Sequential Orbit Determination

Real-time recursive orbit updates, process noise, maneuver handling, state augmentation, onboard orbit determination, asynchronous multi-rate measurements, continuous tracking, and estimator consistency monitoring.

4.3.14 Sequential Orbit Determination

Sequential orbit determination estimates a spacecraft trajectory recursively, updating the state whenever new measurements become available.

Unlike batch orbit determination, which processes an entire observation arc together, sequential OD propagates the current state estimate forward in time and then corrects it using each new observation or measurement packet.

The central idea is

\[\boxed{ \text{Previous Estimate} \rightarrow \text{Propagation} \rightarrow \text{New Measurement} \rightarrow \text{Correction} \rightarrow \text{Updated Estimate} }\]

Sequential OD is particularly important for

A sequential estimator therefore combines

4.3.14.1 Basic Sequential Estimation Problem

Let the estimated state after processing measurements through time t_{k-1} be

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

with covariance

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

The estimator propagates this solution to the next measurement epoch t_k:

\[\boxed{ \hat{\mathbf x}_{k|k-1} = \phi( t_k,t_{k-1}, \hat{\mathbf x}_{k-1|k-1} ) }\]

The covariance is also propagated:

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

A new measurement is then processed to form an updated state

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

and covariance

\[\boxed{ \mathbf P_{k|k}. }\]

4.3.14.2 Meaning of the Sequential Notation

The notation

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

means:

estimate of the state at time k, using measurements only through time k-1.

This is the predicted or a priori estimate.

The notation

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

means:

estimate of the state at time k, after incorporating the measurement at time k.

This is the updated or a posteriori estimate.

Thus,

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

is the fundamental measurement-correction step.

4.3.14.3 Sequential OD as a Recursive Process

The complete recursive sequence is

\[\boxed{ \begin{array}{c} \hat{\mathbf x}_{k-1|k-1}, \mathbf P_{k-1|k-1} \\ \downarrow\\ \text{Propagate} \\ \downarrow\\ \hat{\mathbf x}_{k|k-1}, \mathbf P_{k|k-1} \\ \downarrow\\ \text{Receive Measurement} \\ \downarrow\\ \text{Compute Innovation} \\ \downarrow\\ \text{Measurement Update} \\ \downarrow\\ \hat{\mathbf x}_{k|k}, \mathbf P_{k|k} \end{array} }\]

Then the process repeats for the next measurement.

4.3.14.4 Dynamic Propagation

The spacecraft state is propagated according to

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

where \boldsymbol\theta may contain dynamic parameters.

For a translational orbit state,

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

the propagation satisfies

\[\boxed{ \dot{\mathbf r} = \mathbf v }\]

and

\[\boxed{ \dot{\mathbf v} = \mathbf a( \mathbf r, \mathbf v, t, \boldsymbol\theta ). }\]

4.3.14.5 Predicted State

For nonlinear dynamics,

\[\boxed{ \hat{\mathbf x}_{k|k-1} = \phi( t_k,t_{k-1}, \hat{\mathbf x}_{k-1|k-1} ) }\]

where \phi denotes numerical integration of the orbit dynamics.

No measurement has yet been processed at t_k.

Therefore this estimate depends entirely on

4.3.14.6 Predicted Covariance

For a discrete linearized system,

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

Here,

Thus,

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

4.3.14.7 STM Form of Covariance Propagation

For orbit dynamics,

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

is often obtained from the state transition matrix

\[\boldsymbol\Phi(t_k,t_{k-1}).\]

Therefore,

\[\boxed{ \mathbf P_{k|k-1} = \boldsymbol\Phi_k \mathbf P_{k-1|k-1} \boldsymbol\Phi_k^{T} + \mathbf Q_k. }\]

This connects sequential OD directly with the STM and covariance-analysis concepts developed earlier.

4.3.14.8 Measurement Model

At time t_k, a measurement is modeled as

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

where

\[\mathbf v_k \sim \mathcal N( \mathbf0, \mathbf R_k ).\]

Possible measurements include

4.3.14.9 Predicted Measurement

Given the propagated state,

\[\boxed{ \hat{\mathbf z}_{k|k-1} = h_k( \hat{\mathbf x}_{k|k-1} ) }\]

is the expected measurement.

This must use the same

as the actual sensor observation.

4.3.14.10 Innovation

The measurement innovation is

\[\boxed{ \mathbf y_k = \mathbf z_k - \hat{\mathbf z}_{k|k-1}. }\]

It represents the discrepancy between what the sensor observed and what the propagated state predicted.

A nonzero innovation may result from

4.3.14.11 Measurement Jacobian

For an Extended Kalman Filter,

\[\boxed{ \mathbf H_k = \frac{\partial h_k} {\partial\mathbf x} \bigg|_{\hat{\mathbf x}_{k|k-1}} }\]

describes local measurement sensitivity.

Unlike batch OD, the measurement Jacobian acts directly on the current state because the sequential filter estimates the state at the current epoch.

4.3.14.12 Innovation Covariance

The predicted innovation covariance is

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

This combines

Thus \mathbf S_k provides the natural statistical scale for judging whether a measurement is consistent with the propagated orbit.

4.3.14.13 Kalman Gain

The Kalman gain is

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

It determines how strongly the measurement innovation modifies the state.

Conceptually,

\[\boxed{ \text{Large Predicted Uncertainty} + \text{Accurate Measurement} \rightarrow \text{Large Correction} }\]

whereas

\[\boxed{ \text{High State Confidence} + \text{Noisy Measurement} \rightarrow \text{Small Correction}. }\]

4.3.14.14 State Update

The updated state is

\[\boxed{ \hat{\mathbf x}_{k|k} = \hat{\mathbf x}_{k|k-1} + \mathbf K_k \mathbf y_k. }\]

Thus the innovation is converted into a state correction through the Kalman gain.

4.3.14.15 Covariance Update

The simplified covariance update is

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

A numerically safer form is the Joseph form:

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

4.3.14.16 Why the Measurement Usually Reduces Covariance

A valid measurement adds information.

Therefore the posterior covariance should generally satisfy

\[\boxed{ \mathbf P_{k|k} \preceq \mathbf P_{k|k-1} }\]

under the assumed linear-Gaussian model.

The measurement therefore reduces uncertainty in state combinations that it observes.

4.3.14.17 Extended Kalman Filter for Orbit Determination

Orbit dynamics and many measurement equations are nonlinear.

Therefore sequential OD commonly uses an Extended Kalman Filter.

The EKF cycle is

\[\boxed{ \text{Nonlinear State Propagation} \rightarrow \text{Linearized Covariance Propagation} \rightarrow \text{Nonlinear Measurement Prediction} \rightarrow \text{Linearized Measurement Update}. }\]

4.3.14.18 EKF Linearization Point

The Jacobians are evaluated about the current estimated trajectory.

Thus,

\[\mathbf F_k\]

and

\[\mathbf H_k\]

change with time.

This distinguishes the EKF from a linear time-invariant Kalman filter.

4.3.14.19 EKF Limitations

The EKF relies on first-order linearization.

Problems can arise when

In such cases, alternatives may include

4.3.14.20 Iterated EKF

For a highly nonlinear measurement, the measurement update may be re-linearized iteratively.

Conceptually,

\[\boxed{ \text{Predict} \rightarrow \text{Update} \rightarrow \text{Re-linearize} \rightarrow \text{Update Again} }\]

until the correction stabilizes.

This can improve nonlinear measurement processing without changing the propagation architecture.

4.3.14.21 Unscented Kalman Filter in Sequential OD

The UKF avoids explicit first-order Jacobians for the nonlinear state and measurement mappings.

Sigma points are generated around

\[\hat{\mathbf x}\]

and propagated through

\[f(\cdot)\]

and

\[h(\cdot).\]

The mean and covariance are reconstructed from the propagated sigma points.

4.3.14.22 UKF Advantages and Limitations

Potential advantages include

Limitations include

Thus,

\[\boxed{ \text{No Jacobian} \neq \text{No Modeling Requirement}. }\]

4.3.14.23 Process Noise

The process-noise covariance

\[\boxed{ \mathbf Q_k }\]

represents uncertainty in the propagated dynamics.

Possible sources include

Process noise prevents the estimator from becoming unrealistically confident in an imperfect dynamic model.

4.3.14.24 Why Process Noise Is Essential in Sequential OD

Without process noise,

\[\mathbf Q=0,\]

the filter assumes the dynamic model is exact.

If the real spacecraft experiences unmodeled disturbances, the true error can grow while the filter covariance remains too small.

This leads to filter inconsistency.

Thus,

\[\boxed{ \text{Imperfect Dynamics} \rightarrow \text{Need for Appropriate Process Noise}. }\]

4.3.14.25 Process Noise versus Measurement Noise

These matrices describe different uncertainty sources:

\[\boxed{ \mathbf Q \rightarrow \text{Dynamic / Model Uncertainty} }\]

\[\boxed{ \mathbf R \rightarrow \text{Measurement Uncertainty}. }\]

Increasing \mathbf Q makes the filter more willing to deviate from the propagated model.

Increasing \mathbf R makes the filter trust measurements less.

4.3.14.26 Continuous Process Noise

For

\[\dot{\delta\mathbf x} = \mathbf A \delta\mathbf x + \mathbf G \mathbf w,\]

where

\[E[ \mathbf w(t) \mathbf w^T(\tau) ] = \mathbf Q_c \delta(t-\tau),\]

the corresponding discrete process covariance is

\[\boxed{ \mathbf Q_d = \int_{t_k}^{t_{k+1}} \boldsymbol\Phi(t_{k+1},\tau) \mathbf G \mathbf Q_c \mathbf G^T \boldsymbol\Phi^T(t_{k+1},\tau) \,d\tau. }\]

Continuous noise density should therefore not be inserted directly as a discrete covariance without appropriate conversion.

4.3.14.27 White-Noise Acceleration Model

A common simple OD process model assumes random acceleration uncertainty.

For one Cartesian axis, the corresponding position-velocity covariance over interval \Delta t may have the form

\[\boxed{ \mathbf Q = q_a \begin{bmatrix} \frac{\Delta t^3}{3} & \frac{\Delta t^2}{2} \\[1mm] \frac{\Delta t^2}{2} & \Delta t \end{bmatrix} }\]

for continuous white acceleration spectral density q_a.

This structure can be extended to three dimensions.

4.3.14.28 Process Noise Tuning

If \mathbf Q is too small,

If \mathbf Q is too large,

Process noise should therefore represent expected dynamic-model uncertainty, not be chosen merely to make residuals look small.

4.3.14.29 Covariance Growth Between Measurements

When no measurement is available,

\[\boxed{ \mathbf P_{k|k-1} = \boldsymbol\Phi \mathbf P \boldsymbol\Phi^T + \mathbf Q. }\]

Therefore uncertainty generally grows with propagation time.

Long tracking gaps cause the navigation solution to rely increasingly on the dynamic model.

4.3.14.30 Measurement Update after a Tracking Gap

When observations resume after a long gap, innovations may be larger because covariance and actual trajectory uncertainty have grown.

If the covariance has been modeled correctly, the innovation covariance

\[\mathbf S\]

should also reflect this larger uncertainty.

Thus the filter can still process a large absolute residual without necessarily declaring it an outlier.

4.3.14.31 Continuous Tracking

Sequential OD is naturally suited to continuous or near-continuous measurements.

For example,

\[\boxed{ \text{Propagate} \rightarrow \text{Receive Measurement} \rightarrow \text{Update} \rightarrow \text{Repeat} }\]

may operate continuously during tracking visibility.

This allows navigation uncertainty to remain bounded through repeated measurement corrections.

4.3.14.32 Continuous Tracking Does Not Mean Continuous Measurement

Sensors still produce discrete samples.

Continuous tracking means measurements are regularly available over time.

The estimator itself may propagate continuously or at a chosen internal integration step while processing measurements at discrete epochs.

4.3.14.33 Measurement Rate versus Filter Propagation Rate

These rates need not be equal.

For example,

\[f_{\rm propagation} > f_{\rm measurement}\]

is common.

The estimator may propagate the dynamic model at a high internal rate and apply measurement updates whenever observations arrive.

Therefore,

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

4.3.14.34 Asynchronous Measurements

Different sensors may produce observations at different times.

For example,

The estimator should propagate to each measurement epoch and process the measurement using the corresponding model.

No common fixed sensor rate is mathematically required.

4.3.14.35 Multi-Rate Sequential OD

Suppose sensors operate at

\[f_1,\quad f_2,\quad f_3.\]

Sequential OD can process them as

\[\boxed{ \text{High-Rate Propagation} + \text{Event-Driven Measurement Updates}. }\]

This is especially useful for multisensor navigation systems.

4.3.14.36 Measurement Latency

A measurement may correspond to observation time

\[t_m\]

but arrive at the estimator later.

The estimator must distinguish

\[\boxed{ \text{Measurement Epoch} }\]

from

\[\boxed{ \text{Reception / Processing Epoch}. }\]

Ignoring latency can produce systematic state errors.

4.3.14.37 Delayed Measurements

Possible approaches include

The best method depends on

4.3.14.38 Innovation Gating

Before accepting a measurement, the normalized innovation may be checked.

For vector observations,

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

A measurement may be rejected or flagged if

\[\boxed{ \epsilon_k > \gamma. }\]

The threshold \gamma is chosen from an appropriate statistical confidence level.

4.3.14.39 Why Raw Residual Magnitude Is Not Enough

A residual of 100 m may be acceptable if predicted uncertainty is kilometers.

A residual of 1 m may be highly inconsistent if the filter predicts millimeter-level uncertainty.

Therefore measurement consistency should be evaluated relative to

\[\mathbf S_k,\]

not by residual magnitude alone.

4.3.14.40 Normalized Innovation Squared

For a measurement dimension m,

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

should approximately follow

\[\chi_m^2\]

under a consistent Gaussian model.

This provides an important filter-health diagnostic.

4.3.14.41 Innovation Mean

Ideally,

\[\boxed{ E[\mathbf y_k] \approx \mathbf0. }\]

Persistent nonzero innovation mean may indicate

4.3.14.42 Innovation Whiteness

Innovations should also be approximately temporally white under the classical Kalman assumptions.

Persistent correlation may reveal

Thus innovation history provides information about both the filter and its models.

4.3.14.43 Sequential OD and Maneuvers

A spacecraft maneuver changes the predicted trajectory.

If the maneuver is known, the state propagation should explicitly include it.

For an impulsive maneuver,

\[\boxed{ \mathbf v^+ = \mathbf v^- + \Delta\mathbf v. }\]

For a finite burn,

\[\boxed{ \dot{\mathbf v} = \mathbf a_{\rm natural} + \frac{\mathbf F(t)}{m(t)}. }\]

4.3.14.44 Known Impulsive Maneuver

If \Delta\mathbf v is considered known exactly,

\[\boxed{ \hat{\mathbf v}^{+} = \hat{\mathbf v}^{-} + \Delta\mathbf v_{\rm cmd}. }\]

The covariance may still need modification if execution error is nonzero.

4.3.14.45 Maneuver Execution Uncertainty

Let

\[\Delta\mathbf v_{\rm actual} = \Delta\mathbf v_{\rm cmd} + \delta\Delta\mathbf v.\]

If

\[E[ \delta\Delta\mathbf v \delta\Delta\mathbf v^T ] = \mathbf P_{\Delta v},\]

then the maneuver injects additional velocity covariance.

Conceptually,

\[\boxed{ \mathbf P_{vv}^{+} = \mathbf P_{vv}^{-} + \mathbf P_{\Delta v} }\]

when the maneuver error is independent of previous state error.

4.3.14.46 Maneuver Covariance Mapping

More generally,

\[\boxed{ \mathbf P^{+} = \mathbf M \mathbf P^{-} \mathbf M^T + \mathbf B \mathbf P_{\Delta v} \mathbf B^T. }\]

Here,

4.3.14.47 Unknown Maneuver

If a maneuver occurs unexpectedly, measurements after the event may produce large innovations.

The estimator may handle this through

The appropriate response depends on the mission architecture.

4.3.14.48 Maneuver Detection

A persistent sequence of unexpectedly large innovations can indicate an unmodeled maneuver.

Conceptually,

\[\boxed{ \text{Innovation Change} \rightarrow \text{Possible Dynamic Event} }\]

but innovation changes can also result from sensor problems.

Therefore maneuver detection should distinguish

4.3.14.49 Finite-Burn Sequential Estimation

During a finite maneuver, thrust acceleration may be modeled directly:

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

Possible uncertain burn parameters include

These may be represented through process noise or explicit augmented states.

4.3.14.50 State Augmentation

Sequential OD is not limited to

\[\mathbf r,\mathbf v.\]

The state may be expanded as

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

where \boldsymbol\theta contains uncertain parameters.

4.3.14.51 Common Augmented States in Sequential OD

Possible states include

These states can be updated continuously as measurements arrive.

4.3.14.52 Augmented Dynamics

For augmented state

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

the dynamics are

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

Static parameters may use

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

while slowly varying parameters may use random-walk or Gauss-Markov models.

4.3.14.53 Augmented Covariance

The covariance becomes

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

The cross-covariance blocks are essential because the physical state and parameter uncertainties become coupled.

4.3.14.54 Parameter Correction through Measurements

If the Kalman gain is partitioned as

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

the parameter correction is

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

Thus measurements can update both the orbit and the uncertain model parameters.

4.3.14.55 Sequential Estimation of Ballistic Coefficient

A drag-related parameter may be modeled as

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

As tracking measurements arrive, the filter can adjust the parameter so that the propagated trajectory better matches observations.

However, estimated ballistic parameters may correlate strongly with atmospheric density errors.

4.3.14.56 Sequential Bias Estimation

Suppose a range measurement contains unknown bias:

\[z_\rho = \rho + b_\rho + v.\]

Augment

\[b_\rho\]

into the state.

A random-walk model may be

\[\boxed{ b_{\rho,k+1} = b_{\rho,k} + w_{b,k}. }\]

The estimator then separates orbit error from persistent measurement bias as the measurement geometry evolves.

4.3.14.57 Empirical Acceleration States

Unmodeled dynamic effects may be represented using estimated accelerations:

\[\boxed{ \mathbf a_{\rm emp} = \begin{bmatrix} a_R\\ a_T\\ a_N \end{bmatrix}. }\]

These can be modeled as constant over short intervals, random walks, or Gauss-Markov processes.

They provide flexibility but can absorb genuine modeling errors if overused.

4.3.14.58 State Augmentation and Observability

Adding a state does not make it observable.

If two parameters affect the measurements in nearly identical ways, they may become strongly correlated.

Therefore,

\[\boxed{ \text{Augmentation} \neq \text{Guaranteed Estimability}. }\]

Observability should be assessed before adding large numbers of parameters.

4.3.14.59 Filter Divergence

Filter divergence occurs when the state estimate and covariance no longer represent the true estimation error.

Symptoms may include

Divergence can result from

4.3.14.60 Covariance Consistency

When truth is available, use

\[\boxed{ \epsilon_x = \tilde{\mathbf x}^T \mathbf P^{-1} \tilde{\mathbf x} }\]

to compare actual state error with predicted covariance.

For an n-state consistent Gaussian filter,

\[\epsilon_x \sim \chi_n^2.\]

This is the NEES consistency test.

4.3.14.61 NIS for Operational Monitoring

When truth is not available, NIS uses the innovation:

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

This can be monitored continuously during mission operations.

Persistent NIS outside expected bounds may indicate incorrect filter tuning or modeling.

4.3.14.62 Covariance Inflation

If the filter is believed to be overconfident, covariance may be inflated:

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

This gives subsequent measurements more ability to correct the state.

However, inflation should not replace correction of a known modeling problem.

4.3.14.63 Adaptive Process Noise

Some filters adjust process noise according to observed innovation statistics.

For example,

\[\mathbf Q\]

may be increased when residual behavior indicates larger-than-expected dynamic uncertainty.

Adaptive filtering can improve robustness but should be designed carefully to avoid instability or arbitrary tuning.

4.3.14.64 Filter Reinitialization

If navigation is severely lost, the estimator may require reinitialization from

A robust system should define criteria for when normal recursive correction is no longer sufficient.

4.3.14.65 Initialization

Sequential filters require

\[\boxed{ \hat{\mathbf x}_{0|0} }\]

and

\[\boxed{ \mathbf P_{0|0}. }\]

The initial covariance should realistically represent uncertainty in the initial state.

Too-small initial covariance can cause the filter to reject valid measurements.

4.3.14.66 Initial Transient

During early operation, the filter may undergo a convergence period.

State corrections and covariance reductions may initially be larger.

Performance should therefore distinguish

4.3.14.67 Sequential OD and GNSS

GNSS is particularly suitable for sequential OD.

At each epoch, pseudorange and Doppler measurements can update

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

Dynamic propagation between GNSS updates improves continuity and allows integration with other sensors.

4.3.14.68 Sequential Optical OD

Optical LOS or pixel measurements can be incorporated whenever available.

Because monocular measurements often contain weak instantaneous range information, dynamic propagation across multiple observations provides information accumulation over time.

Thus sequential optical OD gradually improves state observability as geometry evolves.

4.3.14.69 General Sequential Relative Navigation

For a general relative-navigation problem,

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

The relative state is propagated using an appropriate relative-dynamics model and corrected using measurements such as

This applies generally to formation flying and rendezvous/proximity operations.

4.3.14.70 Relative-State Propagation

A generic relative state evolves as

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

The specific model may be

The sequential estimation architecture is unchanged.

4.3.14.71 Relative Measurement Update

For relative observation

\[\mathbf z_k = h_{\rm rel}( \mathbf x_{{\rm rel},k} ) + \mathbf v_k,\]

the innovation is

\[\boxed{ \mathbf y_k = \mathbf z_k - h_{\rm rel}( \hat{\mathbf x}_{{\rm rel},k|k-1} ). }\]

The relative-state filter then updates using the same Kalman logic.

4.3.14.72 Onboard Orbit Determination

Sequential OD is well suited to onboard operation because it processes measurements recursively and does not require storing a complete observation arc.

An onboard OD system may combine

4.3.14.73 Advantages of Onboard OD

Onboard orbit determination can provide

It is particularly important for autonomous missions.

4.3.14.74 Constraints of Onboard OD

Onboard navigation must operate within limits on

Therefore estimator architecture may differ from high-fidelity ground OD.

4.3.14.75 Ground versus Onboard Sequential OD

Ground systems may use

Onboard systems may use simplified models while prioritizing

The two systems can complement each other.

4.3.14.76 Continuous Ground Tracking

Sequential OD can also run on the ground during continuous tracking.

New measurements are incorporated as they are received, producing an updated orbit solution in real time.

This is useful for

4.3.14.77 Sequential OD with Tracking Gaps

If no measurement is available, the filter continues propagation:

\[\boxed{ \hat{\mathbf x}_{k|k-1} = \phi( \hat{\mathbf x} ) }\]

while covariance grows.

When measurements return, the larger covariance naturally gives them increased corrective influence.

4.3.14.78 Tracking Pass Transitions

Ground-based tracking may occur in separate visibility passes.

At the end of one pass, the sequential estimate is propagated until the next pass.

The navigation state and covariance therefore provide continuity between tracking opportunities.

4.3.14.79 Multi-Sensor Sequential Fusion

Measurements from multiple sensors can be processed one after another.

Suppose sensor A and sensor B provide observations at the same nominal epoch.

One may process

\[\boxed{ A \rightarrow B }\]

or combine them into a single stacked measurement update.

The approaches are equivalent under appropriate linear-Gaussian assumptions if correlations are treated consistently.

4.3.14.80 Correlated Sensors

If two measurements share common error,

\[\operatorname{Cov}( \mathbf v_A, \mathbf v_B ) \neq0,\]

they should not be treated as independent updates without accounting for correlation.

Otherwise, the same information may effectively be counted twice.

4.3.14.81 Sequential Processing of Derived Measurements

If one measurement is mathematically derived from another already-used measurement, correlation may be extremely strong.

For example, feeding both

can double-count information unless correlation is modeled.

4.3.14.82 Sensor Dropout

If one sensor becomes unavailable, the estimator can continue using

The covariance will reflect the reduced information.

This is a major advantage of multisensor sequential architectures.

4.3.14.83 Sensor Recovery

When a failed or unavailable sensor returns, its measurements can be reincorporated.

Innovation gating is particularly useful during recovery because the navigation state may have drifted while the sensor was unavailable.

4.3.14.84 State Constraints in Sequential OD

Physical constraints can sometimes be included through

For example,

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

can represent an approximate known relation.

The assigned covariance determines the strength of the constraint.

4.3.14.85 Numerical Conditioning

Sequential covariance matrices may become poorly conditioned due to

Useful techniques include

4.3.14.86 Square-Root Sequential Filters

Instead of propagating \mathbf P, propagate a factor

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

Square-root methods can improve numerical stability and preserve positive definiteness more reliably.

They are useful in high-precision onboard or ground navigation.

4.3.14.87 Information Filter

Instead of covariance, define information matrix

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

and information vector

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

Measurement information is then accumulated additively.

Information filters can be advantageous in some distributed or multisensor estimation architectures.

4.3.14.88 Sequential OD versus Batch OD

Batch OD uses an entire observation arc:

\[\boxed{ \{ \mathbf z_1,\ldots,\mathbf z_N \} \rightarrow \hat{\mathbf x}_0. }\]

Sequential OD processes measurements recursively:

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

They use the same

The principal difference is how measurement information is accumulated.

4.3.14.89 Advantages of Sequential OD

Sequential orbit determination provides

4.3.14.90 Limitations of Sequential OD

Limitations include

The estimator therefore requires continuous health monitoring.

4.3.14.91 Sequential Filter versus Smoother

A filter estimates

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

using only measurements through time k.

A smoother estimates

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

using future measurements as well.

Therefore smoothing generally improves historical state estimates but cannot provide strictly real-time estimates for future measurements that have not yet occurred.

4.3.14.92 Prediction Mode

Sequential OD can operate temporarily without measurements.

Then the estimator performs only

\[\boxed{ \text{State Propagation} + \text{Covariance Propagation}. }\]

This is called prediction or coast mode.

Uncertainty typically grows until another measurement becomes available.

4.3.14.93 Update Mode

When a valid measurement becomes available,

\[\boxed{ \text{Innovation} \rightarrow \text{Gain} \rightarrow \text{State Correction} \rightarrow \text{Covariance Reduction}. }\]

The estimator alternates naturally between prediction and update modes.

4.3.14.94 Real-Time Recursive Orbit Updates

The operational navigation state can therefore be continuously maintained as

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

Every new observation refreshes the solution.

This is the defining feature of real-time sequential orbit determination.

4.3.14.95 Sequential OD Health Monitoring

A practical system should monitor

This allows early detection of estimator problems.

4.3.14.96 Common Mistakes in Sequential Orbit Determination

Common errors include:

  1. Assuming sensor update rate and estimator propagation rate must be identical.
  2. Using incorrect process-noise scaling.
  3. Setting \mathbf Q=0 for imperfect dynamics.
  4. Using unrealistic measurement covariance.
  5. Ignoring maneuver uncertainty.
  6. Processing delayed measurements at arrival time instead of measurement time.
  7. Ignoring sensor bias.
  8. Adding unobservable augmented states.
  9. Treating correlated sensors as independent.
  10. Using raw residual thresholds instead of normalized innovation.
  11. Ignoring covariance consistency.
  12. Using an incorrect measurement Jacobian.
  13. Allowing covariance to lose symmetry or positive definiteness.
  14. Failing to detect unexpected maneuvers.
  15. Overusing covariance inflation instead of correcting model deficiencies.
  16. Confusing estimator rate with measurement rate.
  17. Ignoring tracking gaps.
  18. Double counting derived measurements.
  19. Declaring filter convergence based only on a stable-looking state.
  20. Failing to monitor innovation whiteness and bias.

4.3.14.97 Practical Sequential OD Workflow

A disciplined sequential OD architecture follows

\[\boxed{ \begin{array}{c} \text{Initialize State and Covariance}\\ \downarrow\\ \text{Propagate Dynamics}\\ \downarrow\\ \text{Propagate Covariance}\\ \downarrow\\ \text{Check for Maneuver / Dynamic Event}\\ \downarrow\\ \text{Receive Measurement}\\ \downarrow\\ \text{Associate Correct Measurement Time}\\ \downarrow\\ \text{Predict Measurement}\\ \downarrow\\ \text{Compute Innovation}\\ \downarrow\\ \text{Compute Innovation Covariance}\\ \downarrow\\ \text{Apply Gating / Quality Checks}\\ \downarrow\\ \text{Compute Kalman Gain}\\ \downarrow\\ \text{Update State}\\ \downarrow\\ \text{Update Covariance}\\ \downarrow\\ \text{Monitor Consistency}\\ \downarrow\\ \text{Repeat for Next Measurement} \end{array} }\]

4.3.14.98 General Sequential OD Architecture

The full information flow can be summarized as

\[\boxed{ \begin{array}{c} \text{Previous Navigation Solution}\\ \hat{\mathbf x}_{k-1|k-1}, \mathbf P_{k-1|k-1} \\ \downarrow\\ \text{Dynamics + Maneuver Model}\\ \downarrow\\ \hat{\mathbf x}_{k|k-1}, \mathbf P_{k|k-1} \\ \downarrow\\ \text{New Sensor Measurement}\\ \mathbf z_k \\ \downarrow\\ \text{Measurement Prediction}\\ \hat{\mathbf z}_k \\ \downarrow\\ \text{Innovation} \\ \mathbf y_k \\ \downarrow\\ \text{Estimator Update} \\ \downarrow\\ \hat{\mathbf x}_{k|k}, \mathbf P_{k|k} \\ \downarrow\\ \text{Repeat Continuously} \end{array} }\]

4.3.14.99 Core Sequential-OD Equations

State propagation

\[\boxed{ \hat{\mathbf x}_{k|k-1} = \phi( t_k,t_{k-1}, \hat{\mathbf x}_{k-1|k-1} ) }\]

Covariance propagation

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

Measurement prediction

\[\boxed{ \hat{\mathbf z}_k = h_k( \hat{\mathbf x}_{k|k-1} ) }\]

Innovation

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

Innovation covariance

\[\boxed{ \mathbf S_k = \mathbf H_k \mathbf P_{k|k-1} \mathbf H_k^T + \mathbf R_k }\]

Kalman gain

\[\boxed{ \mathbf K_k = \mathbf P_{k|k-1} \mathbf H_k^T \mathbf S_k^{-1} }\]

State update

\[\boxed{ \hat{\mathbf x}_{k|k} = \hat{\mathbf x}_{k|k-1} + \mathbf K_k \mathbf y_k }\]

Joseph covariance update

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

Normalized innovation squared

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

4.3.14.100 Key Takeaways

Sequential orbit determination maintains an orbit solution continuously by alternating between dynamic prediction and measurement correction.

The basic propagation step is

\[\boxed{ \hat{\mathbf x}_{k|k-1} = \phi( \hat{\mathbf x}_{k-1|k-1} ). }\]

The corresponding uncertainty is propagated through

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

A new measurement is predicted:

\[\boxed{ \hat{\mathbf z}_k = h( \hat{\mathbf x}_{k|k-1} ) }\]

and compared with the actual observation:

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

The innovation is then mapped into a state correction using the Kalman gain:

\[\boxed{ \hat{\mathbf x}_{k|k} = \hat{\mathbf x}_{k|k-1} + \mathbf K_k\mathbf y_k. }\]

The most important distinguishing feature is that sequential OD does not wait for a complete observation arc.

Instead,

\[\boxed{ \text{Measurement Arrives} \rightarrow \text{Navigation Solution Updates Immediately}. }\]

Process noise is essential because real spacecraft dynamics are never modeled perfectly:

\[\boxed{ \mathbf Q \rightarrow \text{Representation of Dynamic Uncertainty}. }\]

Maneuvers must be explicitly incorporated because they alter both the nominal trajectory and the covariance.

State augmentation allows the estimator to track quantities such as

Sequential OD naturally supports

The key engineering distinction is:

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

A navigation system may propagate at a high internal rate while incorporating measurements at slower or asynchronous rates.

Finally, a good sequential OD system must maintain not only an estimated orbit but also a statistically consistent uncertainty estimate:

\[\boxed{ \text{Sequential OD Solution} = \{ \hat{\mathbf x}_{k|k}, \mathbf P_{k|k} \}. }\]

Its performance must therefore be assessed through the state history, covariance, innovation statistics, maneuver handling, sensor availability, and consistency over time.

Next: 4.3.15 Maneuver and Dynamic-Parameter Estimation

Sequential orbit determination propagates and updates the orbit recursively as measurements arrive. The next section focuses specifically on estimating impulsive and finite-burn maneuvers, burn timing, thrust magnitude and direction, empirical accelerations, drag-related parameters, and solar-radiation-pressure parameters.

4.3.15 Maneuver and Dynamic-Parameter Estimation

Impulsive and finite-burn estimation, burn timing, thrust magnitude and direction, empirical accelerations, drag, and SRP parameter estimation.

Continue to Maneuver and Dynamic-Parameter Estimation →