} = \frac{1}{N} \sum_{k=1}^{N} \boldsymbol\omega_{m,k} - \boldsymbol\omega_{\text{known}}. \]

This can significantly reduce the initial bias uncertainty.

4.2.18.11 Accelerometer-Bias Initialization

Similarly,

\[ \mathbf a_m = \mathbf a + \mathbf b_a + \mathbf n_a. \]

The accelerometer bias may initially be set to zero,

\[ \hat{\mathbf b}_{a,0}=\mathbf 0, \]

with covariance

\[ \mathbf P_{b_a,0} = \sigma_{b_a,0}^{2}\mathbf I. \]

Alternatively, calibration data or pre-flight characterization may provide a better initial estimate.

For spacecraft applications, care is required because an accelerometer does not simply measure orbital gravitational acceleration in the same way that a stationary terrestrial accelerometer appears to measure g. Its measurement interpretation depends on the inertial-navigation formulation and the specific force acting on the spacecraft.

4.2.18.12 Calibration-State Initialization

More sophisticated filters may estimate:

\[ \mathbf b_g, \quad \mathbf b_a, \quad \mathbf s_g, \quad \mathbf s_a, \quad \boldsymbol\epsilon_g, \quad \boldsymbol\epsilon_a, \quad \Delta t. \]

These can represent biases, scale factors, misalignment parameters, and timing offsets.

A calibration state should be initialized from the best available information:

\[ \hat{x}_{cal,0} = x_{factory} \]

when factory calibration is available.

Its covariance should reflect the uncertainty remaining after that calibration:

\[ P_{cal,0} = \sigma_{cal}^{2}. \]

This connects initialization directly to the observability discussion of Section 4.2.17: adding a calibration state is useful only if the subsequent sensor geometry and spacecraft motion provide enough information for that state to converge.

4.2.18.13 Initial Covariance Matrix

For an estimator with state error

\[ \delta\mathbf x = \begin{bmatrix} \delta\mathbf r\\ \delta\mathbf v\\ \delta\boldsymbol\theta\\ \delta\mathbf b_g\\ \delta\mathbf b_a \end{bmatrix}, \]

a common initial covariance structure is

\[ \mathbf P_0 = \begin{bmatrix} \mathbf P_r & \mathbf P_{rv} & \cdots\\ \mathbf P_{vr} & \mathbf P_v & \cdots\\ \vdots & \vdots & \ddots \end{bmatrix}. \]

A simplified implementation may initially use

\[ \mathbf P_0 = \operatorname{diag} \left( \sigma_r^2, \sigma_v^2, \sigma_\theta^2, \sigma_{b_g}^2, \sigma_{b_a}^2 \right), \]

with each quantity expanded across the appropriate axes.

But a diagonal covariance implies that the initial errors are uncorrelated. That assumption should be made deliberately rather than automatically.

4.2.18.14 Initial Covariance Too Small

Suppose the actual initial position error is large but

\[ \sigma_{r,0} \]

is specified as extremely small.

The filter effectively says:

My initial position is already very accurate.

Then the measurement correction may be weak because

\[ \mathbf K_k \]

is influenced by the predicted covariance.

Consequences can include:

• slow correction of the initial error,
• innovation rejection,
• inconsistent covariance,
• prolonged navigation bias,
• apparent filter divergence.

This is an overconfident initialization.

4.2.18.15 Initial Covariance Too Large

The opposite case is

\[ \mathbf P_0 \gg \text{actual initial uncertainty}. \]

The filter then places substantial weight on early measurements.

Possible consequences include:

• large initial corrections,
• noisy initial state estimates,
• attitude jumps,
• strong sensitivity to individual measurements,
• large transient covariance reduction.

A deliberately conservative covariance can be useful, but unnecessarily enormous values are not automatically safer.

4.2.18.16 Consistent Initialization

An initialization is statistically consistent when the actual initial estimation error is compatible with the covariance assigned to it.

If

\[ \tilde{\mathbf x}_0 = \mathbf x_0-\hat{\mathbf x}_0, \]

then ideally

\[ E[ \tilde{\mathbf x}_0 \tilde{\mathbf x}_0^T ] \approx \mathbf P_0. \]

This is the fundamental objective.

The covariance should neither hide uncertainty nor invent unrealistic uncertainty.

4.2.18.17 Cross-Covariance at Initialization

Suppose velocity is computed from two noisy position measurements.

Then position and velocity errors may be correlated.

Consequently,

\[ \mathbf P_{rv,0} \neq \mathbf 0. \]

Likewise, if attitude and gyro bias are obtained from the same alignment procedure, their uncertainties may be correlated.

Ignoring these correlations can make the filter statistically inconsistent even if the diagonal variances appear reasonable.

4.2.18.18 Initialization from a Previous Navigation Solution

A spacecraft may already have a valid navigation solution before switching estimator modes.

Then

\[ \hat{\mathbf x}_{new,0} = f(\hat{\mathbf x}_{old}) \]

and

\[ \mathbf P_{new,0} = \mathbf J \mathbf P_{old} \mathbf J^T, \]

where

\[ \mathbf J = \frac{\partial f}{\partial\mathbf x}. \]

This is generally preferable to discarding existing state information.

However, care is needed when the new estimator uses different coordinates, frames, state definitions, or calibration parameters.

4.2.18.19 Cold Start and Warm Start

A cold start occurs when little or no reliable previous navigation information is available.

It may require:

\[ \text{sensor acquisition} \rightarrow \text{coarse state determination} \rightarrow \text{filter initialization}. \]

A warm start uses a previous valid solution:

\[ \hat{\mathbf x}_0 \approx \hat{\mathbf x}_{previous}. \]

Warm starts can substantially reduce convergence time, provided the stored state has not become stale or invalid.

4.2.18.20 Initialization After a Long Data Gap

Suppose the filter has not received external measurements for a long period.

The previous state can be propagated:

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

while covariance grows:

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

After a sufficiently long outage, simply restoring the previous covariance may be inappropriate.

The estimator should account for uncertainty accumulated during the gap.

4.2.18.21 What Is Filter Convergence?

Filter convergence refers to the process by which the state estimate approaches the true state and its uncertainty reaches a statistically meaningful operating level.

Conceptually,

\[ \hat{\mathbf x}_k \rightarrow \mathbf x_k. \]

The estimation error

\[ \tilde{\mathbf x}_k = \mathbf x_k-\hat{\mathbf x}_k \]

should decrease.

At the same time, the covariance generally contracts from its initialization value toward a level determined by sensor noise, process noise, geometry, and system dynamics.

Thus convergence is not merely

\[ \mathbf P_k\downarrow. \]

It should involve both:

\[ \boxed{ \text{state-error reduction} + \text{consistent uncertainty reduction} } \]

4.2.18.22 Filter Transient

The period immediately after initialization is called the filter transient.

During this period,

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

may change rapidly.

Early estimates may therefore have lower confidence than steady-state estimates.

A flight system may explicitly distinguish

\[ \text{INITIALIZING} \]

from

\[ \text{CONVERGED}. \]

The navigation solution should not necessarily be declared mission-ready immediately after the filter begins running.

4.2.18.23 Covariance Contraction

Measurements provide information, so observable state uncertainty generally decreases.

For the Kalman update,

\[ \mathbf P_k^+ = (\mathbf I-\mathbf K_k\mathbf H_k) \mathbf P_k^-. \]

As measurements accumulate,

\[ \sigma_{x,k} = \sqrt{P_{ii,k}} \]

may decrease.

This is covariance contraction.

But covariance reduction is meaningful only if the filter model and noise assumptions are reasonably consistent.

4.2.18.24 Covariance Collapse

A dangerous situation occurs when

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

while the actual estimation error remains significant.

This is often described as covariance collapse or severe overconfidence.

Then

\[ |\tilde{x}_k| \gg \sqrt{P_{ii,k}}. \]

The filter believes it knows the state much more accurately than it actually does.

Consequences can include measurement rejection, poor recovery from modelling errors, and misleading navigation-validity flags.

4.2.18.25 Convergence Is State Dependent

Not all states converge at the same rate.

For example,

\[ \mathbf r \]

may converge rapidly from GNSS,

while

\[ \mathbf b_g \]

may require sustained attitude-reference measurements.

A scale factor may require manoeuvres.

A camera-boresight parameter may require changing target geometry.

Therefore,

\[ \boxed{ T_{\text{conv},i} \neq T_{\text{conv},j} } \]

in general.

A single global statement such as “the filter converged after 10 seconds” can therefore hide important state-specific behaviour.

4.2.18.26 Convergence and Observability

Convergence is closely related to Section 4.2.17.

If a state is unobservable,

\[ \text{measurement history} \nRightarrow x_i, \]

then measurement updates cannot uniquely determine that state.

If a state is weakly observable, it may converge only slowly.

Therefore,

\[ \boxed{ \text{Strong observability} \Rightarrow \text{typically faster convergence} } \]

while

\[ \boxed{ \text{Weak observability} \Rightarrow \text{slow or geometry-dependent convergence} } \]

and

\[ \boxed{ \text{Unobservable state} \Rightarrow \text{no measurement-driven convergence} } \]

unless additional information becomes available.

4.2.18.27 Bias Convergence

Bias states are particularly important.

For a gyro,

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

Gyro measurements alone cannot generally distinguish true angular rate from bias.

An external attitude reference creates information that allows the estimator to identify persistent drift.

Thus the gyro bias may gradually converge:

\[ \hat{\mathbf b}_{g,k} \rightarrow \mathbf b_g. \]

Bias convergence time may be substantially longer than attitude convergence time.

4.2.18.28 Calibration-State Convergence

Suppose the estimator includes an accelerometer scale factor:

\[ a_m = (1+s_a)a+b_a+n_a. \]

If

\[ a\approx0 \]

for a long period, the scale factor has little effect on the measurement.

Consequently, s_a may converge very slowly.

When manoeuvres create significant acceleration,

\[ a\neq0, \]

the scale-factor effect becomes visible.

Therefore,

\[ \boxed{ \text{Calibration convergence} \text{ often requires excitation} } \]

rather than merely longer filter runtime.

4.2.18.29 Convergence and Sensor Geometry

Geometry can strongly affect convergence.

For camera bearing-only relative navigation,

\[ \mathbf z = \begin{bmatrix} u\\v \end{bmatrix}, \]

range may initially be weakly observable.

A lateral manoeuvre creates parallax:

\[ \text{lateral motion} \rightarrow \text{bearing change} \rightarrow \text{range information}. \]

Thus the range estimate may remain uncertain initially and then converge rapidly after the geometry improves.

Convergence time is therefore sometimes a trajectory-dependent quantity.

4.2.18.30 Convergence and Measurement Rate

A sensor providing measurements at

\[ f_m \]

updates per second supplies information more frequently than a slower sensor, although each measurement's quality also matters.

Increasing measurement rate can often accelerate convergence, but only when the measurements contain useful independent information.

Repeated highly correlated measurements do not provide the same information gain as genuinely independent observations.

4.2.18.31 Convergence and Measurement Noise

Suppose

\[ \mathbf R_1 < \mathbf R_2 \]

in an appropriate covariance sense.

The first sensor provides more precise measurements.

All else being equal, it may therefore constrain the state more strongly and reduce covariance more rapidly.

But convergence depends jointly on

\[ \boxed{ \mathbf H,\ \mathbf R,\ \mathbf Q,\ \mathbf F,\ \text{geometry} } \]

not on measurement noise alone.

4.2.18.32 Convergence and Process Noise

Process noise enters the covariance propagation:

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

Larger \mathbf Q tells the estimator that the model is less certain.

The filter therefore tends to retain more uncertainty and rely more strongly on measurements.

Very small \mathbf Q can make the estimator excessively confident in the model.

Thus process-noise tuning influences both convergence behaviour and steady-state uncertainty.

4.2.18.33 Convergence and Filter Gain

At startup, a large covariance may produce a relatively large Kalman gain:

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

The filter then responds strongly to measurements.

As covariance contracts,

\[ \mathbf K_k \]

often settles toward a lower operating level.

This explains why early filter behaviour can look much more aggressive than steady-state behaviour.

4.2.18.34 Innovation During Convergence

The innovation is

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

During initial convergence, relatively large innovations may be expected because the initial state may be inaccurate.

As the estimator converges,

\[ E[\mathbf y_k]\rightarrow\mathbf 0 \]

for a correctly modelled unbiased system.

The innovation sequence should also become statistically compatible with

\[ \mathbf S_k. \]

Innovation behaviour therefore provides an important indication of convergence.

4.2.18.35 Initialization and Innovation Gating

There is an important interaction between initialization and the measurement-validation logic of Section 4.2.15.

Suppose the initial estimate is poor.

Then

\[ \mathbf y_k \]

may be large.

If \mathbf P_0 is simultaneously too small, then

\[ \mathbf S_k \]

may also be too small.

Consequently,

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

can become very large.

The filter may reject precisely the measurements needed to correct its initial error.

This creates a dangerous loop:

\[ \boxed{ \text{Poor initialization} \rightarrow \text{large innovation} \rightarrow \text{measurement rejection} \rightarrow \text{no correction} } \]

Initialization and gating therefore cannot be designed independently.

4.2.18.36 Startup Gating Strategy

During initialization, some systems use modified measurement-validation logic.

Possible approaches include:

• wider startup gates,
• covariance inflation,
• staged sensor acceptance,
• multiple-measurement confirmation,
• coarse alignment before normal filtering,
• temporary initialization modes.

However, completely disabling validation can expose the filter to bad data.

A better philosophy is

\[ \boxed{ \text{Initialization-aware gating} } \]

rather than simply turning fault protection off.

4.2.18.37 Staged Initialization

A complex navigation system does not need to initialize every state simultaneously.

A possible sequence is

\[ \text{Time} \rightarrow \text{Attitude} \rightarrow \text{Position} \rightarrow \text{Velocity} \rightarrow \text{Biases} \rightarrow \text{Calibration states}. \]

For example:

1. establish valid timestamps,
2. acquire star tracker,
3. initialize attitude,
4. acquire GNSS,
5. initialize position and velocity,
6. start IMU propagation,
7. estimate biases,
8. enable additional calibration states.

This can be substantially more robust than activating a large nonlinear filter with poorly known states simultaneously.

4.2.18.38 Filter Initialization State Machine

Flight software can explicitly represent initialization states.

For example,

\[ \boxed{ \text{OFF} \rightarrow \text{ACQUIRE} \rightarrow \text{COARSE INIT} \rightarrow \text{FILTER INIT} \rightarrow \text{CONVERGING} \rightarrow \text{NAV VALID} } \]

Possible failure transitions include

\[ \text{CONVERGING} \rightarrow \text{REINITIALIZE} \]

or

\[ \text{NAV VALID} \rightarrow \text{DEGRADED}. \]

This prevents estimator status from being reduced to a simple “running/not running” flag.

4.2.18.39 Convergence Criteria

A filter should not be declared converged merely because it has operated for a fixed amount of time.

Useful convergence indicators include:

\[ \sigma_{r}<\sigma_{r,\max}, \]

\[ \sigma_v<\sigma_{v,\max}, \]

\[ \sigma_\theta<\sigma_{\theta,\max}, \]

\[ \sigma_{b_g}<\sigma_{b_g,\max}, \]

and innovation statistics within acceptable limits.

A navigation-valid flag may therefore require

\[ \boxed{ C_P \land C_{innov} \land C_{sensor} \land C_{time} } \]

where the conditions represent covariance, innovation, sensor-health, and minimum-duration requirements.

4.2.18.40 Covariance-Based Convergence Criteria

One possible criterion is

\[ \sqrt{P_{ii}} < \sigma_{i,\text{req}}. \]

For example,

\[ \sqrt{P_{rr}} < r_{\text{req}}. \]

But covariance alone is not sufficient.

A badly tuned filter can report very small covariance while being wrong.

Therefore covariance thresholds should normally be combined with other consistency checks.

4.2.18.41 Innovation-Based Convergence Criteria

The mean innovation should approach zero:

\[ \bar{\mathbf y} \approx \mathbf 0. \]

Its statistical magnitude should be compatible with

\[ \mathbf S. \]

NIS can be monitored over a window:

\[ \mathrm{NIS}_k = \mathbf y_k^T \mathbf S_k^{-1} \mathbf y_k. \]

Persistent excessive NIS may indicate that the estimator has not converged or that the assumed models are inconsistent.

4.2.18.42 State-Change Convergence Criterion

Another useful measure is the correction magnitude:

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

During early convergence,

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

may be large.

As the filter stabilizes,

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

should generally become smaller.

This should not be used alone, because an unobservable filter can also produce small corrections.

4.2.18.43 Time-Based Convergence Requirement

Even when statistical criteria are satisfied, a minimum dwell time may be useful:

\[ t_{\text{valid}} > t_{\min}. \]

For example, the system may require all convergence criteria to remain satisfied continuously for

\[ T_{\text{dwell}} \]

before asserting

\[ \text{NAV\_VALID}=1. \]

This prevents rapid toggling caused by temporary favourable measurements.

4.2.18.44 Truth-Based Convergence in Simulation

In simulation, truth is available.

Therefore,

\[ \tilde{\mathbf x}_k = \mathbf x_k^{truth} - \hat{\mathbf x}_k \]

can be computed directly.

A convergence criterion may be

\[ |\tilde{x}_{i,k}| < e_{i,\max}. \]

This is extremely useful for verification.

However, flight software cannot rely on truth-based criteria because the true state is precisely what the estimator is trying to determine.

4.2.18.45 NEES During Convergence Testing

For simulations with known truth,

\[ \mathrm{NEES} = \tilde{\mathbf x}^{T} \mathbf P^{-1} \tilde{\mathbf x}. \]

NEES tests whether the actual state-estimation error is statistically compatible with the predicted covariance.

This is particularly valuable for detecting covariance collapse.

If the actual error remains large while \mathbf P becomes very small, NEES becomes excessive.

Thus NEES provides more information than simply plotting covariance.

4.2.18.46 Monte Carlo Convergence Testing

A single simulation cannot adequately validate filter convergence.

Instead, run

\[ N \]

Monte Carlo cases with variations in:

• initial position error,
• initial velocity error,
• initial attitude error,
• gyro bias,
• accelerometer bias,
• sensor noise,
• measurement dropout,
• timing uncertainty,
• manoeuvre uncertainty,
• initial calibration errors.

For each run, record

\[ T_{\text{conv}}. \]

Then analyse

\[ P(T_{\text{conv}}

This converts convergence from a qualitative plot into a quantitative navigation-performance requirement.

4.2.18.47 Convergence-Time Distribution

Monte Carlo analysis produces a distribution rather than one convergence time.

Useful statistics include

\[ T_{50}, \qquad T_{95}, \qquad T_{99}, \qquad T_{\max}. \]

For example, a requirement might be expressed as

\[ P(T_{\text{conv}}<60\,s)>0.99. \]

This is more meaningful than reporting:

The filter converged in approximately 30 seconds.

4.2.18.48 Initialization Error Envelope

The estimator should be tested over an expected initialization envelope.

For example,

\[ |\delta r_0| \le r_{0,\max}, \]

\[ |\delta v_0| \le v_{0,\max}, \]

\[ |\delta\theta_0| \le \theta_{0,\max}. \]

The objective is to determine the region from which the estimator reliably converges.

This region can be thought of as a practical convergence basin.

4.2.18.49 Nonlinear Filter Convergence Basin

For strongly nonlinear systems, convergence may depend critically on initial error.

An EKF assumes that local linearization is sufficiently accurate:

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

If

\[ \|\delta\mathbf x_0\| \]

is too large, this approximation can be poor.

The filter may then diverge even though the same filter performs well when initialized closer to truth.

This is especially important for attitude, camera measurements, angles, and close-proximity navigation.

4.2.18.50 Large Attitude Initialization Errors

Small-angle error-state formulations often assume

\[ \|\delta\boldsymbol\theta\|\ll1. \]

If the actual initial attitude error is very large, this assumption is violated.

Therefore a spacecraft may first use a global attitude-determination algorithm to obtain a coarse quaternion:

\[ \mathbf q_{\text{coarse}} \]

and only then initialize the error-state filter.

This gives

\[ \boxed{ \text{Global coarse attitude solution} \rightarrow \text{local error-state estimator} } \]

rather than asking a local filter to solve a global ambiguity.

4.2.18.51 relative navigation Relative-State Initialization

For rendezvous and proximity operations, a representative relative-navigation state is

\[ \mathbf x_{rel} = \begin{bmatrix} \mathbf r_{rel}\\ \mathbf v_{rel}\\ \mathbf q_{rel}\\ \boldsymbol\omega_{rel}\\ \mathbf b_g\\ \mathbf b_a \end{bmatrix}. \]

Each component may originate from a different sensor.

For example:

\[ \mathbf r_{rel} \leftarrow \text{GNSS / camera / LiDAR}, \]

\[ \mathbf v_{rel} \leftarrow \text{GNSS / differentiated relative measurements}, \]

\[ \mathbf q_{rel} \leftarrow \text{vision / attitude sensors}, \]

\[ \boldsymbol\omega_{rel} \leftarrow \text{gyro + target tracking}. \]

Initialization is therefore naturally a multi-sensor fusion problem itself.

4.2.18.52 Far-Range relative navigation Initialization

At far range, GNSS may provide absolute chaser and target states.

Relative position can then be formed as

\[ \mathbf r_{rel} = \mathbf r_c-\mathbf r_t. \]

Similarly,

\[ \mathbf v_{rel} = \mathbf v_c-\mathbf v_t. \]

The covariance must also be transformed appropriately.

If the two absolute estimates are independent,

\[ \mathbf P_{rel} = \mathbf P_c+\mathbf P_t. \]

If they are correlated,

\[ \mathbf P_{rel} = \mathbf P_c+\mathbf P_t - \mathbf P_{ct} - \mathbf P_{tc}. \]

This correlation issue can be important when both vehicles use common measurements or common corrections.

4.2.18.53 Mid-Range relative navigation Initialization

At mid range, camera or LiDAR measurements may become available.

Rather than immediately replacing the far-range solution, the estimator can use the existing relative state as a prior:

\[ \hat{\mathbf x}_{mid,0} = \hat{\mathbf x}_{far}. \]

The new sensor then refines the state.

This is often preferable to restarting the estimator from scratch.

The transition becomes

\[ \boxed{ \text{GNSS solution} \rightarrow \text{prior} \rightarrow \text{camera/LiDAR refinement} } \]

4.2.18.54 Close-Range Initialization

Near precision proximity operations, navigation accuracy requirements become much tighter.

The filter may need precise estimates of

\[ \mathbf r_{rel}, \qquad \mathbf v_{rel}, \qquad \mathbf q_{rel}, \qquad \boldsymbol\omega_{rel}. \]

Before entering the final approach or precision proximity operations phase, the system may require:

\[ \sigma_r<\sigma_{r,\max}, \]

\[ \sigma_v<\sigma_{v,\max}, \]

\[ \sigma_\theta<\sigma_{\theta,\max}, \]

\[ \sigma_\omega<\sigma_{\omega,\max}. \]

Thus filter convergence can become a guidance-transition condition.

4.2.18.55 Sensor Handover and Convergence

Suppose the mission transitions

\[ \text{GNSS} \rightarrow \text{camera} \rightarrow \text{LiDAR}. \]

Each transition may create a temporary change in estimation performance.

A new sensor may have:

• different bias,
• different frame definition,
• different latency,
• different update rate,
• different measurement covariance.

Therefore the estimator may experience a secondary convergence transient even though it was previously converged.

Sensor handover should consequently be verified as carefully as initial startup.

4.2.18.56 Sensor Reacquisition

Suppose a star tracker becomes unavailable and later reacquires.

During the outage,

\[ \mathbf P_{\theta} \]

and

\[ \mathbf P_{b_g} \]

may increase.

When measurements return, the innovation may initially be large.

The system must decide whether to:

• accept the measurement directly,
• perform consistency checks,
• temporarily widen gates,
• reinitialize part of the state,
• perform a new coarse alignment.

This is a partial reinitialization problem.

4.2.18.57 Partial Reinitialization

A fault does not always require resetting the entire navigation filter.

Suppose only the attitude solution becomes invalid.

Then the system may reinitialize

\[ \hat{\mathbf q} \]

and the associated covariance while preserving

\[ \hat{\mathbf r}, \qquad \hat{\mathbf v}, \]

if those remain valid.

Thus

\[ \boxed{ \text{Reinitialization should target invalid information} } \]

rather than automatically destroying valid state knowledge.

4.2.18.58 Covariance Inflation

When the estimator becomes uncertain after a sensor outage or suspected inconsistency, covariance can sometimes be deliberately inflated:

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

This reduces estimator overconfidence and allows new measurements to exert greater influence.

Selective inflation may be preferable:

\[ P_{ii} \leftarrow \alpha_i P_{ii}. \]

Only the affected states then become less confident.

Covariance inflation should have a physical or statistical justification rather than being used merely to force a filter to respond.

4.2.18.59 Reinitialization After Filter Divergence

If the estimator diverges severely, continued recursive updates may not recover it.

A recovery sequence may be

\[ \boxed{ \text{Detect} \rightarrow \text{Isolate} \rightarrow \text{Reset} \rightarrow \text{Reinitialize} \rightarrow \text{Reconverge} } \]

Detection may use:

• innovation monitoring,
• NIS,
• sensor disagreement,
• impossible state values,
• covariance limits,
• navigation residuals.

This connects initialization directly with FDIR and robust sensor fusion.

4.2.18.60 Navigation Validity During Reinitialization

During reinitialization,

\[ \text{NAV\_VALID} \]

should not necessarily remain true.

The flight system may instead indicate

\[ \text{NAV\_DEGRADED} \]

or

\[ \text{NAV\_CONVERGING}. \]

Guidance and control can then respond appropriately.

For example, during relative navigation:

\[ \text{navigation invalid} \rightarrow \text{hold / retreat / inhibit approach}. \]

Thus estimator initialization is connected directly to mission safety logic.

4.2.18.61 Initialization and Guidance

Guidance should know whether the navigation state is sufficiently mature.

For example,

\[ \text{Proceed to final approach} \]

may require

\[ \text{NAV\_VALID}=1. \]

The flag itself may depend on

\[ \sigma_r, \quad \sigma_v, \quad \sigma_\theta, \quad \sigma_\omega, \quad \text{sensor health}, \quad \text{innovation consistency}. \]

This prevents guidance from commanding aggressive manoeuvres while the estimator is still converging.

4.2.18.62 Initialization and Control

The controller uses estimated states:

\[ \mathbf u = g(\hat{\mathbf x}). \]

During initial convergence,

\[ \hat{\mathbf x} \]

may contain substantial error.

Large control authority during this period can therefore create unnecessary or unsafe commands.

Possible system-level responses include:

• reduced control gains,
• limited translational thrust,
• attitude-only stabilization,
• hold mode,
• approach inhibition.

Navigation convergence and control-mode design should therefore be coordinated.

4.2.18.63 Initialization and Sensor Biases

An important distinction is

\[ \boxed{ \text{unknown bias} \neq \text{zero bias} } \]

Setting

\[ \hat b_0=0 \]

is merely an initial estimate.

The covariance must express the uncertainty:

\[ P_{b,0} = \sigma_{b,0}^2. \]

If the estimator is initialized with both

\[ \hat b_0=0 \]

and

\[ P_{b,0}\approx0, \]

it is effectively told that the sensor has almost exactly zero bias.

That can prevent correct bias estimation.

4.2.18.64 Initial Bias Error and Position Drift

For inertial navigation, accelerometer bias can integrate into velocity and position error.

Approximately,

\[ \delta v(t) \sim b_a t \]

and

\[ \delta r(t) \sim \frac12b_a t^2. \]

Similarly, gyro bias produces attitude error that can contaminate the transformation of measured acceleration.

Thus bias initialization can have long-term consequences far beyond the bias state itself.

4.2.18.65 Initial Attitude Error and Accelerometer Fusion

Suppose acceleration measured in the body frame is transformed using the estimated attitude:

\[ \hat{\mathbf a}^{I} = \hat{\mathbf C}_{IB}\mathbf a^B. \]

If

\[ \hat{\mathbf C}_{IB} \]

contains an initial attitude error, the acceleration is projected into incorrect inertial directions.

This can create velocity and position errors.

Therefore attitude initialization and translational navigation are coupled.

4.2.18.66 Clock and Timing Initialization

A multi-sensor fusion system may contain clock states:

\[ \Delta t, \qquad \dot{\Delta t}. \]

These also require initialization.

A timing error can cause

\[ \mathbf z(t+\Delta t) \]

to be compared with a prediction at

\[ t. \]

During rapid spacecraft motion, even a small timing error can resemble a position, velocity, or attitude error.

Accurate time synchronization can therefore significantly reduce the burden on filter convergence.

4.2.18.67 Frame Consistency at Initialization

Every initialized state must use the correct frame.

Examples include:

• ECI,
• ECEF,
• LVLH/Hill,
• body frame,
• sensor frame,
• target frame,
• docking frame.

For example,

\[ \mathbf r_{rel}^{H} = \mathbf C_{HI} ( \mathbf r_c^I-\mathbf r_t^I ). \]

Using an ECI relative state as though it were already in the Hill frame can create an apparent filter failure that is actually an initialization-frame error.

Frame verification should therefore be part of initialization testing.

4.2.18.68 Unit Consistency at Initialization

Similarly,

\[ \mathrm{m} \neq \mathrm{km}, \]

\[ \mathrm{rad} \neq \mathrm{deg}. \]

An attitude covariance of

\[ (1^\circ)^2 \]

must be converted correctly if the estimator expects radians:

\[ \sigma_\theta = \frac{\pi}{180}. \]

Then

\[ P_{\theta\theta} = \left(\frac{\pi}{180}\right)^2. \]

Incorrect covariance units can alter the Kalman gain dramatically without producing an obvious software error.

4.2.18.69 Quaternion Sign Consistency

Quaternions satisfy

\[ \mathbf q \equiv -\mathbf q. \]

Both represent the same physical attitude.

But direct subtraction of inconsistent quaternion signs can create an apparently enormous difference.

Before comparing two quaternions, the implementation may enforce

\[ \mathbf q_1^T\mathbf q_2>0. \]

If not,

\[ \mathbf q_2\leftarrow-\mathbf q_2. \]

This is especially important during initialization from multiple attitude sources.

4.2.18.70 Initialization of Relative Attitude

For docking,

\[ \mathbf C_{rel} = \mathbf C_{target}^{T} \mathbf C_{chaser} \]

depending on the chosen convention.

The relative quaternion can then be obtained from

\[ \mathbf C_{rel}. \]

A convention error can invert the relative attitude:

\[ \mathbf q_{TC} \neq \mathbf q_{CT}. \]

Therefore the definition of the relative quaternion must be fixed explicitly before filter initialization.

4.2.18.71 Initial Covariance Transformation Between Frames

Covariance must transform with the state.

If

\[ \mathbf x_B = \mathbf T\mathbf x_A, \]

then

\[ \boxed{ \mathbf P_B = \mathbf T \mathbf P_A \mathbf T^T } \]

for a linear transformation.

For nonlinear transformations,

\[ \mathbf P_B \approx \mathbf J \mathbf P_A \mathbf J^T. \]

Transforming the state but not its covariance creates an internally inconsistent estimator.

4.2.18.72 Initialization of Error-State Filters

An error-state filter separates

\[ \text{nominal state} \]

from

\[ \text{error state}. \]

For example,

\[ \mathbf x = (\mathbf r,\mathbf v,\mathbf q,\mathbf b_g,\mathbf b_a) \]

may be maintained as the nominal state, while

\[ \delta\mathbf x = \begin{bmatrix} \delta\mathbf r\\ \delta\mathbf v\\ \delta\boldsymbol\theta\\ \delta\mathbf b_g\\ \delta\mathbf b_a \end{bmatrix}. \]

At initialization, the nominal state receives the best estimate.

The error state is normally centred around

\[ \delta\hat{\mathbf x}_0=\mathbf0, \]

while

\[ \mathbf P_0 \]

describes the uncertainty around that nominal estimate.

This distinction is essential.

4.2.18.73 Error Injection and Reset

After an error-state update,

\[ \delta\hat{\mathbf x}^{+}, \]

the correction is injected into the nominal state:

\[ \hat{\mathbf x}^{+} = \hat{\mathbf x}^{-} \oplus \delta\hat{\mathbf x}^{+}. \]

For attitude,

\[ \hat{\mathbf q}^{+} = \delta\mathbf q \otimes \hat{\mathbf q}^{-}. \]

The error-state mean is then reset toward zero.

This reset operation may require a corresponding covariance transformation.

Initialization and reset logic must therefore use compatible error definitions.

4.2.18.74 Initialization of UKF Sigma Points

For an Unscented Kalman Filter, sigma points are generated from

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

and

\[ \mathbf P_0. \]

Conceptually,

\[ \chi_0^{(i)} = \hat{\mathbf x}_0 \pm \text{columns of } \sqrt{(n+\lambda)\mathbf P_0}. \]

Therefore unrealistic initial covariance produces unrealistic initial sigma-point dispersion.

For attitude states, sigma points must also respect the chosen manifold representation rather than treating quaternion components carelessly as ordinary Euclidean variables.

4.2.18.75 Particle-Filter Initialization

For a particle filter, initialization means defining the initial probability density.

Particles are drawn approximately as

\[ \mathbf x_0^{(i)} \sim p(\mathbf x_0). \]

If the true state lies outside the support of the initialized particle distribution, the filter may never recover adequately.

Thus particle-filter initialization requires a realistic prior distribution, not merely a mean and covariance.

4.2.18.76 Multimodal Initialization

Sometimes multiple initial states are plausible.

For example, visual pose estimation may produce several candidate orientations or feature correspondences.

A single Gaussian approximation may be inappropriate.

Instead,

\[ p(\mathbf x_0) = \sum_i w_i p_i(\mathbf x_0) \]

may better represent the uncertainty.

This is one reason global initialization and local tracking are sometimes handled by different algorithms.

4.2.18.77 Initialization From Batch Measurements

Instead of initializing from one measurement, several measurements can be processed together.

Given

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

one may solve

\[ \hat{\mathbf x}_0 = \arg\min_{\mathbf x} \sum_{k=1}^{N} \mathbf r_k^T \mathbf R_k^{-1} \mathbf r_k. \]

This provides a batch least-squares initialization.

The recursive filter can then begin from a more accurate state.

4.2.18.78 Batch-to-Recursive Transition

A common architecture is

\[ \boxed{ \text{Batch Initialization} \rightarrow \text{Recursive Filtering} } \]

The batch estimator uses a short measurement window to establish the initial state.

Then an EKF, UKF, or other recursive estimator takes over.

This is particularly useful when the initial geometry requires several measurements before the state becomes observable.

4.2.18.79 Bearing-Only Batch Initialization

A single bearing measurement cannot determine range.

But multiple bearing measurements from changing viewpoints can.

Therefore a camera-based relative navigation system may first collect

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

while the spacecraft moves.

The resulting geometry can provide an initial estimate of

\[ \mathbf r_{rel}. \]

Only then is the recursive relative-navigation filter initialized.

This avoids asking the filter to estimate a poorly constrained initial range from one image.

4.2.18.80 Initialization From LiDAR

LiDAR may directly provide range:

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

Depending on the sensor, it may also provide bearing, point clouds, or relative pose.

This can provide a stronger close-range initialization than bearing-only vision.

However, the LiDAR measurement must still be transformed correctly from the sensor frame into the navigation frame.

4.2.18.81 Initialization From Vision Pose

A camera observing known target features may estimate relative pose:

\[ \mathbf T_{TC} = \begin{bmatrix} \mathbf C_{TC} & \mathbf r_{TC}\\ 0 & 1 \end{bmatrix}. \]

This can initialize both

\[ \mathbf r_{rel} \]

and

\[ \mathbf q_{rel}. \]

But the covariance of a pose solution can be highly anisotropic.

For example, depth may be much less accurate than lateral position.

Therefore,

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

should reflect the geometry of the vision solution rather than being assumed isotropic.

4.2.18.82 Initialization Under Sensor Dropout

Suppose the preferred initialization sensor is unavailable.

The navigation architecture should define fallback paths.

For example,

\[ \text{Star tracker unavailable} \rightarrow \text{Sun sensor + magnetometer coarse attitude}. \]

Or

\[ \text{GNSS unavailable} \rightarrow \text{propagated orbit + optical navigation}. \]

A robust system therefore has an initialization hierarchy, not just one nominal initialization procedure.

4.2.18.83 Initialization and Redundancy

With redundant sensors,

\[ S_1,\quad S_2,\quad S_3, \]

the system can compare their measurements before accepting an initial solution.

For example,

\[ \|\mathbf z_1-\mathbf z_2\|<\gamma. \]

Agreement between independent sensors can increase confidence in initialization.

Disagreement can trigger FDIR before the estimator is allowed to enter normal navigation.

4.2.18.84 Initialization and Sensor Health

A measurement should generally not initialize a state merely because it is numerically available.

The sensor may need to satisfy

\[ \text{HEALTHY}=1, \]

\[ \text{VALID}=1, \]

\[ \text{TIME\_SYNC}=1. \]

Thus,

\[ \boxed{ \text{Measurement available} \neq \text{measurement suitable for initialization} } \]

This distinction is especially important after startup or reacquisition.

4.2.18.85 Initialization and Measurement Quality

Some sensors provide internal quality metrics.

Examples include:

• GNSS fix status,
• satellite geometry,
• star-tracker solution quality,
• feature count,
• reprojection error,
• LiDAR return quality,
• target-detection confidence.

These metrics can influence

\[ \mathbf P_0 \]

or determine whether initialization is accepted at all.

A fixed initial covariance may therefore be inferior to a quality-dependent covariance.

4.2.18.86 Adaptive Initial Covariance

Instead of assigning one fixed value,

\[ \mathbf P_0 = \mathbf P_{fixed}, \]

the covariance may depend on measurement quality:

\[ \mathbf P_0 = f(q_{\text{sensor}}). \]

For example, poor camera geometry may produce

\[ \sigma_{\rho,0} \gg \sigma_{xy,0}. \]

This gives the filter a more realistic description of its initial uncertainty.

4.2.18.87 Initialization and Constraints

Physical constraints can also help initialization.

Examples include

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

known docking geometry,

\[ \rho>0, \]

known orbital altitude ranges, or bounded sensor biases.

Constraints can prevent physically impossible initial states.

However, constraints should not be used to hide genuine uncertainty.

4.2.18.88 Initialization and Priors

Bayesian estimation treats initialization as specifying a prior:

\[ p(\mathbf x_0). \]

For a Gaussian prior,

\[ p(\mathbf x_0) = \mathcal N( \hat{\mathbf x}_0, \mathbf P_0 ). \]

The first measurements then update this prior into a posterior.

Thus the familiar EKF quantities

\[ \hat{\mathbf x}_0,\mathbf P_0 \]

are simply a Gaussian representation of prior knowledge.

4.2.18.89 Informative and Weak Priors

A small covariance represents an informative prior.

A large covariance represents a weak prior.

Conceptually,

\[ P_{ii,0}\downarrow \Rightarrow \text{strong prior confidence}, \]

while

\[ P_{ii,0}\uparrow \Rightarrow \text{weak prior confidence}. \]

The prior should reflect actual knowledge, not how strongly the designer wants the filter to trust a particular source.

4.2.18.90 Convergence Toward Steady-State Covariance

For a linear time-invariant system with appropriate observability and noise assumptions, the covariance may approach a steady solution:

\[ \mathbf P_k \rightarrow \mathbf P_\infty. \]

Correspondingly,

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

In spacecraft navigation, however, geometry, sensor availability, and dynamics often vary with time.

Therefore a single steady-state covariance may not exist.

Instead,

\[ \mathbf P_k \]

can change continuously throughout the mission.

4.2.18.91 Mission-Phase-Dependent Convergence

Different mission phases can produce very different convergence behaviour.

For spacecraft relative navigation:

\[ \text{Far Range} \rightarrow \text{GNSS-dominated}, \]

\[ \text{Mid Range} \rightarrow \text{vision + dynamics}, \]

\[ \text{Close Range} \rightarrow \text{LiDAR/vision-dominated}, \]

\[ \text{Precision Proximity} \rightarrow \text{high-accuracy relative pose}. \]

Each phase may therefore require its own:

• initialization logic,
• covariance limits,
• convergence criteria,
• sensor-validity criteria,
• maximum convergence time.

4.2.18.92 Reinitialization at Mission-Phase Transition

A phase transition does not always require resetting the estimator.

Often it is preferable to preserve the existing state and simply introduce new measurements.

However, reinitialization may be appropriate if:

• the state definition changes,
• the reference frame changes,
• the estimator architecture changes,
• a new relative target becomes the reference,
• a previous state has become invalid.

The key question is:

\[ \boxed{ \text{What information remains valid across the transition?} } \]

Only invalid information should be discarded.

4.2.18.93 Convergence Under Measurement Dropout

Suppose a filter is still converging when an important sensor drops out.

The covariance may stop contracting and begin growing:

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

Whether convergence can continue depends on the remaining sensor set and observability.

Thus:

\[ \boxed{ \text{Sensor dropout} \rightarrow \text{new observability} \rightarrow \text{new convergence behaviour} } \]

This links Sections 4.2.16, 4.2.17, and 4.2.18 directly.

4.2.18.94 Divergence Versus Slow Convergence

These should not be confused.

Slow convergence means

\[ \|\tilde{\mathbf x}_k\| \downarrow \]

but slowly.

Divergence means the estimation error grows or fails to remain bounded:

\[ \|\tilde{\mathbf x}_k\| \uparrow. \]

A weakly observable bias may converge slowly without indicating a filter failure.

A rapidly growing position error, by contrast, may indicate genuine divergence.

4.2.18.95 False Convergence

A filter can appear converged because

\[ \mathbf P_k \]

is small and state updates are small.

But if the actual state error remains large, the apparent convergence is false.

Possible causes include:

• wrong sensor model,
• incorrect frame,
• wrong measurement sign,
• unmodelled bias,
• incorrect covariance,
• measurement double counting,
• timing error.

Therefore

\[ \boxed{ \text{small covariance} \neq \text{proof of correct convergence} } \]

4.2.18.96 Filter Consistency After Convergence

After convergence, the estimator should remain statistically consistent.

For truth-based simulation,

\[ \mathrm{NEES} \]

can be monitored.

For flight data,

\[ \mathrm{NIS} \]

and innovation whiteness can be examined.

A filter that initially converges but subsequently becomes inconsistent still has a navigation-design problem.

Convergence is therefore not a one-time event.

4.2.18.97 Innovation Whiteness After Convergence

For a well-modelled filter, the innovation should ideally be approximately zero-mean and temporally uncorrelated:

\[ E[\mathbf y_k]\approx0, \]

\[ E[\mathbf y_k\mathbf y_{k-\tau}^T] \approx0, \qquad \tau\neq0. \]

Persistent correlation may indicate that the estimator is missing dynamics or bias states.

Thus innovation whiteness can reveal whether apparent convergence is genuinely healthy.

4.2.18.98 Convergence of Individual Covariance Eigenmodes

Examining only diagonal elements of

\[ \mathbf P \]

can hide coupled uncertainty.

Eigenvalue decomposition gives

\[ \mathbf P = \mathbf V\boldsymbol\Lambda\mathbf V^T. \]

Each eigenvector represents an uncertainty direction in state space.

A slowly decreasing eigenvalue may identify a weakly constrained combination of states.

This connects covariance convergence directly to observability singular directions.

4.2.18.99 Numerical Conditioning During Initialization

Very different state scales can produce poor numerical conditioning.

For example,

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

while

\[ b_g\sim10^{-5}\text{ rad/s}. \]

Corresponding covariance entries may differ by many orders of magnitude.

State scaling, square-root filtering, or numerically stable covariance implementations may therefore be important.

A mathematically valid \mathbf P_0 can still create numerical difficulties if poorly scaled.

4.2.18.100 Positive Semidefinite Initial Covariance

A covariance matrix must satisfy

\[ \mathbf P_0 = \mathbf P_0^T \]

and

\[ \mathbf P_0\succeq0. \]

Negative eigenvalues are physically meaningless for a covariance matrix.

Implementation checks should therefore verify:

\[ \lambda_i(\mathbf P_0)\ge0. \]

Small negative eigenvalues caused by numerical roundoff may require careful numerical handling, but a substantially indefinite covariance indicates an initialization error.

4.2.18.101 Joseph-Form Covariance Update

During convergence, numerical covariance quality can be improved using the Joseph form:

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

This form better preserves covariance symmetry and positive semidefiniteness under finite-precision arithmetic than some simplified implementations.

This is particularly useful when covariance changes rapidly during startup.

4.2.18.102 Square-Root Filters and Initialization

A square-root filter represents

\[ \mathbf P = \mathbf S\mathbf S^T. \]

Initialization therefore requires a valid factorization of

\[ \mathbf P_0. \]

For example,

\[ \mathbf S_0 = \operatorname{chol}(\mathbf P_0). \]

If Cholesky factorization fails, \mathbf P_0 may not be positive definite or may be numerically ill-conditioned.

This provides a useful initialization diagnostic.

4.2.18.103 Initialization of Process-Noise States

Some estimators augment the state with slowly varying disturbances:

\[ \mathbf x = \begin{bmatrix} \cdots\\ \mathbf a_{dist}\\ \boldsymbol\tau_{dist} \end{bmatrix}. \]

Their initial estimates may be zero:

\[ \hat{\mathbf a}_{dist,0}=0, \]

but their covariance should represent the expected uncertainty.

These states often converge only if the sensor suite can distinguish external disturbances from sensor biases and modelling errors.

4.2.18.104 Initialization of Orbital Parameters

Instead of Cartesian position and velocity, an estimator may use orbital elements or other orbital-state representations.

Then initialization may involve

\[ (\mathbf r,\mathbf v) \rightarrow (a,e,i,\Omega,\omega,\nu). \]

The associated covariance requires a Jacobian transformation:

\[ \mathbf P_{OE} = \mathbf J \mathbf P_{rv} \mathbf J^T. \]

Near singular orbital-element configurations, alternative state representations may be preferable.

4.2.18.105 Initialization in Hill Coordinates

For spacecraft relative navigation,

\[ \mathbf x_H = \begin{bmatrix} x&y&z&\dot x&\dot y&\dot z \end{bmatrix}^T. \]

If absolute inertial states are known, relative Hill coordinates can be constructed.

But relative velocity requires more than simply rotating

\[ \mathbf v_c-\mathbf v_t. \]

Because the Hill frame rotates,

\[ \mathbf v_{rel}^{H} = \mathbf C_{HI} (\mathbf v_c^I-\mathbf v_t^I) - \boldsymbol\omega_{HI}^{H} \times \mathbf r_{rel}^{H}. \]

This term is important for correct relative navigation initialization.

4.2.18.106 Initialization and Dynamic-Model Consistency

The initial state should be compatible with the dynamic model.

For example, if an relative navigation filter assumes Clohessy-Wiltshire dynamics around a circular reference orbit, its initial state and reference mean motion

\[ n \]

must correspond to that reference orbit.

Otherwise the filter begins with a model mismatch from the first propagation step.

Initialization therefore includes relevant model parameters, not just navigation states.

4.2.18.107 Initialization of Environmental Parameters

Some filters estimate or use parameters such as:

• atmospheric density scale,
• drag coefficient,
• solar-radiation-pressure coefficient,
• gravity-model corrections.

If these are part of the estimated state,

\[ \mathbf x = [\cdots,C_D,C_R,\rho_s,\ldots]^T, \]

their priors and covariances must be initialized.

Their convergence may be much slower than position or velocity because their effects accumulate gradually through dynamics.

4.2.18.108 Initialization and Ballistic Coefficient

For orbit determination or re-entry estimation,

\[ BC = \frac{m}{C_DA} \]

may be estimated.

An initial value

\[ \widehat{BC}_0 \]

can come from vehicle design data, previous orbit determination, or another estimator.

Its covariance should reflect uncertainty in

\[ m,\ C_D,\ A \]

and model mismatch.

Because ballistic coefficient affects the dynamics indirectly through drag, convergence strongly depends on atmospheric density and orbital regime.

4.2.18.109 Convergence Under Model Mismatch

Even a perfectly initialized filter can converge incorrectly if

\[ f_{\text{model}} \neq f_{\text{truth}}. \]

Examples include:

• incorrect gravity,
• unmodelled drag,
• thruster execution errors,
• target manoeuvres,
• flexible-body effects,
• incorrect sensor lever arms.

The estimator may compensate by shifting bias or calibration states.

Therefore convergence of the numerical filter does not automatically prove that the underlying physical model is correct.

4.2.18.110 Initialization of Sensor Lever Arms

Suppose a GNSS antenna or camera is offset from the spacecraft centre of mass by

\[ \mathbf r_{SB}. \]

The sensor position is

\[ \mathbf r_S = \mathbf r_B + \mathbf C\mathbf r_{SB}. \]

If the lever arm is ignored during initialization, the filter may interpret the geometric offset as a navigation error.

For precision relative navigation, sensor lever arms can become significant relative to centimetre-level requirements.

4.2.18.111 Initialization of Sensor Misalignment

A sensor mounting matrix may be represented as

\[ \mathbf C_{SB}. \]

If misalignment is estimated,

\[ \delta\boldsymbol\theta_{SB} \]

requires an initial estimate and covariance.

A reasonable prior may come from mechanical alignment measurements.

Poorly initialized mounting states can contaminate attitude or relative-pose estimates until sufficient motion makes the misalignment observable.

4.2.18.112 Initialization of Camera Intrinsics

Vision-based navigation may depend on

\[ f_x,\quad f_y,\quad c_x,\quad c_y \]

and distortion parameters.

These are normally calibrated before flight.

If they are estimated online, their initialization influences the conversion

\[ (X,Y,Z) \rightarrow (u,v). \]

Poor focal-length initialization, for example, can resemble range or depth error.

Calibration-state convergence should therefore be analysed jointly with navigation-state observability.

4.2.18.113 Initialization and Target Model

Pose estimation may require known target geometry:

\[ \mathbf p_i^T. \]

Errors in target dimensions or feature coordinates can produce systematic pose errors.

Therefore initialization should include verification that the correct target model has been loaded.

This becomes particularly important when a chaser may rendezvous with multiple target types.

4.2.18.114 Initialization of Data Association

Before a vision or LiDAR filter can initialize relative pose, measurements must correspond to the correct physical features.

Incorrect initial data association can create a highly confident but incorrect state.

Thus

\[ \boxed{ \text{Correct association} \rightarrow \text{valid initialization} } \]

is a prerequisite for feature-based navigation.

Multiple-hypothesis initialization may be appropriate when association is ambiguous.

4.2.18.115 Initialization and Outlier Rejection

The first measurements can themselves contain outliers.

Therefore initialization should not blindly trust a single sample.

Possible approaches include:

• temporal averaging,
• median statistics,
• RANSAC,
• multi-sensor agreement,
• quality thresholds,
• batch robust estimation.

This connects the robust-fusion principles of Section 4.2.16 with initialization.

4.2.18.116 Initialization From Averaged Measurements

For a stationary or slowly varying quantity,

\[ \bar{\mathbf z} = \frac1N \sum_{k=1}^{N} \mathbf z_k. \]

If measurement noise is independent,

\[ \operatorname{Var}(\bar z) = \frac{\sigma^2}{N}. \]

Thus averaging can provide a better initialization estimate.

But it is only valid if the underlying state remains sufficiently constant during the averaging window.

4.2.18.117 Initialization Latency

Collecting more measurements improves initial accuracy but delays estimator availability.

This creates a trade:

\[ \boxed{ \text{Initialization accuracy} \leftrightarrow \text{Initialization latency} } \]

For time-critical relative navigation or autonomous recovery, waiting too long for a highly accurate initialization may itself be undesirable.

The required initialization quality should therefore be linked to mission timing.

4.2.18.118 Fast Acquisition Versus Fine Convergence

A useful architecture separates

\[ \text{fast acquisition} \]

from

\[ \text{fine convergence}. \]

The first stage produces a solution adequate for basic stabilization or hold.

The second stage gradually reaches the precision required for approach or precision proximity operations.

For example,

\[ \boxed{ \text{Coarse NAV valid} \rightarrow \text{Fine NAV valid} } \]

can be more operationally useful than a single binary validity state.

4.2.18.119 Multi-Level Navigation Validity

A spacecraft may define

\[ \text{NAV\_LEVEL}_0 = \text{invalid}, \]

\[ \text{NAV\_LEVEL}_1 = \text{coarse}, \]

\[ \text{NAV\_LEVEL}_2 = \text{nominal}, \]

\[ \text{NAV\_LEVEL}_3 = \text{precision}. \]

Different guidance modes can require different levels.

For example,

\[ \text{Hold} \Rightarrow \text{LEVEL}_1, \]

while

\[ \text{Precision Proximity} \Rightarrow \text{LEVEL}_3. \]

This directly connects estimator convergence to mission autonomy.

4.2.18.120 Convergence Monitoring by State Group

Instead of one global flag, monitor groups:

\[ C_{trans}, \qquad C_{att}, \qquad C_{bias}, \qquad C_{cal}. \]

For example,

\[ C_{trans} = (\sigma_r

Similarly,

\[ C_{att} = (\sigma_\theta<\theta_{req}) \land (\sigma_\omega<\omega_{req}). \]

This reveals which part of the navigation solution is preventing mission readiness.

4.2.18.121 Initialization Failure Modes

Common initialization failures include:

• wrong state units,
• wrong coordinate frame,
• incorrect quaternion convention,
• covariance too small,
• covariance too large,
• invalid covariance matrix,
• poor sensor geometry,
• uninitialized bias states,
• timestamp mismatch,
• incorrect sensor health status,
• outlier used as initial state,
• stale previous solution,
• incorrect target identity,
• unobservable augmented states.

These should be explicitly tested rather than discovered only during integrated mission simulations.

4.2.18.122 Filter Convergence Failure Modes

Convergence may fail because of:

• insufficient observability,
• excessive initial error,
• nonlinear linearization failure,
• incorrect Q,
• incorrect R,
• incorrect measurement Jacobian,
• wrong measurement model,
• sensor bias,
• data association failure,
• persistent outliers,
• timing errors,
• covariance inconsistency.

A convergence problem is therefore not automatically solved by changing Kalman gains.

The underlying cause must be identified.

4.2.18.123 Initialization Verification Tests

A useful verification campaign should vary

\[ \delta\mathbf x_0. \]

Test cases should include:

• nominal initialization,
• maximum expected position error,
• maximum expected velocity error,
• maximum attitude error,
• maximum bias error,
• combined errors,
• delayed sensor acquisition,
• sensor dropout during initialization,
• outlier at startup,
• corrupted previous solution,
• frame transformation errors.

The objective is to establish the practical convergence envelope.

4.2.18.124 Covariance Initialization Verification

For each Monte Carlo run i,

\[ \tilde{\mathbf x}_0^{(i)} = \mathbf x_0^{truth,(i)} - \hat{\mathbf x}_0^{(i)}. \]

The empirical initial covariance is

\[ \mathbf P_{emp} = \frac{1}{N-1} \sum_i ( \tilde{\mathbf x}_0^{(i)} - \bar{\tilde{\mathbf x}}_0 ) ( \tilde{\mathbf x}_0^{(i)} - \bar{\tilde{\mathbf x}}_0 )^T. \]

Compare

\[ \mathbf P_{emp} \]

with the implemented

\[ \mathbf P_0. \]

This provides a quantitative way to validate initial covariance assumptions.

4.2.18.125 Convergence Requirement Example

A relative-navigation filter requirement might state:

\[ P \left( T_{\text{conv}}<30\,s \right) \ge0.99 \]

for all initial errors within the specified initialization envelope.

A close-range requirement might additionally require

\[ \sigma_r<0.01\text{ m}, \]

\[ \sigma_v<0.005\text{ m/s}, \]

\[ \sigma_\theta<0.1^\circ. \]

The exact numerical values are mission-specific.

The important point is that convergence should be specified quantitatively.

4.2.18.126 Example: Attitude Filter Initialization

Consider

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

The nominal quaternion is initialized from the star tracker:

\[ \hat{\mathbf q}_0 = \mathbf q_{ST}. \]

Set

\[ \delta\hat{\boldsymbol\theta}_0 = \mathbf0. \]

Gyro bias may initially be

\[ \hat{\mathbf b}_{g,0} = \mathbf0. \]

The covariance may be

\[ \mathbf P_0 = \begin{bmatrix} \sigma_\theta^2\mathbf I & \mathbf0\\ \mathbf0 & \sigma_{b_g}^2\mathbf I \end{bmatrix}. \]

The filter then uses gyro propagation and star-tracker updates to refine attitude and estimate bias.

4.2.18.127 Example: GNSS/IMU Initialization

A navigation state may be

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

A possible initialization is

\[ \hat{\mathbf r}_0 = \mathbf r_{GNSS}, \]

\[ \hat{\mathbf v}_0 = \mathbf v_{GNSS}, \]

\[ \hat{\mathbf q}_0 = \mathbf q_{ST}, \]

\[ \hat{\mathbf b}_{g,0} = \mathbf b_{g,cal}, \]

\[ \hat{\mathbf b}_{a,0} = \mathbf b_{a,cal}. \]

Then

\[ \mathbf P_0 \]

is assembled from GNSS, attitude, and calibration uncertainties.

This is much more physically meaningful than selecting arbitrary diagonal values.

4.2.18.128 Example: Camera-Only relative navigation Initialization

Suppose only bearing measurements are initially available.

The camera gives

\[ u=f_x\frac{X}{Z}+c_x, \]

\[ v=f_y\frac{Y}{Z}+c_y. \]

A single image does not determine Z uniquely.

Therefore range may require:

• a prior range estimate,
• target apparent size,
• known target geometry,
• multiple images,
• relative motion.

A reasonable architecture may be

\[ \text{coarse prior range} + \text{camera bearing} \rightarrow \text{initial state} \]

followed by motion-induced range convergence.

4.2.18.129 Example: LiDAR-Assisted relative navigation Initialization

Suppose LiDAR provides range and camera provides bearing.

Then

\[ \rho = \|\mathbf r_{rel}\|, \]

and

\[ \hat{\mathbf u} = \frac{\mathbf r_{rel}} {\|\mathbf r_{rel}\|}. \]

Relative position can be initialized approximately as

\[ \hat{\mathbf r}_{rel,0} = \rho\hat{\mathbf u}. \]

This demonstrates a central sensor-fusion principle:

\[ \boxed{ \text{Complementary sensors can simplify initialization} } \]

as well as improve steady-state estimation.

4.2.18.130 Example: Sensor Reacquisition During relative navigation

Suppose LiDAR is temporarily lost during final approach.

The estimator propagates using dynamics and IMU.

Then

\[ \mathbf P_{rel} \]

grows.

When LiDAR returns, the system first validates the measurement.

If the residual is reasonable, normal fusion resumes.

If the residual is very large, the system may enter

\[ \text{REACQUIRE} \]

rather than immediately applying a large correction.

This protects the navigation state against false target reacquisition.

4.2.18.133 Initialization and Abort Logic

If the navigation filter fails to converge before a mission deadline,

\[ T_{\text{conv}} > T_{\max}, \]

the system should not wait indefinitely.

Possible actions include:

\[ \text{Hold}, \]

\[ \text{Retreat}, \]

\[ \text{Reinitialize}, \]

or

\[ \text{Abort}. \]

Thus maximum convergence time can become a mission-level safety requirement.

4.2.18.134 Convergence-Aware Guidance

Guidance can also help the estimator converge.

Suppose range is weakly observable during a radial camera-only approach.

A lateral manoeuvre can improve geometry.

Therefore guidance may deliberately generate

\[ \mathbf u_{excite} \]

to improve estimation.

This gives

\[ \boxed{ \text{Guidance} \rightarrow \text{better observability} \rightarrow \text{faster convergence} } \]

which is an important concept for autonomous relative navigation.

4.2.18.135 Initialization and Autonomous Navigation

For autonomous spacecraft, initialization logic must answer:

1. Which sensors are available?
2. Which sensors are healthy?
3. What states can currently be initialized?
4. What uncertainty should be assigned?
5. Is the geometry adequate?
6. Has enough data been collected?
7. Is the resulting state statistically plausible?
8. Can recursive filtering begin?
9. Has the filter converged?
10. Is the navigation solution accurate enough for the current mission phase?

This shows why initialization is a complete subsystem behaviour rather than one line of estimator code.

4.2.18.136 Filter Initialization Architecture

A practical architecture can be represented as

\[ \boxed{ \begin{array}{c} \text{Sensor Acquisition}\\ \downarrow\\ \text{Health / Validity Checks}\\ \downarrow\\ \text{Time Synchronization}\\ \downarrow\\ \text{Frame Transformation}\\ \downarrow\\ \text{Coarse State Determination}\\ \downarrow\\ \hat{\mathbf x}_0,\mathbf P_0\\ \downarrow\\ \text{Recursive Estimator}\\ \downarrow\\ \text{Convergence Monitor}\\ \downarrow\\ \text{NAV VALID} \end{array} } \]

Parallel paths should monitor:

\[ \text{FDIR}, \qquad \text{sensor quality}, \qquad \text{innovation consistency}. \]

4.2.18.137 Initialization Verification Architecture

For simulation and V&V:

\[ \boxed{ \begin{array}{c} \text{Truth Initial State}\\ + \text{Random Initial Error}\\ \downarrow\\ \text{Initialization Algorithm}\\ \downarrow\\ \text{Filter}\\ \downarrow\\ \text{Truth Error + Covariance + Innovation}\\ \downarrow\\ \text{Convergence Detection}\\ \downarrow\\ T_{\text{conv}} \end{array} } \]

Repeating this through Monte Carlo produces the statistical convergence performance of the navigation system.

4.2.18.138 What Should Be Plotted During Convergence Testing?

At minimum, plot:

• true state and estimated state,
• state estimation error,
• 1\sigma, 2\sigma, or 3\sigma covariance bounds,
• innovations,
• NIS,
• Kalman gain where useful,
• bias estimates,
• sensor availability,
• measurement acceptance/rejection,
• navigation-valid flags,
• mission-mode transitions.

For example,

\[ \tilde r_x \]

should be plotted together with

\[ \pm3\sqrt{P_{xx}}. \]

A statistically consistent estimator should not systematically violate its predicted uncertainty bounds.

4.2.18.139 Convergence Plots for Bias States

For gyro bias,

\[ \tilde{\mathbf b}_g = \mathbf b_g^{truth} - \hat{\mathbf b}_g. \]

Plot

\[ \tilde b_{g,x} \]

with

\[ \pm3\sqrt{P_{b_gx,b_gx}}. \]

This reveals whether the bias estimate truly converges or whether the filter merely reports decreasing uncertainty.

Bias-state plots are particularly valuable because these states may converge much more slowly than attitude.

4.2.18.140 Convergence Plots for relative navigation

For spacecraft relative navigation, useful quantities include

\[ \|\tilde{\mathbf r}_{rel}\|, \]

\[ \|\tilde{\mathbf v}_{rel}\|, \]

\[ \|\delta\boldsymbol\theta_{rel}\|, \]

\[ \|\tilde{\boldsymbol\omega}_{rel}\|. \]

These should be compared with both covariance predictions and mission requirements.

For example,

\[ \|\tilde{\mathbf r}_{rel}\| < r_{\text{req}} \]

may be required before final approach.

4.2.18.141 Initialization Test Matrix

A useful test matrix can vary:

4.2.18.142 Initialization Success Rate

Across Monte Carlo runs,

\[ P_{init} = \frac{N_{successful\ initialization}} {N_{total}}. \]

Similarly,

\[ P_{conv} = \frac{N_{converged}} {N_{total}}. \]

These are different metrics.

A filter may initialize successfully but fail to converge within the required time.

Therefore both should be reported.

4.2.18.143 Maximum Recoverable Initial Error

Testing can identify approximately

\[ r_{0,\max}, \qquad v_{0,\max}, \qquad \theta_{0,\max} \]

for reliable convergence.

This helps define upstream sensor requirements.

For example, if the relative-navigation filter reliably converges only when

\[ |\delta r_0|<10\text{ m}, \]

then the acquisition system must provide an initial range estimate inside that envelope.

This creates a direct link between filter design and sensor requirements.

4.2.18.144 Initialization Requirement Flow-Down

A mission-level requirement such as

\[ \text{navigation valid within }T_{\max} \]

can flow down to:

\[ \text{initial position accuracy}, \]

\[ \text{initial attitude accuracy}, \]

\[ \text{sensor acquisition time}, \]

\[ \text{minimum measurement rate}, \]

\[ \text{maximum sensor bias}, \]

\[ \text{required geometry}. \]

Initialization analysis therefore contributes directly to system requirements.

4.2.18.145 Initialization and Sensor Selection

Two sensors with similar steady-state accuracy may have very different acquisition behaviour.

For example:

• one sensor may provide an immediate absolute measurement,
• another may require motion before the state becomes observable.

Therefore sensor trade studies should consider

\[ \boxed{ \text{steady-state accuracy} + \text{initialization capability} + \text{convergence time} } \]

rather than steady-state accuracy alone.

4.2.18.146 Initialization and Update Scheduling

During convergence, some measurements may be more useful than others.

A scheduler may prioritize:

\[ \text{absolute attitude}, \]

\[ \text{absolute position}, \]

or other measurements that rapidly constrain large initial uncertainty.

After convergence, scheduling priorities may change toward power saving or routine tracking.

Thus estimator maturity can influence sensor scheduling.

4.2.18.147 Initialization and Computational Load

Batch initialization, image processing, feature matching, and large covariance updates may create a temporary computational peak.

Therefore initialization must be tested for:

• CPU load,
• memory,
• execution latency,
• measurement queueing,
• real-time deadlines.

A theoretically excellent initialization algorithm is not useful if it prevents the flight computer from meeting real-time requirements.

4.2.18.148 Initialization and Delayed Measurements

Suppose a measurement generated at

\[ t_m \]

arrives at

\[ t_k>t_m. \]

If it is incorrectly treated as a current-time measurement, the initial state may be biased.

Possible solutions include:

• state buffering,
• delayed-state updates,
• propagation to a common epoch,
• timestamp-aware initialization.

All measurements used to initialize the filter should refer to a consistent estimation epoch.

4.2.18.149 Common-Epoch Initialization

Suppose GNSS position is available at t_1 and star-tracker attitude at t_2.

Before constructing

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

the measurements should be referred to a common time

\[ t_0. \]

This may require propagation:

\[ \mathbf r(t_1) \rightarrow \mathbf r(t_0), \]

\[ \mathbf q(t_2) \rightarrow \mathbf q(t_0). \]

Otherwise the initial state can contain an internal time inconsistency.

4.2.18.150 Initialization of Asynchronous Sensor Fusion

A spacecraft may have:

\[ f_{\text{IMU}}=100\text{ Hz}, \]

\[ f_{\text{GNSS}}=10\text{ Hz}, \]

\[ f_{\text{camera}}=5\text{ Hz}, \]

\[ f_{\text{LiDAR}}=20\text{ Hz}. \]

Initialization does not require simultaneous measurements.

But the estimator must maintain correct timestamps and propagate all information to the appropriate estimation epoch.

Thus asynchronous initialization is fundamentally a timing and propagation problem.

4.2.18.151 Initialization and Measurement Correlation

Suppose two initial estimates originate from the same underlying measurement.

Treating them as independent can double-count information.

For example,

\[ \hat{\mathbf r}_1 \]

and

\[ \hat{\mathbf r}_2 \]

may both derive from the same GNSS solution.

Combining them as independent measurements produces an artificially small covariance.

This is the same correlation problem encountered in decentralized sensor fusion.

4.2.18.152 Initialization From Federated Filters

In a federated architecture, local filters may provide

\[ \hat{\mathbf x}_i,\mathbf P_i. \]

The master filter can use these to establish its initial state.

But cross-correlation must be handled correctly.

If the local estimates are correlated, simple information addition

\[ \mathbf P^{-1} = \sum_i\mathbf P_i^{-1} \]

can become overconfident.

Thus initialization inherits the architectural correlation issues discussed earlier in sensor fusion.

4.2.18.153 Initialization and Covariance Intersection

When cross-correlation is unknown, covariance intersection may provide a conservative fusion approach.

For two estimates,

\[ \mathbf P^{-1} = \omega\mathbf P_1^{-1} + (1-\omega)\mathbf P_2^{-1}, \]

with

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

This can be useful when initializing from multiple estimates whose correlation cannot be reliably reconstructed.

4.2.18.154 Convergence and Adaptive Sensor Fusion

During convergence, the estimator may dynamically change measurement weights.

For example, if camera quality improves as range decreases,

\[ \mathbf R_{\text{camera}} \]

may be reduced.

Then the camera contributes more strongly to the solution.

Adaptive weighting can improve convergence, but the quality-to-covariance mapping must be validated carefully.

4.2.18.155 Convergence and Robust Estimation

Large startup residuals do not necessarily indicate sensor faults.

A robust estimator must distinguish between:

\[ \text{large residual due to initial state error} \]

and

\[ \text{large residual due to bad measurement}. \]

This is difficult because both can produce similar innovations.

Multi-sensor consistency, temporal persistence, and initialization mode information can help separate them.

4.2.18.156 Initialization and FDIR Interaction

FDIR should know whether the filter is:

\[ \text{INITIALIZING}, \quad \text{CONVERGING}, \quad \text{NOMINAL}. \]

Thresholds appropriate for steady-state operation may produce false alarms during startup.

However, startup mode should not suppress genuine faults.

The FDIR design should therefore include mode-dependent but bounded thresholds.

4.2.18.157 Convergence and Navigation Integrity

Navigation integrity asks whether the system can bound its error with sufficiently high confidence.

During convergence,

\[ \mathbf P \]

may still be too large for the mission.

Therefore a numerically stable filter may still be unavailable for safety-critical guidance.

This gives

\[ \boxed{ \text{Estimator running} \neq \text{navigation operationally valid} } \]

and

\[ \boxed{ \text{navigation valid} \neq \text{precision navigation valid} } \]

unless the required uncertainty and integrity criteria are satisfied.

4.2.18.158 Protection Levels During Convergence

A system may derive protection levels such as

\[ PL_r = K\sigma_r. \]

Even if the estimated position is near the desired trajectory, guidance may remain inhibited if

\[ PL_r > AL_r, \]

where AL_r is the alert limit.

Thus uncertainty convergence can be tied directly to integrity monitoring.

4.2.18.159 Convergence and Fault Detection Sensitivity

At startup, covariance may be large.

Then innovation covariance

\[ \mathbf S \]

is also often large.

A given residual may therefore produce a smaller NIS.

As the filter converges,

\[ \mathbf S \]

contracts, and the same residual becomes statistically more significant.

Therefore fault-detection sensitivity naturally changes during convergence.

This is another reason to analyse initialization and FDIR together.

4.2.18.160 Initialization and Sensor Saturation

If a sensor begins operation outside its valid range, its measurement may not be suitable for initialization.

Examples include:

• camera target outside FOV,
• LiDAR beyond range,
• gyro saturation,
• star tracker blinded,
• GNSS unavailable.

The initialization manager should therefore understand sensor operating envelopes.

A numerical value alone does not guarantee a physically valid measurement.

4.2.18.161 Initialization After Safe Mode

After spacecraft safe mode, attitude, rates, and orbital state may be known with different levels of confidence.

A recovery sequence may be:

\[ \text{Sun acquisition} \rightarrow \text{rate damping} \rightarrow \text{coarse attitude} \rightarrow \text{star tracker acquisition} \rightarrow \text{navigation reinitialization}. \]

This is a practical example of staged filter initialization.

4.2.18.162 Initialization After Processor Reset

After a flight-computer reset, some state information may survive in non-volatile memory.

The estimator should determine whether the stored state is still usable.

If the stored solution is

\[ \hat{\mathbf x}(t_s) \]

and the current time is

\[ t_0, \]

it must be propagated to

\[ t_0. \]

Its covariance must also grow appropriately.

Simply restoring the old state and covariance unchanged can create severe overconfidence.

4.2.18.163 Initialization After Target Reacquisition

During relative navigation, the target may temporarily leave the camera FOV.

After reacquisition, the detected object must first be confirmed as the correct target.

Then the new measurement is compared with the propagated relative state.

If consistent,

\[ \text{tracking resumes}. \]

If not,

\[ \text{target association / navigation reset} \]

may be required.

Thus target identity is part of relative-navigation initialization.

4.2.18.164 Convergence After Sensor Configuration Change

Suppose the filter switches from

\[ \{\text{GNSS, IMU}\} \]

to

\[ \{\text{Camera, LiDAR, IMU}\}. \]

The state may remain continuous, but covariance behaviour can change sharply.

Some states may become much more observable.

Others may lose absolute references.

The estimator should therefore be monitored for reconvergence after configuration changes, not only after initial startup.

4.2.18.165 Repeated Convergence Events

A spacecraft mission may experience many convergence events:

\[ \text{startup}, \]

\[ \text{sensor acquisition}, \]

\[ \text{sensor handover}, \]

\[ \text{post-manoeuvre}, \]

\[ \text{post-outage}, \]

\[ \text{safe-mode recovery}, \]

\[ \text{target reacquisition}. \]

Thus convergence should be treated as a recurring system behaviour.

4.2.18.166 Convergence After Manoeuvres

Large thruster burns can temporarily increase uncertainty because of:

• thrust magnitude error,
• thrust-direction error,
• valve timing,
• MIB,
• misalignment,
• centre-of-mass uncertainty.

After the burn,

\[ \mathbf P^- \]

may increase.

Subsequent navigation measurements reconstrain the state.

The time required to recover required accuracy is a form of post-manoeuvre convergence time.

4.2.18.167 relative navigation Thruster-Execution Example

Suppose the commanded relative velocity change is

\[ \Delta\mathbf v_c. \]

Actual execution is

\[ \Delta\mathbf v_a = \Delta\mathbf v_c + \delta\Delta\mathbf v. \]

After the burn,

\[ \mathbf v_{rel}^{+} = \mathbf v_{rel}^{-} + \Delta\mathbf v_a. \]

If the estimator assumes perfect execution, the unknown

\[ \delta\Delta\mathbf v \]

appears as an immediate velocity estimation error.

Process-noise modelling or explicit manoeuvre uncertainty should therefore be included so that the filter can reconverge correctly.

4.2.18.168 Initialization and Manoeuvre Covariance

At an impulsive manoeuvre,

\[ \mathbf P^{+} = \mathbf F_m \mathbf P^{-} \mathbf F_m^T + \mathbf Q_m, \]

where

\[ \mathbf Q_m \]

represents manoeuvre-execution uncertainty.

If

\[ \mathbf Q_m=0 \]

despite uncertain thrust, the filter may become overconfident immediately after the burn.

This can lead to measurement rejection and slow recovery.

4.2.18.169 Convergence After Docking Contact

At physical contact, the dynamics change abruptly.

Before contact:

\[ \text{free-flying relative dynamics}. \]

After contact:

\[ \text{contact / coupled-body dynamics}. \]

A navigation filter based on free-flight dynamics may no longer be appropriate.

The estimator may therefore transition to a new mode or state model.

This is another example where initialization and convergence accompany a change in physical regime.

4.2.18.170 Initialization of Target Angular Motion

For spacecraft relative navigation with a rotating target,

\[ \mathbf x = [ \mathbf r_{rel}, \mathbf v_{rel}, \mathbf q_{rel}, \boldsymbol\omega_t ]^T. \]

Target angular rate may initially be uncertain.

Camera feature motion over several frames can estimate

\[ \boldsymbol\omega_t. \]

Therefore relative-attitude initialization may require a short tracking period rather than one image.

4.2.18.171 Initialization and Tumble Estimation

For an uncooperative tumbling target, the initial state may include:

\[ \mathbf q_t, \qquad \boldsymbol\omega_t. \]

The attitude can sometimes be estimated from one pose observation, but angular rate requires temporal information.

Approximately,

\[ \boldsymbol\omega_t \sim \frac{\Delta\boldsymbol\theta}{\Delta t}. \]

Thus the estimator may require several observations before declaring target rotational state initialized.

4.2.18.172 Initialization and Feature Map Building

For unknown targets, vision navigation may need to estimate both spacecraft state and feature locations.

Then

\[ \mathbf x = \begin{bmatrix} \mathbf x_{nav}\\ \mathbf p_1\\ \mathbf p_2\\ \vdots \end{bmatrix}. \]

Initialization becomes a SLAM-like problem.

Feature states must be initialized with realistic depth uncertainty, especially for monocular vision.

This substantially increases the importance of observability and covariance consistency.

4.2.18.173 Initialization of New Features

When a new camera feature first appears, its bearing may be known but its depth may not.

Therefore the feature should not necessarily be initialized with an artificially precise Cartesian position.

Possible representations include inverse depth:

\[ \lambda = \frac1\rho. \]

This allows large depth uncertainty to be represented more naturally.

The general lesson is:

\[ \boxed{ \text{state representation affects initialization quality} } \]

4.2.18.174 Initialization and Parameter Constraints

Some parameters have known physical bounds.

For example,

\[ s_g\in[s_{\min},s_{\max}], \]

or

\[ C_D>0. \]

An unconstrained Gaussian estimator can temporarily produce values outside physically meaningful ranges during convergence.

Possible solutions include transformed states, constrained filtering, or bounded initialization.

The method should preserve statistical consistency as much as possible.

4.2.18.175 Initialization and Normalization Constraints

Several states require normalization.

For quaternions,

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

For direction vectors,

\[ \|\mathbf u\|=1. \]

If such states are initialized from noisy measurements, normalization should be performed carefully.

The covariance should ideally remain consistent with the constrained representation.

4.2.18.176 Initialization of Relative Range Rate

If range measurements are available,

\[ \rho_k, \]

range rate can be approximated as

\[ \dot\rho \approx \frac{\rho_k-\rho_{k-1}} {\Delta t}. \]

But this can be noisy.

A better approach may be to initialize range and allow the dynamic filter to estimate range rate over several measurements.

This is another example of deciding which states must be known immediately and which can converge online.

4.2.18.177 Initialization Quality Versus State Dimension

As state dimension increases,

\[ n_x\uparrow, \]

the initialization problem becomes more difficult.

Each additional state requires:

• an initial estimate,
• an initial variance,
• possible cross-covariance,
• sufficient observability,
• convergence verification.

Therefore,

\[ \boxed{ \text{Do not augment the filter merely because a parameter exists} } \]

unless its estimation is operationally useful and sufficiently observable.

4.2.18.178 Initialization and Schmidt/Consider Parameters

If a parameter is uncertain but poorly observable, it does not always need to be estimated as a full state.

It may instead be treated as a consider parameter.

Its uncertainty is propagated into navigation uncertainty without necessarily attempting to estimate it.

This can avoid poor convergence caused by excessive state augmentation.

4.2.18.179 Initialization and Fixed Parameters

A parameter should remain fixed when:

• it is accurately known,
• online estimation provides little benefit,
• it is weakly observable,
• estimation would destabilize more important states.

For example, a well-calibrated camera intrinsic parameter may be better treated as fixed than estimated continuously.

Initialization design therefore includes deciding what not to estimate.

4.2.18.180 Convergence and State Correlation

During convergence, covariance develops off-diagonal terms.

For example,

\[ P_{r,b_a} \neq0. \]

This means position uncertainty and accelerometer-bias uncertainty have become correlated.

Such correlations are not necessarily undesirable; they represent the estimator's understanding of how errors are coupled.

Artificially zeroing them can destroy information and harm consistency.

4.2.18.181 Covariance Ellipsoid During Convergence

For a three-dimensional position covariance

\[ \mathbf P_r, \]

the uncertainty ellipsoid is determined by

\[ (\mathbf r-\hat{\mathbf r})^T \mathbf P_r^{-1} (\mathbf r-\hat{\mathbf r}) = c. \]

During convergence, this ellipsoid may:

• shrink,
• rotate,
• become more or less elongated.

This gives more insight than looking only at scalar RMS position uncertainty.

For spacecraft relative navigation, a long radial uncertainty axis and small lateral axes may reveal weak range observability.

4.2.18.182 Convergence in Information Form

Define

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

As useful measurements accumulate, information generally increases:

\[ \mathbf Y_k \uparrow. \]

Convergence can therefore also be viewed as accumulation of information.

This links filter convergence directly to the information and observability concepts introduced in Section 4.2.17.

4.2.18.183 Information Gain per Measurement

The information added by a measurement is related to

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

Measurements with favourable geometry and low noise provide greater information.

Thus faster convergence can be achieved by:

• improving geometry,
• reducing measurement noise,
• increasing useful measurement rate,
• adding complementary sensors.

This gives a direct engineering interpretation of convergence speed.

4.2.18.184 Convergence and Diminishing Returns

Early measurements may greatly reduce large initial uncertainty.

Later measurements may provide smaller incremental improvement.

Conceptually,

\[ \Delta\mathbf P_1 > \Delta\mathbf P_{100} \]

in some systems.

Therefore convergence curves often show rapid initial contraction followed by a slower approach to an operating uncertainty level.

This is normal behaviour.

4.2.18.185 Convergence Plateau

If covariance stops decreasing at

\[ P_{ii}>P_{req}, \]

the filter has reached a plateau above the required performance.

Possible causes include:

• sensor noise floor,
• process noise,
• weak observability,
• poor geometry,
• bias uncertainty,
• insufficient measurement rate.

Simply waiting longer may not solve the problem.

The navigation architecture itself may need improvement.

4.2.18.186 Convergence Versus Accuracy Floor

The steady-state accuracy floor is different from convergence time.

A filter may converge rapidly to

\[ \sigma_r=1\text{ m}, \]

but the mission may require

\[ 0.01\text{ m}. \]

That is not a convergence-time problem.

It is an accuracy capability problem.

Therefore filter evaluation should distinguish:

\[ \boxed{ \text{Can it converge?} } \]

\[ \boxed{ \text{How fast does it converge?} } \]

\[ \boxed{ \text{What accuracy does it converge to?} } \]

4.2.18.187 Convergence and Sensor Requirements

Suppose the navigation requirement is

\[ \sigma_r<1\text{ cm}. \]

The convergence analysis can determine what combination of:

\[ \sigma_{\text{sensor}}, \]

\[ f_{\text{update}}, \]

\[ \text{geometry}, \]

and

\[ T_{\text{available}} \]

is required to achieve it.

Thus convergence analysis can flow upward into sensor selection and mission design.

4.2.18.188 Convergence and Navigation Availability

Define navigation availability as the fraction of mission time during which required navigation performance is satisfied.

If every sensor handover causes a long reconvergence period, availability decreases.

Therefore

\[ \boxed{ \text{short convergence time} \rightarrow \text{higher navigation availability} } \]

especially for missions with frequent sensor transitions.

4.2.18.189 Convergence and Graceful Degradation

After losing a sensor, the estimator may converge to a new, less accurate operating level rather than fail completely.

For example,

\[ \sigma_r: 0.01\text{ m} \rightarrow 0.10\text{ m}. \]

If this still satisfies hold-mode requirements but not precision proximity operations requirements, the navigation system can degrade gracefully.

This is better than representing the estimator as simply valid or invalid.

4.2.18.190 Initialization and Redundant Estimators

Safety-critical systems may run multiple estimator channels.

For example,

\[ F_1,\quad F_2. \]

They may use different sensors or algorithms.

At initialization, their states and covariances may differ.

Cross-comparison can detect abnormal convergence:

\[ \|\hat{\mathbf x}_1-\hat{\mathbf x}_2\| > \gamma. \]

This can provide an additional navigation-integrity monitor.

4.2.18.191 Convergence Monitoring Without Truth

In flight, true state is unavailable.

Therefore convergence must be inferred from quantities such as:

• covariance,
• innovation statistics,
• sensor agreement,
• filter-to-filter comparison,
• state-correction magnitude,
• physical constraints,
• temporal stability.

No single metric is perfect.

A robust navigation-validity decision usually combines several indicators.

4.2.18.192 Convergence Monitor Logic

A conceptual monitor might require

\[ C_1: \sigma_r

\[ C_2: \sigma_v

\[ C_3: \mathrm{NIS}\in\text{acceptable range}, \]

\[ C_4: N_{\text{healthy sensors}}\ge N_{\min}, \]

\[ C_5: t_{\text{stable}}>T_{\text{dwell}}. \]

Then

\[ \boxed{ \text{NAV\_CONVERGED} = C_1\land C_2\land C_3\land C_4\land C_5 } \]

The exact logic is mission dependent.

4.2.18.193 Hysteresis in Convergence Flags

Without hysteresis, the convergence flag may toggle rapidly.

For example, enter NAV VALID when

\[ \sigma_r<0.01\text{ m} \]

but exit only when

\[ \sigma_r>0.015\text{ m}. \]

This creates

\[ \boxed{ \text{entry threshold} < \text{exit threshold} } \]

and avoids chatter.

Dwell time can provide additional protection.

4.2.18.194 Initialization Confidence Levels

Initialization may be classified as:

\[ \text{LOW}, \quad \text{MEDIUM}, \quad \text{HIGH}. \]

For example, a propagated state after a long outage may be LOW confidence, while a fresh GNSS + star-tracker solution may be HIGH confidence.

These levels can map to different initial covariances and mission permissions.

4.2.18.195 Initialization Metadata

A good flight-software implementation should record:

• initialization source,
• initialization timestamp,
• state epoch,
• covariance source,
• sensor health,
• sensor quality,
• frame,
• units,
• initialization mode,
• reinitialization reason.

This metadata is extremely valuable during testing and anomaly investigation.

4.2.18.196 Convergence Logging

Useful logs include:

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

\[ \mathbf P_k, \]

\[ \mathbf y_k, \]

\[ \mathbf S_k, \]

\[ \mathbf K_k, \]

sensor validity, gating decisions, and convergence flags.

Without these quantities, diagnosing slow convergence or divergence becomes much harder.

4.2.18.197 Initialization and HIL Testing

Hardware-in-the-loop tests should verify real sensor behaviour during:

• power-up,
• delayed acquisition,
• asynchronous timestamps,
• packet loss,
• sensor reset,
• reacquisition,
• measurement jumps.

Simulation often assumes clean startup behaviour that real hardware does not provide.

Therefore HIL testing is particularly valuable for initialization logic.

4.2.18.198 Initialization and SIL Testing

Software-in-the-loop testing allows systematic Monte Carlo variation of:

\[ \delta\mathbf x_0, \quad \mathbf b_0, \quad \mathbf R, \quad \mathbf Q, \quad \text{timing}, \quad \text{dropout}. \]

Thousands of runs can be used to characterize convergence probability and timing before hardware testing.

SIL is therefore an ideal environment for establishing the convergence envelope.

4.2.18.199 Initialization and Unit Tests

Individual software functions should be tested for:

• quaternion normalization,
• covariance symmetry,
• covariance positive semidefiniteness,
• frame conversion,
• unit conversion,
• timestamp propagation,
• covariance transformation,
• reset logic.

These tests prevent low-level implementation errors from being mistaken for estimator-design problems.

4.2.18.200 Initialization and Regression Testing

Once a filter has demonstrated acceptable convergence, those cases should become regression tests.

A software change should not unexpectedly alter:

\[ T_{\text{conv}}, \]

\[ P_{\text{conv}}, \]

or steady-state accuracy.

This is especially important when tuning Q, R, gating thresholds, or sensor models.

4.2.18.201 Tuning Initial Covariance

Initial covariance should not be tuned solely by trial and error.

A better process is:

\[ \boxed{ \text{Characterize initial error} \rightarrow \text{construct }P_0 \rightarrow \text{Monte Carlo test} \rightarrow \text{check consistency} } \]

If empirical initial errors are known, they provide the strongest basis for P_0.

4.2.18.202 Tuning Process Noise for Convergence

If the filter cannot adapt to initial model error, increasing appropriate components of

\[ \mathbf Q \]

may help.

But increasing all process noise indiscriminately can degrade steady-state accuracy.

Process noise should correspond to actual unmodelled dynamics or state evolution.

The objective is physical realism, not merely faster convergence.

4.2.18.203 Tuning Measurement Noise for Convergence

Artificially reducing

\[ \mathbf R \]

makes measurements more influential.

This may appear to accelerate convergence.

But if the actual measurement noise is larger, the filter becomes inconsistent.

Thus

\[ \boxed{ \text{Do not fake convergence by underestimating }R } \]

Measurement covariance should represent actual sensor uncertainty.

4.2.18.204 Tuning Bias Covariance

Suppose gyro bias uncertainty is too small:

\[ P_{b_g,0}\ll \text{actual bias variance}. \]

The filter may resist changing its bias estimate.

Attitude error can then persist.

Increasing P_{b_g,0} to a realistic level gives the estimator permission to adjust the bias.

This illustrates how state-specific covariance tuning affects convergence.

4.2.18.205 Convergence and Bias Random Walk

Bias dynamics may be modelled as

\[ \dot{\mathbf b}_g = \mathbf w_{bg}. \]

Then

\[ \mathbf Q_{bg} \]

determines how rapidly the estimator expects bias to change.

Too little bias process noise can make the filter slow to track real drift.

Too much can make the bias estimate noisy.

Thus bias convergence is controlled by both

\[ P_{b_g,0} \]

and

\[ Q_{b_g}. \]

4.2.18.206 Convergence of Constant Versus Time-Varying Parameters

A constant parameter model assumes

\[ \dot p=0. \]

A random-walk model assumes

\[ \dot p=w_p. \]

These produce different convergence behaviour.

A constant parameter may converge strongly and then become very confident.

A random-walk parameter retains nonzero uncertainty because it is allowed to change.

The dynamic model should match the physical parameter behaviour.

4.2.18.207 Initialization and Model Switching

Some filters use different dynamic models in different modes.

For example,

\[ M_1=\text{coasting}, \]

\[ M_2=\text{thruster burn}, \]

\[ M_3=\text{contact}. \]

Each switch changes covariance propagation.

The estimator may need mode-specific uncertainty injection or partial reinitialization.

Thus convergence should be verified across model transitions.

4.2.18.208 Initialization in Multiple-Model Filters

An interacting multiple-model estimator may initialize several models:

\[ \hat{\mathbf x}_0^{(j)}, \quad \mathbf P_0^{(j)}, \quad \mu_0^{(j)}. \]

Here

\[ \mu_0^{(j)} \]

is the initial probability of model j.

Poor initialization of model probabilities can delay identification of the correct motion regime.

Therefore initialization extends beyond physical state variables.

4.2.18.209 Convergence of Target-Motion Models

For an uncooperative relative navigation target, possible models may include:

• constant velocity,
• constant acceleration,
• orbital relative dynamics,
• manoeuvring target.

The estimator may require several observations before determining which model best explains the target motion.

Thus target-state convergence and model-identification convergence can occur simultaneously.

4.2.18.210 Initialization and Learning-Based Measurements

If a neural network provides pose or feature estimates, its output may initialize a conventional filter.

The network output should ideally include or support an uncertainty estimate:

\[ \hat{\mathbf z}_{NN}, \qquad \mathbf R_{NN}. \]

A highly accurate mean prediction with poorly calibrated uncertainty can still produce bad filter initialization.

The fusion system needs both estimate quality and uncertainty quality.

4.2.18.211 Initialization and Uncertainty Calibration

Suppose a vision system reports

\[ \sigma_z=1\text{ cm} \]

but its actual error is closer to

\[ 10\text{ cm}. \]

Using the reported covariance directly makes the filter overconfident.

Therefore sensor uncertainty models should be calibrated using representative data.

This is particularly important when measurement uncertainty is produced by learned algorithms.

4.2.18.212 Convergence and Environmental Conditions

Sensor quality can vary with:

• lighting,
• Sun angle,
• target reflectivity,
• GNSS geometry,
• thermal conditions,
• vibration.

Therefore convergence testing should cover environmental conditions, not only initial state errors.

A filter that converges rapidly under ideal lighting may behave very differently under glare or low-contrast conditions.

4.2.18.213 Convergence and Thermal Bias

IMU biases may change after power-up as the sensor warms.

Then

\[ b_g(t) \]

and

\[ b_a(t) \]

may not initially be stationary.

The estimator may appear to converge and then continue adjusting as temperature changes.

Temperature-dependent calibration or sufficient bias process noise may be required.

Thus initialization testing should include realistic thermal startup behaviour.

4.2.18.214 Initialization and Vibration

During thruster activity or mechanism operation, vibration can increase sensor noise.

If initialization occurs during such a period,

\[ \mathbf R_{effective} > \mathbf R_{quiet}. \]

Using a quiet-condition covariance may make the estimator overconfident.

The initialization manager may therefore delay precision initialization until vibration decreases or use mode-dependent measurement covariance.

4.2.18.215 Initialization and Sensor Warm-Up

Some sensors require warm-up or acquisition time.

Examples include:

• IMU thermal stabilization,
• GNSS acquisition,
• star tracker acquisition,
• camera exposure stabilization.

Therefore

\[ \text{POWERED} \neq \text{READY FOR INITIALIZATION}. \]

The sensor interface should provide readiness and quality information.

4.2.18.216 Initialization Sequence Design

A practical sequence might be:

\[ \boxed{ \begin{array}{c} \text{Power Sensors}\\ \downarrow\\ \text{Wait for Ready}\\ \downarrow\\ \text{Synchronize Time}\\ \downarrow\\ \text{Check Health}\\ \downarrow\\ \text{Collect Measurements}\\ \downarrow\\ \text{Coarse State Estimate}\\ \downarrow\\ \text{Construct }P_0\\ \downarrow\\ \text{Initialize Filter}\\ \downarrow\\ \text{Monitor Convergence}\\ \downarrow\\ \text{Declare Navigation Valid} \end{array} } \]

Each transition should have explicit success and failure conditions.

4.2.18.217 Initialization Timeout

Every acquisition or convergence phase should have a maximum allowable duration:

\[ T_{\text{init}}

If the timeout is exceeded,

\[ \text{fallback / hold / retry / abort} \]

should be triggered.

Otherwise a failed initialization process can leave the spacecraft indefinitely waiting for a condition that will never occur.

4.2.18.218 Retry Logic

A failed initialization may be retried.

For example,

\[ \text{INIT FAIL} \rightarrow \text{sensor reset} \rightarrow \text{reacquire} \rightarrow \text{INIT}. \]

But retry count should generally be bounded:

\[ N_{\text{retry}}\le N_{\max}. \]

Repeated failure can indicate a persistent sensor or geometry problem and should escalate to FDIR.

4.2.18.219 Initialization and Safe Hold

During relative navigation, failed initialization should normally prevent continued approach.

A safe response can be

\[ \boxed{ \text{NAV not converged} \rightarrow \text{HOLD} } \]

until required navigation performance is restored.

If uncertainty continues increasing,

\[ \text{HOLD} \rightarrow \text{RETREAT}. \]

This is the operational consequence of filter initialization and convergence design.

4.2.18.220 Convergence Requirement by relative navigation Phase

A useful conceptual structure is:

Far Range

Moderate relative-state uncertainty may be acceptable.

Mid Range

Relative position and velocity must converge sufficiently for controlled approach.

Close Range

Centimetre-level position and low relative-velocity uncertainty may be required.

Precision Proximity

Position, velocity, relative attitude, and angular rate must all satisfy stringent uncertainty limits.

Thus,

\[ \boxed{ \text{Required convergence level increases as range decreases} } \]

4.2.18.221 Example relative navigation Convergence State Machine

A conceptual sequence is

\[ \boxed{ \begin{array}{c} \text{ACQUIRE TARGET}\\ \downarrow\\ \text{COARSE REL NAV}\\ \downarrow\\ \text{REL FILTER INIT}\\ \downarrow\\ \text{REL NAV CONVERGING}\\ \downarrow\\ \text{REL NAV VALID}\\ \downarrow\\ \text{FINAL APPROACH ENABLED}\\ \downarrow\\ \text{PRECISION NAV CONVERGED}\\ \downarrow\\ \text{PRECISION NAV READY} \end{array} } \]

At any stage,

\[ \text{sensor fault} \]

or

\[ \text{loss of convergence} \]

can transition the mission back to hold or retreat.

4.2.18.222 Filter Initialization Checklist

Before starting recursive estimation, verify:

• state vector definition is correct,
• all state units are correct,
• all frames are correct,
• timestamps are synchronized,
• quaternion is normalized,
• initial position is plausible,
• initial velocity is plausible,
• attitude convention is correct,
• bias states have realistic priors,
• calibration states have realistic priors,
• \mathbf P_0 is symmetric,
• \mathbf P_0\succeq0,
• cross-correlations are handled,
• sensor health is valid,
• measurement quality is sufficient,
• geometry is adequate,
• measurement gates are appropriate for startup.

4.2.18.223 Convergence Checklist

After initialization, verify:

• estimation errors decrease in simulation,
• covariance contracts appropriately,
• covariance does not collapse unrealistically,
• innovations approach zero mean,
• innovation statistics are consistent,
• bias states converge,
• calibration states converge where expected,
• required states are observable,
• sensor handovers do not cause divergence,
• reinitialization works,
• convergence time satisfies requirements,
• Monte Carlo success rate satisfies requirements,
• navigation-valid logic is correct.

4.2.18.224 Common Misconception: Small Covariance Means Convergence

Not necessarily.

\[ \mathbf P\downarrow \]

only means that the filter believes its uncertainty is decreasing.

True convergence requires the actual estimation error to be compatible with that covariance.

In simulation this can be checked with NEES and truth-error plots.

In flight it must be inferred through innovations, redundant measurements, physical consistency, and integrity monitoring.

4.2.18.225 Common Misconception: Large Initial Covariance Is Always Safe

Not necessarily.

A very large covariance can produce aggressive early measurement corrections.

It can also create numerical-conditioning problems.

The objective is not

\[ \mathbf P_0\rightarrow\infty. \]

It is

\[ \boxed{ \mathbf P_0 \approx \text{real initial uncertainty} } \]

with an appropriate safety margin.

4.2.18.226 Common Misconception: Zero Bias Is a Neutral Initialization

Only the estimated mean is neutral:

\[ \hat b_0=0. \]

If

\[ P_{b,0} \]

is also nearly zero, the estimator is being told that zero bias is known with high confidence.

Therefore

\[ \boxed{ \hat b_0=0 \text{ requires realistic }P_{b,0} } \]

when the actual bias is uncertain.

4.2.18.227 Common Misconception: Filter Runtime Equals Convergence

A filter running for 100 seconds is not necessarily more converged than one running for 10 seconds.

Convergence depends on information.

If the required state is weakly observable,

\[ 100\text{ s} \]

of poor geometry may provide less useful information than

\[ 10\text{ s} \]

of highly informative manoeuvring.

Thus

\[ \boxed{ \text{Convergence depends on information, not time alone} } \]

4.2.18.228 Common Misconception: Every State Must Converge Before Navigation Is Useful

Not always.

A mission may require only a subset of states for a particular mode.

For example, coarse attitude may be sufficient for Sun pointing even if gyro scale factors have not fully converged.

Therefore navigation validity can be mission-mode dependent.

The important question is whether the states required by the current function are sufficiently accurate and reliable.

4.2.18.229 Common Misconception: Reinitialization Means Reset Everything

A complete reset can discard valuable information.

If only one sensor or state group is invalid, partial reinitialization may preserve the rest of the navigation solution.

Thus

\[ \boxed{ \text{Reset only what has lost validity} } \]

when the architecture permits it.

4.2.18.230 Common Misconception: Filter Convergence Is Only a Navigation Issue

Convergence affects:

• guidance,
• control,
• sensor scheduling,
• FDIR,
• mission sequencing,

-

• abort logic,
• autonomy.

Therefore filter convergence is a system-level spacecraft behaviour.

4.2.18.231 Practical Design Principle

A good initialization strategy should answer three questions:

\[ \boxed{ \text{What do I know?} } \]

\[ \boxed{ \text{How well do I know it?} } \]

\[ \boxed{ \text{What information will make the remaining states converge?} } \]

These correspond respectively to

\[ \hat{\mathbf x}_0, \qquad \mathbf P_0, \qquad \text{future observability}. \]

4.2.18.232 Complete Initialization and Convergence Chain

The complete conceptual chain is

\[ \boxed{ \begin{array}{c} \text{Sensor Acquisition}\\ \downarrow\\ \text{Sensor Health and Quality}\\ \downarrow\\ \text{Time / Frame Alignment}\\ \downarrow\\ \text{Coarse State Determination}\\ \downarrow\\ \hat{\mathbf x}_0,\mathbf P_0\\ \downarrow\\ \text{Filter Propagation + Measurement Updates}\\ \downarrow\\ \text{Innovation / Covariance Monitoring}\\ \downarrow\\ \text{State-by-State Convergence}\\ \downarrow\\ \text{Navigation Validity}\\ \downarrow\\ \text{Guidance / Control Permission} \end{array} } \]

Parallel paths monitor

\[ \boxed{ \text{FDIR + Sensor Availability + Observability + Integrity} } \]

throughout the process.

4.2.18.233 Key Takeaways

1. A navigation filter requires both an initial state and initial covariance.

\[ \boxed{ \hat{\mathbf x}_0,\mathbf P_0 } \]

2. Initial covariance should represent real uncertainty.

Too small creates overconfidence.

Too large can create aggressive transients and poor numerical conditioning.

3. Position, velocity, attitude, biases, timing states, and calibration parameters may all require separate initialization strategies.

4. Attitude initialization often requires a coarse global solution before a local error-state filter is appropriate.

5. Bias initialization is not equivalent to assuming zero bias.

\[ \hat b_0=0 \]

must be accompanied by realistic uncertainty.

6. Convergence means more than covariance reduction.

\[ \boxed{ \text{Convergence} = \text{state accuracy} + \text{uncertainty consistency} } \]

7. Different states converge at different rates.

Biases and calibration parameters can converge much more slowly than directly measured states.

8. Observability controls whether convergence is possible.

\[ \text{unobservable} \Rightarrow \text{cannot converge from measurements}. \]

9. Geometry and spacecraft manoeuvres can strongly influence convergence.

This is particularly important for camera-based relative navigation.

10. Innovation gating must account for initialization transients.

Poor initialization combined with overly tight gating can prevent the filter from correcting itself.

11. Sensor handovers and reacquisition can create new convergence transients.

Convergence is therefore not only a startup issue.

12. Partial reinitialization can preserve valid navigation information after faults.

13. Convergence should be tested statistically with Monte Carlo simulations.

Important metrics include

\[ P_{\text{conv}}, \qquad T_{95}, \qquad T_{99}. \]

14. Truth-error, covariance, NIS, NEES, innovation, and bias histories should all be examined during V&V.

15. Navigation validity should be based on quantitative convergence criteria rather than filter runtime alone.

16. In relative navigation, navigation convergence should gate mission progression.

\[ \boxed{ \text{Navigation not converged} \Rightarrow \text{do not proceed to precision approach} } \]

17.

18. Initialization and convergence interact directly with observability, robust fusion, FDIR, guidance, control, sensor selection, and mission autonomy.

The central engineering idea is:

\[ \boxed{ \text{A filter is ready not when it starts running, but when the available information has made the required state sufficiently accurate, observable, consistent, and trustworthy.} } \]

Next: 4.2.19 Sensor-Fusion Tuning and Performance Assessment

After initialization and convergence are established, the next step is to evaluate estimator tuning and performance across process-noise selection, measurement-noise selection, consistency metrics, residual analysis, sensitivity studies, Monte Carlo statistics, and practical navigation performance assessment.