Smoothing and Retrodiction

4.3.7 Smoothing and Retrodiction

Filtering estimates the state using measurements available up to the current time. Smoothing goes one step further: it uses measurements obtained after a state epoch to improve the estimate of that earlier state.

The key distinction is

\[ \boxed{ \text{Filtering: use past + present measurements} } \]

versus

\[ \boxed{ \text{Smoothing: use past + present + future measurements} } \]

Suppose a filter estimates the spacecraft state at time t_k as

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

This means:

\[ \boxed{ \hat{\mathbf x}_{k|k} = \text{estimate of } \mathbf x_k \text{ using measurements through time } k. } \]

If additional measurements become available through a later time N, we can form

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

which means:

estimate the state at time k using all measurements available through time N.

This is a smoothed estimate.

Because additional measurements contain information about the same dynamic trajectory,

\[ \boxed{ \text{Future Measurements} \rightarrow \text{Additional Information About Past States}. } \]

This is why smoothing is useful for:

• post-flight trajectory reconstruction,
• orbit determination,
• maneuver reconstruction,
• sensor calibration,
• fault investigation,
• navigation-performance analysis,
• scientific-data processing,
• relative-navigation reconstruction,
• offline estimation after an RPO or docking sequence,
• validation of onboard filtering.

Smoothing is especially powerful because it exploits the same spacecraft dynamics that connect states across time.

4.3.7.1 Filtering versus Smoothing

A filter estimates the current state using only measurements that have already arrived.

Using standard notation,

\[ \boxed{ \hat{\mathbf x}_{k|k} = E[ \mathbf x_k | \mathbf z_1,\ldots,\mathbf z_k ]. } \]

A smoother estimates that same state using additional future measurements:

\[ \boxed{ \hat{\mathbf x}_{k|N} = E[ \mathbf x_k | \mathbf z_1,\ldots,\mathbf z_N ], \qquad N>k. } \]

Thus,

\[ \boxed{ \hat{\mathbf x}_{k|N} \text{ generally contains more information than } \hat{\mathbf x}_{k|k}. } \]

The difference is not that smoothing changes the physical trajectory.

Instead, it improves our estimate of the trajectory by using information that was not available at time k.

4.3.7.2 Why Future Measurements Help Past States

Suppose the dynamic model is

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

The state at time k influences the state at later times.

Therefore,

\[ \mathbf x_k \rightarrow \mathbf x_{k+1} \rightarrow \mathbf x_{k+2} \rightarrow \cdots \]

and later measurements contain indirect information about earlier states.

For example,

\[ \mathbf z_{k+2} \]

depends on

\[ \mathbf x_{k+2}, \]

which itself depends dynamically on

\[ \mathbf x_k. \]

Therefore,

\[ \boxed{ \mathbf z_{k+2} \text{ carries information about } \mathbf x_k. } \]

Filtering cannot use that information at time k because the future measurement does not yet exist.

Smoothing can use it later.

4.3.7.3 Conditional-Estimation Notation

The notation

\[ \boxed{ \hat{\mathbf x}_{i|j} } \]

means

estimate of state at time i, using measurements through time j.

Several important cases follow.

Prediction

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

means estimate at time k using measurements only through k-1.

Filtering

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

means estimate at time k using measurements through k.

Smoothing

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

means estimate at time k using later measurements through N.

This notation is extremely useful when distinguishing prediction, filtering, and smoothing.

4.3.7.4 Prediction, Filtering, and Smoothing

The three estimation modes can be summarized as

\[ \boxed{ \begin{aligned} \text{Prediction:} \quad & \hat{\mathbf x}_{k|k-1}\\ \text{Filtering:} \quad & \hat{\mathbf x}_{k|k}\\ \text{Smoothing:} \quad & \hat{\mathbf x}_{k|N} \end{aligned} } \]

with

\[ N>k. \]

Conceptually,

\[ \boxed{ \text{Prediction} \rightarrow \text{Current Estimate} \rightarrow \text{Retrospective Improvement}. } \]

4.3.7.5 Retrodiction

The term retrodiction refers to estimating a state at an earlier time using information available at a later time.

Suppose the estimator is currently at t_N, but we wish to improve the state at

\[ t_k

Then the problem is

\[ \boxed{ \mathbf z_{1:N} \rightarrow \hat{\mathbf x}_{k|N}. } \]

This is sometimes described as backward estimation or retrospective estimation.

Retrodiction is closely related to smoothing, although terminology varies depending on the estimation architecture.

4.3.7.6 Why Retrodiction Is Useful

Retrodiction is important when one wants to determine what the spacecraft was doing at an earlier epoch more accurately than was possible in real time.

Typical applications include:

• reconstructing the precise orbit after a mission segment,
• determining the actual state during a maneuver,
• improving attitude history,
• identifying the time and magnitude of an event,
• reconstructing relative position and velocity during a rendezvous sequence,
• calibrating sensors using later data,
• analysing anomalies.

Thus,

\[ \boxed{ \text{Real-Time Estimate} \neq \text{Best Possible Post-Processed Estimate}. } \]

4.3.7.7 General Smoothing Problem

Suppose the dynamic model is

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

and the measurement model is

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

Filtering seeks

\[ p( \mathbf x_k | \mathbf z_{1:k} ). \]

Smoothing seeks

\[ \boxed{ p( \mathbf x_k | \mathbf z_{1:N} ), \qquad N>k. } \]

This is the probabilistic definition of smoothing.

4.3.7.8 Fundamental Smoothing Principle

The smoothing posterior can be understood conceptually as

\[ \boxed{ \text{Filtered Information at } k + \text{Future Information from } k+1:N \rightarrow \text{Smoothed State at } k. } \]

Therefore,

\[ \boxed{ p( \mathbf x_k | \mathbf z_{1:N} ) \propto p( \mathbf x_k | \mathbf z_{1:k} ) \times \text{Future Information}. } \]

A smoother combines information arriving from both temporal directions.

Engineering interpretation: The future-information term is not a new physical observation at time k; it is statistical information carried backward through the dynamic correlation between states. This is the conceptual heart of smoothing.

4.3.7.9 Forward and Backward Information Flow

Filtering moves forward:

\[ \boxed{ t_0 \rightarrow t_1 \rightarrow t_2 \rightarrow \cdots \rightarrow t_N. } \]

Smoothing adds a backward-information pass:

\[ \boxed{ t_N \rightarrow t_{N-1} \rightarrow \cdots \rightarrow t_0. } \]

Thus,

\[ \boxed{ \text{Smoothing} = \text{Forward Filtering} + \text{Backward Information Propagation}. } \]

This does not mean the physical dynamics are reversed. Rather, information is propagated backward through statistical relationships between consecutive states.

4.3.7.10 Types of Smoothing

Three commonly used smoothing formulations are:

1. Fixed-interval smoothing
2. Fixed-lag smoothing
3. Fixed-point smoothing

They differ mainly in which past states are being improved and how much future information is used.

4.3.7.11 Fixed-Interval Smoothing

In fixed-interval smoothing, the complete measurement interval is known:

\[ \boxed{ t_0 \le t \le t_N. } \]

All measurements

\[ \mathbf z_{1:N} \]

are used to estimate every state in the interval:

\[ \boxed{ \hat{\mathbf x}_{k|N}, \qquad k=0,\ldots,N. } \]

This is primarily an offline method.

Typical use:

\[ \boxed{ \text{Complete Data Set} \rightarrow \text{Best Reconstructed Trajectory Over Entire Interval}. } \]

4.3.7.12 Fixed-Interval Example

Suppose tracking data are available over a two-hour orbital arc.

A forward filter processes the data from the beginning to the end.

Then a smoother runs backward from the final epoch and improves the state estimates throughout the two-hour interval.

Thus,

\[ \boxed{ \{ \hat{\mathbf x}_{0|N}, \hat{\mathbf x}_{1|N}, \ldots, \hat{\mathbf x}_{N|N} \} } \]

provides a reconstructed trajectory using the full data arc.

4.3.7.13 Fixed-Lag Smoothing

Fixed-lag smoothing estimates a state with a constant delay L.

At current time k, estimate

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

Thus the system intentionally waits for L future measurements or time steps before finalizing the state at k-L.

Conceptually,

\[ \boxed{ \text{Small Delay} \rightarrow \text{Access to Future Information} \rightarrow \text{Improved Estimate}. } \]

This provides a compromise between real-time filtering and full offline smoothing.

4.3.7.14 Fixed-Lag Tradeoff

Increasing the lag L gives the estimator access to more future information.

Therefore estimation accuracy can improve.

But increasing lag also increases:

• latency,
• storage,
• computational workload.

Thus,

\[ \boxed{ \text{Larger Lag} \rightarrow \text{Potentially Better Estimate} + \text{Greater Delay}. } \]

Fixed-lag smoothing is useful when some latency is acceptable but full post-processing is not.

4.3.7.15 Fixed-Point Smoothing

Fixed-point smoothing repeatedly improves the estimate of one particular past state.

Suppose the state at time t_j is of special interest.

As later observations arrive,

\[ \boxed{ \hat{\mathbf x}_{j|j}, \quad \hat{\mathbf x}_{j|j+1}, \quad \hat{\mathbf x}_{j|j+2}, \quad \ldots } \]

are progressively refined.

This can be useful for estimating:

• maneuver-state conditions,
• event-state conditions,
• calibration states,
• a state at a specific reference epoch.

4.3.7.16 Comparison of Smoother Types

The best architecture depends on mission requirements.

4.3.7.17 Linear Dynamic Model

Consider the linear stochastic system

\[ \boxed{ \mathbf x_{k+1} = \mathbf F_k\mathbf x_k + \mathbf w_k } \]

with

\[ \mathbf w_k \sim \mathcal N( \mathbf0, \mathbf Q_k ). \]

Measurements are

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

with

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

For this system, a common fixed-interval smoother is the Rauch–Tung–Striebel smoother.

4.3.7.18 Rauch–Tung–Striebel Smoother

The RTS smoother operates in two stages:

\[ \boxed{ \text{Forward Kalman Filtering} + \text{Backward RTS Smoothing}. } \]

The forward filter stores quantities such as:

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

\[ \mathbf P_{k|k}, \]

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

\[ \mathbf P_{k+1|k}. \]

Then the backward pass begins from the final filtered state.

4.3.7.19 Forward Filtering Pass

The forward pass computes standard filtered quantities:

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

and

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

It also computes one-step prediction:

\[ \boxed{ \hat{\mathbf x}_{k+1|k} = \mathbf F_k \hat{\mathbf x}_{k|k} } \]

and

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

These quantities are stored for the backward pass.

4.3.7.20 Final Condition for the Backward Pass

At the final epoch N, there are no later measurements within the smoothing interval.

Therefore,

\[ \boxed{ \hat{\mathbf x}_{N|N}^{s} = \hat{\mathbf x}_{N|N} } \]

and

\[ \boxed{ \mathbf P_{N|N}^{s} = \mathbf P_{N|N}. } \]

The superscript s indicates smoothed quantities.

The algorithm then proceeds backward from

\[ N-1 \]

to

\[ 0. \]

4.3.7.21 RTS Smoother Gain

The smoother gain is

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

This gain determines how information from the improved state at k+1 modifies the filtered state at k.

It is the backward-information analogue of the filtering gain.

Engineering interpretation: The gain is large when the filtered state at k is strongly correlated with the predicted state at k+1 and the prediction uncertainty is sufficiently well conditioned.

4.3.7.22 Physical Interpretation of the Smoother Gain

The smoother gain depends on:

• uncertainty in the filtered state at time k,
• dynamic coupling between k and k+1,
• predicted uncertainty at time k+1.

Thus,

\[ \boxed{ \mathbf G_k \rightarrow \text{How Future-State Improvement Is Mapped Back to the Earlier State}. } \]

If the earlier and later states are strongly dynamically correlated, future information can significantly improve the earlier estimate.

4.3.7.23 RTS State Update

The smoothed state is

\[ \boxed{ \hat{\mathbf x}_{k|N} = \hat{\mathbf x}_{k|k} + \mathbf G_k \left( \hat{\mathbf x}_{k+1|N} - \hat{\mathbf x}_{k+1|k} \right). } \]

This equation has a very useful interpretation.

The term

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

is the improvement made to the next state by using future measurements.

The smoother maps that improvement backward through

\[ \mathbf G_k. \]

Therefore,

\[ \boxed{ \text{Smoothed Earlier State} = \text{Filtered Earlier State} + \text{Back-Propagated Future Correction}. } \]

4.3.7.24 RTS Covariance Update

The smoothed covariance is

\[ \boxed{ \mathbf P_{k|N} = \mathbf P_{k|k} + \mathbf G_k \left( \mathbf P_{k+1|N} - \mathbf P_{k+1|k} \right) \mathbf G_k^T. } \]

Because typically

\[ \mathbf P_{k+1|N} \preceq \mathbf P_{k+1|k}, \]

the correction term generally reduces uncertainty.

Thus,

\[ \boxed{ \mathbf P_{k|N} \preceq \mathbf P_{k|k} } \]

under the appropriate linear-Gaussian assumptions.

Engineering interpretation: The covariance update mirrors the state correction: if the next smoothed state is more certain than the next predicted state, that reduction is mapped backward into the earlier epoch.

4.3.7.25 Why Smoothed Covariance Is Usually Smaller

Filtering has access only to measurements through time k.

Smoothing additionally uses measurements from

\[ k+1,\ldots,N. \]

These provide extra information.

Therefore,

\[ \boxed{ \text{Additional Future Information} \rightarrow \text{Reduced Past-State Uncertainty}. } \]

However, covariance reduction is meaningful only if the dynamic and measurement uncertainty models are themselves realistic.

4.3.7.26 RTS Backward Recursion

The complete backward recursion is

\[ \boxed{ \begin{aligned} \mathbf G_k &= \mathbf P_{k|k} \mathbf F_k^T \mathbf P_{k+1|k}^{-1}, \\ \hat{\mathbf x}_{k|N} &= \hat{\mathbf x}_{k|k} + \mathbf G_k ( \hat{\mathbf x}_{k+1|N} - \hat{\mathbf x}_{k+1|k} ), \\ \mathbf P_{k|N} &= \mathbf P_{k|k} + \mathbf G_k ( \mathbf P_{k+1|N} - \mathbf P_{k+1|k} ) \mathbf G_k^T. \end{aligned} } \]

for

\[ k=N-1,\ldots,0. \]

4.3.7.27 Complete RTS Architecture

The full architecture is

\[ \boxed{ \begin{array}{c} \text{Forward Filter}\\ \downarrow\\ \{ \hat{\mathbf x}_{k|k}, \mathbf P_{k|k}, \hat{\mathbf x}_{k+1|k}, \mathbf P_{k+1|k} \} \\ \downarrow\\ \text{Final Filtered State at }N\\ \downarrow\\ \text{Backward RTS Pass}\\ \downarrow\\ \mathbf G_{N-1}, \mathbf G_{N-2}, \ldots \\ \downarrow\\ \text{Smoothed States}\\ \hat{\mathbf x}_{k|N} \\ \downarrow\\ \text{Smoothed Covariances}\\ \mathbf P_{k|N} \end{array} } \]

4.3.7.28 Difference Between Filter Gain and Smoother Gain

The filter gain is typically

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

It maps measurement innovation into a current-state correction.

The smoother gain is

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

It maps a future-state improvement into a past-state correction.

Therefore,

\[ \boxed{ \mathbf K_k: \text{measurement space} \rightarrow \text{state space} } \]

while

\[ \boxed{ \mathbf G_k: \text{future-state correction} \rightarrow \text{past-state correction}. } \]

4.3.7.29 Smoothing and Dynamic Correlation

Smoothing works because adjacent states are statistically related.

The cross-covariance between

\[ \mathbf x_k \]

and

\[ \mathbf x_{k+1} \]

allows information to propagate backward.

For a linear model,

\[ \boxed{ \operatorname{Cov} ( \mathbf x_k,\mathbf x_{k+1} ) = \mathbf P_{k|k} \mathbf F_k^T. } \]

This term appears directly in the smoother gain.

Thus,

\[ \boxed{ \text{Smoothing Relies on State-to-State Correlation Created by Dynamics}. } \]

Engineering interpretation: This cross-covariance is the bridge that allows information to travel backward. Without temporal correlation, later measurements could not improve earlier states.

4.3.7.30 Process Noise and Smoothing

If

\[ \mathbf Q_k \]

is small, states at adjacent times are strongly tied by the dynamic model.

Future measurements can therefore strongly improve past states.

If

\[ \mathbf Q_k \]

is large, the connection between consecutive states is weaker.

Therefore,

\[ \boxed{ \mathbf Q\uparrow \rightarrow \text{Weaker Temporal Correlation} \rightarrow \text{Less Backward Information Transfer}. } \]

This is physically sensible: if the dynamics are highly uncertain, later measurements tell us less about earlier states.

4.3.7.31 Measurement Noise and Smoothing

Measurement noise also affects smoothing.

If later measurements are precise,

\[ \mathbf R\downarrow, \]

they strongly improve future states.

Those improvements can then propagate backward.

If later measurements are very noisy,

\[ \mathbf R\uparrow, \]

their contribution to earlier-state improvement is weaker.

Thus,

\[ \boxed{ \text{Smoothing Benefit Depends on Both} \mathbf Q \text{ and } \mathbf R. } \]

4.3.7.32 Nonlinear Smoothing

Spacecraft systems are often nonlinear:

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

and

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

A nonlinear smoother may use local linearization.

For example, an extended RTS smoother uses

\[ \boxed{ \mathbf F_k = \left. \frac{\partial f_k} {\partial\mathbf x} \right|_{\hat{\mathbf x}_{k|k}}. } \]

The smoother then applies RTS-type backward equations using the locally linearized dynamics.

4.3.7.33 Extended RTS Smoother

For nonlinear dynamics, forward processing may use an EKF.

The predicted state is

\[ \hat{\mathbf x}_{k+1|k} = f_k( \hat{\mathbf x}_{k|k} ). \]

The Jacobian is

\[ \mathbf F_k = \left. \frac{\partial f_k} {\partial\mathbf x} \right|_{\hat{\mathbf x}_{k|k}}. \]

Then a backward smoother uses

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

The resulting smoother is only locally valid because the underlying Gaussian-linear approximation is local.

4.3.7.34 Iterated Nonlinear Smoothing

For strongly nonlinear problems, one forward-backward pass may not be sufficient.

A possible architecture is

\[ \boxed{ \text{Forward Filter} \rightarrow \text{Backward Smoother} \rightarrow \text{Update Reference Trajectory} \rightarrow \text{Repeat}. } \]

This resembles differential correction and nonlinear batch estimation.

Thus smoothing and nonlinear least-squares estimation are closely related.

Engineering interpretation: Iterated smoothing is effectively a repeated relinearization process. Each pass can update the nominal trajectory, producing improved Jacobians and a better local Gaussian approximation.

4.3.7.35 Sigma-Point Smoothing

If a UKF or sigma-point filter is used, smoothing can also be performed using propagated sigma-point cross-covariances.

This avoids explicit first-order Jacobian linearization.

The key idea remains the same:

\[ \boxed{ \text{Future-State Improvement} \rightarrow \text{Past-State Improvement Through Cross-Covariance}. } \]

Sigma-point smoothers can be useful when nonlinear effects are stronger than a first-order approximation captures well.

4.3.7.36 Particle Smoothing

For strongly non-Gaussian or multimodal systems, particle filtering can be extended to particle smoothing.

Instead of representing uncertainty only by mean and covariance, smoothing uses weighted samples of entire state histories.

This can represent:

• multimodal states,
• nonlinear ambiguity,
• non-Gaussian uncertainty.

However, computational cost can be much greater.

4.3.7.37 Smoothing versus Backward Integration

Smoothing is not the same as simply integrating the equations of motion backward.

Backward integration produces a dynamically propagated state.

Smoothing combines:

• forward filtered information,
• future measurements,
• uncertainty,
• state correlations.

Therefore,

\[ \boxed{ \text{Backward Dynamics Alone} \neq \text{Statistical Smoothing}. } \]

4.3.7.38 Backward Information Filtering

An alternative to RTS smoothing is to construct a backward information filter.

One set of information is propagated forward and another backward.

The two are then combined.

Conceptually,

\[ \boxed{ \text{Forward Information} + \text{Backward Information} \rightarrow \text{Smoothed State}. } \]

This is sometimes called a two-filter smoothing formulation.

4.3.7.39 Two-Filter Smoothing Concept

The forward filter provides

\[ p( \mathbf x_k | \mathbf z_{1:k} ). \]

A backward processor provides information from

\[ \mathbf z_{k+1:N}. \]

Then

\[ \boxed{ p( \mathbf x_k | \mathbf z_{1:N} ) \propto p( \mathbf x_k | \mathbf z_{1:k} ) \times p( \mathbf z_{k+1:N} | \mathbf x_k ). } \]

This gives the probabilistic basis of forward/backward smoothing.

4.3.7.40 Filtering Error versus Smoothing Error

Define filtering error

\[ \tilde{\mathbf x}_{k|k} = \mathbf x_k - \hat{\mathbf x}_{k|k}. \]

Define smoothing error

\[ \tilde{\mathbf x}_{k|N} = \mathbf x_k - \hat{\mathbf x}_{k|N}. \]

Under consistent linear-Gaussian assumptions,

\[ \boxed{ E[ \tilde{\mathbf x}_{k|N} \tilde{\mathbf x}_{k|N}^T ] \preceq E[ \tilde{\mathbf x}_{k|k} \tilde{\mathbf x}_{k|k}^T ]. } \]

Thus smoothing should not increase the theoretical minimum uncertainty when valid future information is added.

4.3.7.41 Smoothed Trajectory

The output of a smoother is not merely one improved state.

It may be the entire sequence

\[ \boxed{ \hat{\mathbf X}_{0:N} = \{ \hat{\mathbf x}_{0|N}, \hat{\mathbf x}_{1|N}, \ldots, \hat{\mathbf x}_{N|N} \}. } \]

This constitutes a reconstructed trajectory.

Associated covariances are

\[ \boxed{ \mathbf P_{0|N}, \mathbf P_{1|N}, \ldots, \mathbf P_{N|N}. } \]

4.3.7.42 Trajectory Reconstruction

Trajectory reconstruction aims to estimate the best possible state history after all relevant data are available.

Conceptually,

\[ \boxed{ \text{Stored Sensor Data} + \text{Dynamic Model} + \text{Smoothing} \rightarrow \text{Reconstructed State History}. } \]

This is useful when the post-processed trajectory is more important than the original real-time estimate.

4.3.7.43 Orbit Reconstruction

A spacecraft orbit can be reconstructed using measurements collected over a complete arc.

Examples may include:

• GNSS measurements,
• range,
• Doppler,
• angular tracking,
• radiometric data.

Later tracking observations can improve the estimated orbit at earlier epochs.

Thus,

\[ \boxed{ \text{Full Tracking Arc} \rightarrow \text{Smoothed Orbit History}. } \]

4.3.7.44 General Relative-Navigation Reconstruction

Consider a general rendezvous or proximity-operation scenario.

During the operation, a real-time estimator may provide

\[ \hat{\mathbf r}_{rel,k|k}, \qquad \hat{\mathbf v}_{rel,k|k}. \]

After the complete measurement history is available, smoothing can produce

\[ \boxed{ \hat{\mathbf r}_{rel,k|N}, \qquad \hat{\mathbf v}_{rel,k|N}. } \]

This can improve reconstruction of:

• relative trajectory,
• approach geometry,
• maneuver response,
• hold-point behaviour,
• sensor performance.

This is a general application and does not refer to any specific RPO implementation.

4.3.7.45 Maneuver Reconstruction

Suppose a maneuver occurs near time t_m.

Before the maneuver, state uncertainty exists.

After the maneuver, later measurements provide information about the resulting trajectory.

Smoothing can use those later measurements to improve:

• pre-maneuver state,
• post-maneuver state,
• maneuver magnitude,
• maneuver timing,

if these quantities are included in the estimation formulation.

Thus,

\[ \boxed{ \text{Pre-Event Data} + \text{Post-Event Data} \rightarrow \text{Improved Event Reconstruction}. } \]

4.3.7.46 Sensor Bias Reconstruction

If sensor bias is included in the state,

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

smoothing can improve estimates of the past bias history.

Later measurements may help distinguish whether an earlier residual was caused by:

• true state error,
• sensor bias,
• dynamic-model error.

This can be valuable for calibration and post-flight analysis.

4.3.7.47 Fault Reconstruction

Suppose a sensor or actuator fault occurs temporarily.

A real-time filter may not immediately identify the source.

Later measurements may reveal a consistent trajectory pattern.

Smoothing may then improve reconstruction of:

• fault onset time,
• magnitude,
• duration,
• state consequences.

However, smoothing only helps if the fault or uncertainty is represented adequately in the estimator model.

4.3.7.48 Event-Time Estimation

Sometimes the unknown is not only a state but also an event time

\[ t_e. \]

Examples include:

• maneuver start,
• impulse epoch,
• sensor reset,
• disturbance onset.

The complete measurement history may constrain t_e better than measurements available at the event time itself.

Thus retrospective estimation can be especially useful for event-time reconstruction.

4.3.7.49 Smoothing and Batch Estimation

Batch estimation and smoothing both use measurements across an interval.

They are closely related.

A batch estimator may solve for a reference state or complete trajectory using all measurements simultaneously.

A smoother typically begins from a recursive filter and then performs a backward pass.

Under linear-Gaussian assumptions, the two approaches can yield equivalent state estimates when based on the same models and information.

Therefore,

\[ \boxed{ \text{Batch Estimation} \Longleftrightarrow \text{Smoothing} } \]

under appropriate formulations.

Engineering interpretation: Equivalence with batch estimation requires consistent models, priors, noise assumptions, and treatment of the same measurements. Different linearization or data-handling choices can lead to different practical results.

4.3.7.50 Conceptual Difference from Batch Estimation

Although they can produce equivalent results, the computational viewpoints differ.

Batch viewpoint

\[ \boxed{ \text{All Measurements} \rightarrow \text{One Global Estimation Problem}. } \]

Smoothing viewpoint

\[ \boxed{ \text{Forward Recursive Processing} + \text{Backward Refinement}. } \]

Batch estimation emphasizes global optimization.

Smoothing emphasizes recursive information reuse.

4.3.7.51 Smoothing and Factor Graphs

A trajectory estimation problem can also be represented as a factor graph.

State nodes are

\[ \mathbf x_0,\mathbf x_1,\ldots,\mathbf x_N. \]

Dynamic factors connect consecutive states.

Measurement factors connect states to observations.

Solving the graph uses information from the entire trajectory.

Conceptually,

\[ \boxed{ \text{Factor-Graph Optimization} \approx \text{Batch Smoothing}. } \]

This is another modern interpretation of trajectory smoothing.

4.3.7.52 Dynamic Constraints in Smoothing

Dynamics impose relationships such as

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

A smoother uses these constraints to transfer information between neighbouring times.

If \mathbf Q_k is very small, the trajectory is constrained strongly to the dynamic model.

If \mathbf Q_k is large, the trajectory can depart more freely from nominal dynamics.

4.3.7.53 Deterministic versus Stochastic Trajectory Smoothing

If process noise is zero,

\[ \mathbf Q=0, \]

the trajectory is forced to follow the deterministic dynamic model exactly.

With process noise,

\[ \mathbf Q>0, \]

the smoothed trajectory is allowed to deviate according to dynamic uncertainty.

Thus,

\[ \boxed{ \text{Deterministic Smoothing} = \text{Exact Dynamic Constraint} } \]

while

\[ \boxed{ \text{Stochastic Smoothing} = \text{Soft Dynamic Constraint}. } \]

4.3.7.54 Process-Noise Interpretation

Process noise should represent uncertainty in the transition model.

Examples include:

• unmodeled accelerations,
• atmospheric variability,
• actuator uncertainty,
• unmodeled environmental disturbances,
• slowly varying parameters.

If \mathbf Q is too small, the smoother may force the reconstructed trajectory too strongly through an inaccurate model.

If \mathbf Q is too large, the trajectory may become unnecessarily flexible.

4.3.7.55 Measurement Noise Interpretation

Measurement covariance determines how strongly observations influence the reconstructed trajectory.

Small

\[ \mathbf R \]

means observations are trusted strongly.

Large

\[ \mathbf R \]

means observations contribute less information.

Incorrect measurement covariance can therefore distort the smoothed trajectory just as it distorts filtered estimation.

4.3.7.56 Smoothing Does Not Remove Bad Models

Smoothing can reduce random estimation uncertainty.

However, it cannot automatically correct:

• wrong force models,
• wrong sensor models,
• frame errors,
• timing errors,
• unmodeled biases,
• incorrect covariance assumptions.

Instead, a smoother may distribute the model mismatch over the reconstructed trajectory.

Thus,

\[ \boxed{ \text{Smoother} \neq \text{Model-Error Corrector}. } \]

Engineering interpretation: A smoother can make model mismatch look deceptively smooth because the inconsistency is redistributed over multiple epochs. Residual structure and truth-based validation remain essential.

4.3.7.57 Residual Analysis after Smoothing

Once the smoothed trajectory is available, recompute measurement residuals:

\[ \boxed{ \mathbf r_{k,s} = \mathbf z_k - h_k( \hat{\mathbf x}_{k|N} ). } \]

Compare these with filtering residuals.

Residual analysis should examine:

• bias,
• RMS,
• time trends,
• periodic behaviour,
• sensor dependence,
• outliers.

A smooth trajectory is not automatically a physically correct trajectory.

4.3.7.58 Smoothed Residuals versus Filter Innovations

A filter innovation is based on the prior prediction:

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

A smoothed residual uses a state that has already benefited from future measurements.

Therefore these quantities have different statistical meanings.

They should not be compared as though they were identical diagnostics.

4.3.7.59 Innovation Independence and Smoothing

Filtering innovations are expected to be approximately white under a properly modeled linear-Gaussian filter.

Smoothed residuals are generally correlated differently because the measurement itself and future measurements influence the smoothed state.

Therefore filter-consistency tests based on innovation statistics should not simply be applied unchanged to smoothed residuals.

4.3.7.60 Cross-Covariance in Smoothing

Smoothers often require cross-covariance between neighbouring states.

For linear propagation,

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

This expresses the statistical coupling between the current filtered state and the next predicted state.

Cross-covariance is central to backward correction.

4.3.7.61 Smoothed Cross-Covariance

Some applications require covariance between smoothed states at different epochs:

\[ \boxed{ \mathbf P_{k,k+1|N}. } \]

These terms are useful for:

• parameter estimation,
• expectation-maximization,
• trajectory uncertainty,
• estimating process-noise statistics.

Smoothing therefore involves more than individual state covariances when full trajectory statistics are required.

4.3.7.62 State-History Storage

RTS smoothing requires forward-pass quantities from many epochs.

Therefore memory is needed to store quantities such as:

• filtered state,
• filtered covariance,
• predicted state,
• predicted covariance,
• transition matrices.

This is one of the main practical differences from a purely online filter.

4.3.7.63 Memory versus Recalculation

Instead of storing all forward-pass quantities, some systems may reconstruct them by rerunning the forward filter.

This trades

\[ \boxed{ \text{Memory} \leftrightarrow \text{Computation}. } \]

The preferred strategy depends on available onboard or ground-processing resources.

4.3.7.64 Numerical Stability

Like filtering, smoothing involves repeated covariance operations.

Numerical issues can include:

• loss of symmetry,
• poor conditioning,
• unstable matrix inversion,
• negative covariance eigenvalues.

Stable implementations should avoid explicit matrix inversion where possible.

For example, rather than forming

\[ \mathbf P_{k+1|k}^{-1}, \]

solve the corresponding linear system.

Engineering interpretation: For long arcs and weakly conditioned problems, square-root or factorized implementations are often preferable because they better preserve positive semidefiniteness and reduce round-off damage.

4.3.7.65 Square-Root Smoothing

Square-root smoothing propagates matrix factors rather than covariance matrices directly.

If

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

the smoother operates with \mathbf S or related factors.

Advantages include:

• improved numerical stability,
• better preservation of positive semidefiniteness,
• improved behaviour for ill-conditioned problems.

4.3.7.66 State Scaling

If a state contains variables with very different magnitudes,

\[ \mathbf x = [ \text{position}, \text{velocity}, \text{bias}, \text{coefficient} ]^T, \]

numerical conditioning may be poor.

State scaling can improve both filtering and smoothing.

Define

\[ \boxed{ \mathbf x = \mathbf S_x \tilde{\mathbf x}. } \]

All covariance and dynamic matrices must then be transformed consistently.

4.3.7.67 Missing Measurements

Smoothing can handle gaps in measurement availability.

During a gap, the state is connected through dynamics.

Later measurements may partially reduce uncertainty within the gap.

Thus,

\[ \boxed{ \text{Future Measurements Can Improve States Inside Earlier Measurement Gaps}. } \]

The improvement depends on process noise and dynamic coupling.

4.3.7.68 Long Measurement Gaps

For a long gap, the dynamic model becomes increasingly important.

If process uncertainty accumulates,

\[ \mathbf Q \]

reduces the strength with which later observations can determine states far back in the gap.

Therefore smoothing cannot fully overcome long intervals of weak dynamics and missing observations.

4.3.7.69 Irregular Measurement Times

Measurements need not occur at uniform intervals.

The forward filter and backward smoother can operate using the actual time-dependent transition model:

\[ \boxed{ \mathbf F(t_{k+1},t_k). } \]

Accurate timestamp handling remains essential.

4.3.7.70 Delayed Measurements

Smoothing is naturally useful when measurements arrive late.

Instead of forcing a delayed observation into the present-time update, the system can use the actual historical measurement epoch and reconstruct the affected interval.

This is closely related to out-of-sequence measurement processing.

4.3.7.71 Fixed-Lag Smoothing for Delayed Measurements

A fixed-lag architecture can preserve a recent history of states.

If a delayed measurement arrives within the lag window, it may be incorporated into the corresponding past epoch and the recent state history recomputed.

Thus fixed-lag smoothing can provide practical support for delayed or asynchronous sensors.

4.3.7.72 Real-Time versus Post-Processed Accuracy

A real-time filter is optimized for timeliness.

A smoother is optimized for retrospective accuracy.

Therefore,

\[ \boxed{ \text{Real-Time Navigation} \rightarrow \text{Filter} } \]

while

\[ \boxed{ \text{Post-Processed Best Estimate} \rightarrow \text{Smoother}. } \]

Neither is universally better; they serve different operational purposes.

4.3.7.73 Accuracy versus Latency

Fixed-lag smoothing exposes an important systems tradeoff:

\[ \boxed{ \text{More Future Information} \rightarrow \text{Better Estimate} + \text{More Delay}. } \]

For guidance and control, large delays may be unacceptable.

For trajectory monitoring or navigation-quality reporting, a short lag may be acceptable.

4.3.7.74 Smoothing for Sensor Calibration

Suppose an unknown sensor calibration parameter affects a long measurement arc.

Future measurements can help improve estimates of the parameter and, consequently, earlier states.

This makes smoothing useful in joint

\[ \boxed{ \text{State} + \text{Calibration Parameter} } \]

estimation.

Examples include:

• sensor biases,
• scale factors,
• alignment terms.

4.3.7.75 Smoothing Dynamic Parameters

Dynamic parameters may also be included.

For example,

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

Smoothing can refine the parameter history or constant parameter estimate using the full data interval.

However,

\[ \boxed{ \text{Smoothing} \neq \text{Automatic Parameter Identifiability}. } \]

The measurements still need sufficient sensitivity.

4.3.7.76 Constant versus Time-Varying Parameters

A constant parameter may use

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

A slowly varying parameter may use

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

Smoothing can use later measurements to improve historical estimates of \theta.

This is particularly useful for slowly varying environmental or sensor parameters.

4.3.7.77 Smoothing and Bias Separation

Suppose a state error and sensor bias produce similar effects locally.

Over a longer interval, their temporal signatures may differ.

A smoother can exploit this additional temporal information.

Therefore,

\[ \boxed{ \text{Longer Temporal Context} \rightarrow \text{Potentially Better State/Bias Separation}. } \]

But if the signatures remain indistinguishable, smoothing cannot create identifiability.

4.3.7.78 Smoothing and Observability

A state component may be weakly observable at time k, but later geometry may become more informative.

Future measurements can then improve the past estimate through smoothing.

Thus,

\[ \boxed{ \text{Future Observability} \rightarrow \text{Improved Earlier-State Estimate}. } \]

This is one of the strongest conceptual reasons smoothing works well in dynamic systems.

Engineering interpretation: This illustrates a key distinction between instantaneous and dynamic observability: information may become available later and still improve an earlier state through the dynamic model.

4.3.7.79 Changing Geometry

In orbit and relative-navigation problems, measurement geometry changes with time.

A weak geometry at one epoch may be followed by a stronger geometry later.

Smoothing transfers the later information backward.

Therefore,

\[ \boxed{ \text{Changing Geometry Over Time} \rightarrow \text{Retrospective Improvement of Weak Earlier Geometry}. } \]

4.3.7.80 Initial-State Improvement

Early states often have larger uncertainty because relatively few measurements have been processed.

A fixed-interval smoother can substantially improve these early states because the full later data set becomes available.

Thus the improvement from smoothing is often greatest near the beginning of an observation arc.

4.3.7.81 End-of-Interval Behaviour

At the final time N,

\[ \boxed{ \hat{\mathbf x}_{N|N} = \hat{\mathbf x}_{N|N}^{s}. } \]

No future measurements exist within the interval.

Therefore there is no smoothing improvement at the final state itself.

The benefit increases as we move backward into the interval.

4.3.7.82 Boundary Effects

Smoothing performance may differ near the beginning and end of an interval.

States in the middle often benefit from measurements on both sides.

States near the boundaries have less information from one temporal direction.

Therefore uncertainty may vary across the smoothed trajectory even when sensor quality is constant.

4.3.7.83 Smoothing Window Selection

For fixed-lag or moving-window smoothers, the chosen interval length affects performance.

A longer window provides more information but requires more:

• memory,
• computation,
• delay.

The window should be long enough to capture meaningful dynamic information without violating system latency or computational requirements.

4.3.7.84 Sliding-Window Estimation

A sliding-window estimator maintains only a recent state history:

\[ \boxed{ \mathbf x_{k-L}, \ldots, \mathbf x_k. } \]

Measurements within the window are optimized or smoothed jointly.

As time advances, old states are marginalized or removed.

This is closely related to fixed-lag smoothing.

4.3.7.85 Sliding Window versus RTS

RTS smoothing typically assumes a completed interval and performs a backward pass.

Sliding-window estimation repeatedly optimizes a moving recent interval.

Thus,

\[ \boxed{ \text{RTS} \rightarrow \text{Classical Forward/Backward Smoothing} } \]

while

\[ \boxed{ \text{Sliding Window} \rightarrow \text{Repeated Local Batch Smoothing}. } \]

Both exploit future information relative to older states.

4.3.7.86 Marginalization in Sliding Windows

When an old state leaves the estimation window, its information should not simply be discarded.

Its effect may be compressed into a prior factor on the remaining states.

This is called marginalization.

Conceptually,

\[ \boxed{ \text{Remove Old State} + \text{Preserve Its Information} \rightarrow \text{Marginalized Prior}. } \]

This connects smoothing to factor-graph and optimization-based navigation methods.

4.3.7.87 Smoothing and Maximum A Posteriori Trajectory Estimation

Instead of estimating one state, one can estimate the entire state sequence:

\[ \boxed{ \mathbf X = \{ \mathbf x_0, \mathbf x_1, \ldots, \mathbf x_N \}. } \]

A MAP trajectory estimator solves

\[ \boxed{ \hat{\mathbf X}_{MAP} = \arg\max_{\mathbf X} p( \mathbf X | \mathbf z_{1:N} ). } \]

This is a probabilistic formulation of batch smoothing.

4.3.7.88 MAP Smoothing Cost

Under Gaussian transition and measurement errors,

\[ \boxed{ \begin{aligned} J(\mathbf X) ={}& \| \mathbf x_0-\bar{\mathbf x}_0 \|_{\mathbf P_0^{-1}}^2 \\ &+ \sum_{k=0}^{N-1} \| \mathbf x_{k+1} - f_k(\mathbf x_k) \|_{\mathbf Q_k^{-1}}^2 \\ &+ \sum_{k=1}^{N} \| \mathbf z_k - h_k(\mathbf x_k) \|_{\mathbf R_k^{-1}}^2. \end{aligned} } \]

This equation is extremely important.

It combines:

• prior consistency,
• dynamic consistency,
• measurement consistency.

Thus,

\[ \boxed{ \text{Trajectory Smoothing} = \text{Balance Prior + Dynamics + Measurements}. } \]

Engineering interpretation: This cost is the batch-MAP view of smoothing. It makes clear that a smoothed trajectory is a compromise among prior belief, dynamic consistency, and measurement fit.

4.3.7.89 Dynamic Residual

Define a dynamic residual:

\[ \boxed{ \mathbf r_{f,k} = \mathbf x_{k+1} - f_k(\mathbf x_k). } \]

The process covariance

\[ \mathbf Q_k \]

determines how strongly deviations from the model are penalized.

This makes dynamics themselves behave like probabilistic constraints between state nodes.

4.3.7.90 Measurement Residual in Batch Smoothing

The measurement residual remains

\[ \boxed{ \mathbf r_{z,k} = \mathbf z_k-h_k(\mathbf x_k). } \]

The complete smoothing objective therefore contains both

\[ \mathbf r_{f,k} \]

and

\[ \mathbf r_{z,k}. \]

This makes the trajectory simultaneously consistent with dynamics and observations.

4.3.7.91 Deterministic Dynamics as a Limiting Case

If

\[ \mathbf Q_k\rightarrow\mathbf0, \]

deviations from dynamics become strongly penalized.

The trajectory approaches

\[ \boxed{ \mathbf x_{k+1} = f_k(\mathbf x_k) } \]

exactly.

Thus deterministic batch orbit determination can be viewed as a limiting case of stochastic trajectory smoothing.

4.3.7.92 Large Process Noise Limit

If

\[ \mathbf Q_k \]

becomes very large, the dynamic constraint becomes weak.

Then neighbouring states can move more independently to fit measurements.

This may reduce residuals but can produce physically unrealistic trajectories if process noise is exaggerated.

Therefore,

\[ \boxed{ \text{More Flexible Trajectory} \neq \text{More Accurate Physical Trajectory}. } \]

4.3.7.93 Smoothing and Overfitting

A smoother with too much process flexibility or too many estimated parameters may follow measurement noise.

This can produce an apparently excellent fit while degrading physical interpretability.

Therefore the balance among

\[ \mathbf Q, \qquad \mathbf R, \qquad \text{model structure} \]

is critical.

4.3.7.94 Smoothing and Outliers

Outliers can influence both forward filtering and backward smoothing.

Because smoothing propagates information across time, one bad measurement may affect states before and after its epoch.

Possible approaches include:

• outlier gating before smoothing,
• robust cost functions,
• iterative reweighting,
• sensor-quality flags.

Outlier treatment should be physically and statistically justified.

4.3.7.95 Robust Smoothing

A robust smoother may replace quadratic measurement losses with robust functions:

\[ \boxed{ J = \sum_k \rho( \mathbf r_{z,k} ). } \]

Examples include Huber or heavy-tailed probabilistic losses.

This can reduce the influence of corrupted measurements.

However, the resulting problem is generally nonlinear and iterative.

4.3.7.96 Model Discontinuities

Events such as

• impulsive maneuvers,
• stage changes,
• contact events,
• sensor resets

introduce discontinuities.

The smoother should represent these explicitly.

For an impulsive maneuver,

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

Ignoring the discontinuity may cause the smoother to distribute the event incorrectly across nearby states.

4.3.7.97 Multiple-Arc Smoothing

A long trajectory may be divided into arcs.

Each arc may have its own state while sharing global parameters.

This is useful when:

• maneuvers separate segments,
• dynamic models change,
• sensor availability changes,
• different uncertainty assumptions apply.

Arc boundaries can be connected by transition constraints.

4.3.7.98 Smoothing with Known Maneuvers

If a commanded maneuver is known,

\[ \Delta\mathbf v_c, \]

the state transition can include it explicitly.

Any maneuver uncertainty may be represented in process covariance.

This allows post-maneuver measurements to refine pre- and post-maneuver states consistently.

4.3.7.99 Smoothing with Unknown Maneuvers

An unknown maneuver may be represented as an estimated parameter:

\[ \boxed{ \boldsymbol{\xi} = \begin{bmatrix} \mathbf X\\ \Delta\mathbf v \end{bmatrix}. } \]

The full trajectory before and after the event then provides information about the maneuver.

This can be solved through batch or smoothing-based estimation.

4.3.7.100 Smoothing and Time-Bias Estimation

Measurement timing errors can be estimated jointly if their effect is observable.

Suppose

\[ t_{\text{meas}} = t+\Delta t. \]

A time offset changes the state at which the observation should be evaluated.

Using the full trajectory can help distinguish timing error from state error.

4.3.7.101 Frame Consistency

The smoothed trajectory may involve multiple sensor frames.

All residuals must be evaluated consistently.

For example,

\[ \text{inertial state} \rightarrow \text{sensor frame} \rightarrow \text{predicted observation}. \]

A frame error can create systematic structure that a smoother may incorrectly absorb into the trajectory.

4.3.7.102 Smoothing Attitude

Smoothing is not limited to translational states.

Attitude may also be smoothed.

However, quaternion states require proper manifold-aware error representation.

One should not generally apply unconstrained Euclidean smoothing directly to quaternion components.

Instead, attitude corrections can be represented through local rotation vectors or multiplicative error states.

4.3.7.103 Quaternion Smoothing

Suppose the filtered attitude is

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

A local attitude error may be represented by

\[ \delta\boldsymbol{\theta}_k. \]

The smoother can operate on the local error coordinates and then reconstruct the quaternion through multiplication.

Thus,

\[ \boxed{ \text{Quaternion Nominal State} + \text{Smoothed Local Rotation Error} \rightarrow \text{Smoothed Attitude}. } \]

4.3.7.104 Relative Pose Smoothing

In close-proximity navigation, a state may include:

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

Smoothing can improve the complete relative pose history using the full measurement arc.

This can be useful for post-analysis of:

• approach geometry,
• relative attitude,
• docking alignment,
• sensor performance.

Again, this is a general application.

4.3.7.105 Smoothing for Ground Processing

Ground systems often have more:

• processing power,
• memory,
• time,
• complete measurement archives

than onboard systems.

Therefore sophisticated smoothing can often be performed on the ground even when only a recursive filter is practical onboard.

A common architecture is

\[ \boxed{ \text{Onboard Filter} \rightarrow \text{Stored Telemetry} \rightarrow \text{Ground Smoother}. } \]

4.3.7.106 Onboard Fixed-Lag Smoothing

Some spacecraft may support limited onboard smoothing.

A short-lag smoother can improve:

• delayed sensor integration,
• navigation quality,
• recent trajectory estimates.

However, guidance and control must account for the fact that a smoothed state may correspond to an earlier epoch.

4.3.7.107 Time Tag of a Smoothed Estimate

A critical point is that

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

is an improved estimate of the state at

\[ k-L, \]

not the current state at k.

Therefore it cannot be sent directly to a controller expecting a current state without proper propagation.

Thus,

\[ \boxed{ \text{Accurate Past State} \neq \text{Current Navigation State}. } \]

Engineering interpretation: Time tagging is operationally critical. A smoothed state can be more accurate but still unusable for immediate control if it corresponds to an earlier epoch and is not propagated forward consistently.

4.3.7.108 Propagating a Smoothed Past State Forward

A smoothed state at t_j may be propagated forward:

\[ \boxed{ \hat{\mathbf x}_{k} = \phi( t_k,t_j, \hat{\mathbf x}_{j|N} ). } \]

This can provide an improved current estimate if the dynamics and process-noise treatment are handled consistently.

However, covariance and correlation must also be propagated correctly.

4.3.7.109 Covariance Propagation from a Smoothed State

If deterministic linearized propagation is appropriate,

\[ \boxed{ \mathbf P_k = \boldsymbol{\Phi}_{k,j} \mathbf P_{j|N} \boldsymbol{\Phi}_{k,j}^T } \]

plus process-noise contributions if applicable.

This allows uncertainty from a smoothed reference epoch to be mapped elsewhere along the trajectory.

4.3.7.110 Retrodiction from Current State

A simpler retrodiction problem may ask:

Given a current estimated state and covariance, what was the state at an earlier epoch?

Backward propagation can provide a nominal answer.

But if measurements across the interval are available, smoothing is usually statistically more complete because it uses all relevant information rather than only the current state estimate.

4.3.7.111 Smoothing and Measurement Reprocessing

A major advantage of smoothing is the ability to incorporate:

• improved calibration,
• corrected timestamps,
• revised sensor models,
• updated environmental models.

The complete measurement archive can then be reprocessed.

This can yield a substantially improved trajectory compared with the original real-time navigation solution.

4.3.7.112 Reprocessing after Calibration Update

Suppose a sensor bias is discovered after the mission segment.

The corrected measurement model can be applied to the stored data.

Then filtering and smoothing can be rerun.

Thus,

\[ \boxed{ \text{Archived Measurements} + \text{Improved Model} \rightarrow \text{Improved Reconstructed Trajectory}. } \]

This is one reason storing raw sensor data is valuable.

4.3.7.113 Filter/Smoother Consistency Check

Comparing

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

with

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

can reveal how much later information changed the state estimate.

Define

\[ \boxed{ \Delta\mathbf x_{s,k} = \hat{\mathbf x}_{k|N} - \hat{\mathbf x}_{k|k}. } \]

Large systematic smoothing corrections may indicate:

• weak earlier observability,
• model mismatch,
• poor filtering assumptions,
• delayed information.

This can be a useful diagnostic.

4.3.7.114 Smoothed Correction History

Plotting

\[ \Delta\mathbf x_{s,k} \]

versus time can reveal when future data contributed strongly.

For example:

• large corrections near an event,
• large corrections during measurement gaps,
• persistent bias-like corrections.

This is useful when analysing navigation performance.

4.3.7.115 Smoothed Covariance Reduction

A useful metric is

\[ \boxed{ \Delta\mathbf P_k = \mathbf P_{k|k} - \mathbf P_{k|N}. } \]

This indicates how much covariance reduction was achieved through future information.

Eigenvalue comparison can reveal which state directions benefited most from smoothing.

4.3.7.116 Position-Uncertainty Ellipsoid

For a three-dimensional position covariance,

\[ \mathbf P_{rr}, \]

eigenvalue decomposition gives

\[ \boxed{ \mathbf P_{rr} = \mathbf V \boldsymbol{\Lambda} \mathbf V^T. } \]

The eigenvectors define uncertainty-ellipsoid directions.

The eigenvalues define squared principal-axis uncertainty.

Comparing filtered and smoothed ellipsoids gives a geometric visualization of smoothing benefit.

4.3.7.117 Smoothing Performance Metrics

Useful metrics include:

• RMS state error,
• final or average covariance,
• covariance reduction,
• maximum smoothing correction,
• residual RMS,
• normalized residual statistics,
• error-envelope consistency,
• reconstruction accuracy around maneuvers or events.

Performance should be assessed over the full interval, not at only one epoch.

4.3.7.118 Monte Carlo Validation

In simulation, run many cases with random:

• initial errors,
• process noise,
• measurement noise,
• sensor biases,
• parameter uncertainties.

For each run, compare:

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

and

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

against truth.

A consistent smoother should show statistically credible uncertainty reduction without becoming overconfident.

Engineering interpretation: Monte Carlo validation should compare both error reduction and covariance credibility. A smoother that reduces RMS error but reports overconfident covariance is still statistically inconsistent.

4.3.7.119 Smoothing Can Reveal Filter Deficiencies

If smoothed corrections are consistently large, possible explanations include:

• filter covariance too small,
• process noise too small,
• measurement model errors,
• delayed sensor information,
• weak real-time observability.

Thus the smoother is not just a reconstruction tool; it can also diagnose real-time estimator performance.

4.3.7.120 Smoother Consistency

A smoother that reports very small covariance but still has large truth error is inconsistent.

Therefore,

\[ \boxed{ \text{Lower Covariance} \neq \text{Automatically Better Estimation}. } \]

Covariance reduction must be supported by actual error reduction in simulation or by credible statistical diagnostics.

4.3.7.121 Numerical Implementation of RTS Gain

Although written as

\[ \mathbf G_k = \mathbf P_{k|k} \mathbf F_k^T \mathbf P_{k+1|k}^{-1}, \]

one should usually avoid explicitly computing

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

Instead solve a linear system.

This improves numerical stability.

4.3.7.122 Positive Semidefinite Covariance

After smoothing,

\[ \mathbf P_{k|N} \]

should remain symmetric positive semidefinite.

Finite-precision arithmetic may introduce small asymmetry.

Numerical checks should include:

\[ \boxed{ \mathbf P \approx \mathbf P^T } \]

and nonnegative eigenvalues within numerical tolerance.

4.3.7.123 Cholesky and Square-Root Methods

For high-precision applications, factorized covariance methods may be preferred.

If

\[ \mathbf P = \mathbf L\mathbf L^T, \]

factor-based propagation avoids some numerical problems associated with direct covariance arithmetic.

This is particularly important when uncertainties span several orders of magnitude.

4.3.7.124 Smoothing with Large State Dimensions

Large smoothing problems may involve:

• many trajectory states,
• biases,
• calibration parameters,
• environmental parameters.

Direct dense-matrix methods may become expensive.

Sparse smoothing methods exploit the fact that dynamics usually connect only nearby states.

4.3.7.125 Block-Tridiagonal Structure

In Markov dynamic models, each state is directly connected mainly to neighbouring states.

This often creates block-tridiagonal structure in the batch information matrix.

Such structure can be exploited for efficient smoothing.

This is one reason trajectory smoothing can remain computationally practical even for long state histories.

4.3.7.126 Relation to Factor-Graph Sparsity

A factor graph visually exposes the same structure:

\[ \boxed{ \mathbf x_k \leftrightarrow \mathbf x_{k+1} } \]

through dynamics, with measurements attached locally.

Sparse linear algebra then exploits this local connectivity.

RTS smoothing can be viewed as one efficient elimination strategy for a linear-Gaussian chain.

Engineering interpretation: The block-tridiagonal structure explains why long trajectory-smoothing problems can often be solved efficiently: each state is locally coupled to only a few neighbours.

4.3.7.127 Smoothing as Information Redistribution

Filtering accumulates information only forward.

Smoothing redistributes that accumulated information across the trajectory.

Thus,

\[ \boxed{ \text{Smoother Does Not Create New Measurements}. } \]

Instead,

\[ \boxed{ \text{Smoother Uses Existing Measurements More Completely Across Time}. } \]

This is a fundamental interpretation.

4.3.7.128 Causality

Filtering is causal:

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

uses no future data.

Full smoothing is noncausal:

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

uses future data.

Therefore a fixed-interval smoother cannot normally replace a real-time navigation filter for immediate control decisions.

4.3.7.129 Fixed-Lag Smoothing and Near-Causality

A fixed-lag smoother introduces finite delay.

For lag L,

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

becomes available at time k.

Thus it is causal with respect to its delayed output but not with respect to the original state epoch.

This distinction matters in real-time systems.

4.3.7.130 Smoothing and Guidance/Control

Guidance and control typically require a current-state estimate.

Therefore they usually consume

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

rather than

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

Smoothed estimates are more commonly used for:

• monitoring,
• reconstruction,
• calibration,
• offline analysis,
• delayed supervisory logic.

4.3.7.131 Smoothing in System Identification

If unknown dynamic parameters are estimated, smoothing can improve the state trajectory used for parameter identification.

A better state reconstruction may yield better parameter estimates.

This leads to iterative procedures such as:

\[ \boxed{ \text{State Smoothing} \rightarrow \text{Parameter Update} \rightarrow \text{State Smoothing Again}. } \]

4.3.7.132 Expectation-Maximization Connection

A common probabilistic parameter-estimation method is Expectation-Maximization, or EM.

Conceptually:

E-step

Estimate or smooth the state trajectory using current parameter values.

M-step

Update parameters using the smoothed trajectory.

Thus,

\[ \boxed{ \text{Smoothing} \rightarrow \text{Parameter Re-estimation} } \]

forms an important part of some system-identification algorithms.

4.3.7.133 Smoothing and Sensor Alignment Estimation

Suppose an unknown sensor alignment matrix depends on parameter vector

\[ \boldsymbol{\theta}_{align}. \]

The full trajectory provides measurements over changing geometry.

This can make alignment parameters more observable than they would be over a short interval.

Smoothing and batch estimation can therefore support post-calibration of alignment parameters.

4.3.7.134 Smoothing and Timing Calibration

Similarly, timing offsets may become identifiable because the state evolves over time.

A measurement timestamp shift changes predicted geometry.

Using a long, dynamically rich trajectory can provide strong timing information.

Thus,

\[ \boxed{ \text{Dynamic Variation} \rightarrow \text{Timing Sensitivity} \rightarrow \text{Possible Time-Bias Estimation}. } \]

4.3.7.135 General RPO Interpretation

In a general RPO problem, the real-time estimator must provide current relative navigation for guidance and control.

After the operation, a smoother may use the full sensor history to reconstruct:

\[ \mathbf r_{rel}(t), \qquad \mathbf v_{rel}(t), \qquad \mathbf q_{rel}(t), \qquad \boldsymbol{\omega}_{rel}(t). \]

The smoother may help answer questions such as:

• What was the best estimated separation at a given time?
• What relative velocity existed before or after a maneuver?
• How did measurement geometry affect navigation uncertainty?
• Were later measurements consistent with earlier filter estimates?

These are general estimation questions and not specific to any particular spacecraft.

4.3.7.136 Post-Event Navigation Reconstruction

For docking, rendezvous, flyby, formation flight, or close approach, a smoothed trajectory can provide a high-quality reference for:

• performance assessment,
• sensor validation,
• control-performance analysis,
• reconstruction of approach geometry.

Again, the purpose is retrospective estimation, not real-time control.

4.3.7.137 Smoothing and Truth Data

During simulation or ground testing, true state may be known.

Then smoothing performance can be directly assessed:

\[ \boxed{ \mathbf e_{smooth,k} = \mathbf x_{true,k} - \hat{\mathbf x}_{k|N}. } \]

Compare this with filtering error:

\[ \boxed{ \mathbf e_{filter,k} = \mathbf x_{true,k} - \hat{\mathbf x}_{k|k}. } \]

This provides a direct measure of smoothing benefit.

4.3.7.138 Normalized Estimation Error

A normalized consistency measure can use

\[ \boxed{ \epsilon_k = \mathbf e_k^T \mathbf P_k^{-1} \mathbf e_k. } \]

For a smoother, use

\[ \boxed{ \epsilon_{s,k} = \mathbf e_{s,k}^T \mathbf P_{k|N}^{-1} \mathbf e_{s,k}. } \]

This helps determine whether the covariance reduction is statistically justified.

4.3.7.139 Smoothing before Parameter Analysis

When analysing forces or parameters from reconstructed trajectory data, it is often preferable to use a smoothed trajectory rather than a noisy filtered one.

However, one must account for the correlation structure introduced by smoothing.

Smoothed states are not independent observations.

4.3.7.140 Correlation between Smoothed States

Because the entire data set contributes to multiple state estimates,

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

and

\[ \hat{\mathbf x}_{j|N} \]

are statistically correlated.

Ignoring these correlations in downstream parameter estimation can lead to overconfidence.

Therefore full covariance or cross-covariance may be required.

4.3.7.141 Smoothing and Interpolation

Smoothing estimates states at selected epochs.

A continuous trajectory may then be reconstructed through:

• dynamic propagation,
• interpolation,
• dense integration output.

Interpolation error should remain below the required trajectory-accuracy level.

4.3.7.142 Continuous-Time Smoothing

For continuous-time dynamics,

\[ \dot{\mathbf x} = f( \mathbf x,t ) + \mathbf G(t)\mathbf w(t), \]

measurements may occur at discrete times.

Continuous-discrete smoothers propagate information across continuous dynamics while applying observations at their measurement epochs.

The principle remains the same: future observations improve earlier continuous-time states.

4.3.7.143 State Transition Matrix in Smoothing

For linearized continuous dynamics,

\[ \boxed{ \delta\mathbf x(t_{k+1}) = \boldsymbol{\Phi}(t_{k+1},t_k) \delta\mathbf x(t_k). } \]

Therefore the transition matrix

\[ \boldsymbol{\Phi} \]

plays the same role as

\[ \mathbf F_k \]

in discrete smoothing.

The RTS gain can therefore be written conceptually using propagated covariance and STM relationships.

4.3.7.144 Sensitivity of Smoothing to Dynamic Model

Because smoothing transfers information through dynamics, its performance is sensitive to dynamic-model accuracy.

If the transition model is incorrect, future measurements may be mapped backward incorrectly.

Therefore,

\[ \boxed{ \text{Better Future Measurements} \text{ cannot completely compensate for} \text{Incorrect Dynamic Coupling}. } \]

4.3.7.145 Smoothing with Model Switching

If dynamics change between modes, different transition models may apply:

\[ f^{(1)}, \quad f^{(2)}, \quad \ldots \]

The smoother must know or estimate the mode sequence.

Examples include:

• thrusting versus coasting,
• sensor modes,
• attitude-control modes.

Incorrect mode assumptions can distort reconstruction.

4.3.7.146 Smoothing around Maneuver Boundaries

A maneuver boundary should generally be represented explicitly.

Otherwise a smoother may attempt to fit the discontinuity by gradually bending the trajectory.

This can produce misleading pre- and post-maneuver state estimates.

Thus event modelling is essential for accurate smoothing.

4.3.7.147 Smoothing and State Constraints

Constraints may also apply during smoothing.

Examples include:

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

parameter positivity, or physical bounds.

Possible approaches include:

• constrained optimization,
• local-error representations,
• projection,
• parameter transformation.

The constraint should be incorporated consistently throughout the trajectory.

4.3.7.148 Smoothing and Prior Information

The state at the beginning of the smoothing interval may have prior

\[ \boxed{ \mathbf x_0 \sim \mathcal N( \bar{\mathbf x}_0, \mathbf P_0 ). } \]

This prior influences the entire trajectory.

If it is strong, early states may remain closely tied to the initial estimate.

If it is weak, measurements and dynamics dominate more strongly.

4.3.7.149 Prior versus Future Data

At an early epoch, the smoothed estimate may reflect both:

\[ \boxed{ \text{Initial Prior Information} + \text{Future Measurement Information}. } \]

These contributions should be distinguished when interpreting uncertainty.

A small smoothed covariance may result partly from a strong prior rather than from measurement data alone.

4.3.7.150 Smoothing with Multiple Sensors

Suppose the trajectory is observed by:

• GNSS,
• IMU,
• star tracker,
• optical camera,
• ranging sensor.

The forward filter integrates the sensors as they arrive.

The smoother then redistributes their combined information across time.

This can improve states at epochs when one or more sensors were temporarily weak or unavailable.

4.3.7.151 Asynchronous Multi-Sensor Smoothing

Sensors may have different rates and timestamps.

The smoother must retain the correct association between:

\[ \mathbf z_k \]

and the state epoch where that measurement was generated.

Time alignment remains as important in smoothing as in filtering.

4.3.7.152 Correlated Multi-Sensor Errors

If sensors share common errors, smoothing should not treat them as completely independent.

Examples include:

• common clock,
• common environmental corrections,
• shared preprocessing.

Ignoring correlation can produce unrealistically small smoothed covariance.

4.3.7.153 Double Counting in Smoothing

If an already-fused estimate is later included as though it were an independent raw measurement, information can be counted twice.

The result may appear highly precise but be statistically overconfident.

Thus,

\[ \boxed{ \text{Information Provenance Matters in Smoothing}. } \]

Engineering interpretation: Double counting is especially dangerous in smoothing because an already-fused estimate may contain information from many measurements that also appear separately elsewhere in the graph or batch problem.

4.3.7.154 Offline Reprocessing Advantage

Unlike onboard filtering, offline smoothing can often afford:

• more accurate force models,
• better ephemerides,
• refined calibration,
• more computationally intensive solvers.

Therefore the final reconstructed trajectory may differ from the real-time estimate because both the data set and the models have improved.

4.3.7.155 Smoothing as a Validation Reference

A high-quality post-processed smoothed trajectory can serve as a reference for evaluating:

• onboard filter performance,
• navigation latency,
• sensor quality,
• control execution.

It is not necessarily absolute truth, but it can be a better post-processed estimate.

4.3.7.156 Common Failure Modes

Common smoothing failures include:

• incorrect forward filter,
• inconsistent covariance,
• wrong transition matrices,
• underestimated process noise,
• incorrect measurement covariance,
• ignored measurement correlations,
• bad timestamps,
• wrong frames,
• unmodeled maneuvers,
• numerical instability,
• weak observability,
• excessive parameter augmentation,
• outliers.

Smoothing amplifies the importance of correct temporal modelling because information is transferred across the entire interval.

4.3.7.157 Diagnosing a Poor Smoothed Solution

Check systematically:

1. Does the forward filter behave consistently?
2. Are stored predicted and filtered states correct?
3. Is the transition model correct?
4. Are process-noise terms correct?
5. Are timestamps correct?
6. Are measurement frames consistent?
7. Are smoother-gain dimensions and signs correct?
8. Is covariance symmetric?
9. Does smoothing reduce actual truth error in simulation?
10. Are smoothing corrections physically plausible?
11. Are maneuver boundaries modelled?
12. Are later measurements causing unexpected large past corrections?
13. Is the solution sensitive to Q or R?
14. Is the final smoothed state equal to the final filtered state for fixed-interval RTS?

That last check is especially useful:

\[ \boxed{ \hat{\mathbf x}_{N|N}^{s} = \hat{\mathbf x}_{N|N}. } \]

Engineering interpretation: The final-state equality is a powerful RTS sanity check because there is no future information beyond the end of the interval. If it fails, indexing or recursion logic is likely wrong.

4.3.7.158 Practical Fixed-Interval Smoothing Workflow

A disciplined workflow is:

1. Define state and parameters.
2. Define dynamics.
3. Define process noise.
4. Define measurement models.
5. Define measurement covariances.
6. Initialize state and covariance.
7. Run the forward filter.
8. Store filtered state at every epoch.
9. Store filtered covariance.
10. Store one-step predictions.
11. Store predicted covariance.
12. Store transition matrices or required cross-covariances.
13. Check forward-filter consistency.
14. Initialize smoother at the final epoch.
15. Move backward from N-1 to 0.
16. Compute smoother gain.
17. Compute smoothed state.
18. Compute smoothed covariance.
19. Preserve covariance symmetry.
20. Recompute residuals using the smoothed trajectory.
21. Compare filtered and smoothed states.
22. Examine covariance reduction.
23. Examine smoothing-correction history.
24. Check maneuvers and discontinuities.
25. Validate against truth if available.
26. Test sensitivity to Q.
27. Test sensitivity to R.
28. Test sensitivity to initial conditions.
29. Check numerical stability.
30. Store the final reconstructed trajectory.

4.3.7.159 Complete Fixed-Interval Architecture

\[ \boxed{ \begin{array}{c} \text{Initial State and Covariance} \\ \downarrow\\ \text{Forward Filter} \\ \downarrow\\ \hat{\mathbf x}_{0|0}, \hat{\mathbf x}_{1|1}, \ldots, \hat{\mathbf x}_{N|N} \\ + \\ \mathbf P_{0|0}, \mathbf P_{1|1}, \ldots, \mathbf P_{N|N} \\ + \\ \hat{\mathbf x}_{k+1|k}, \mathbf P_{k+1|k} \\ \downarrow\\ \text{Initialize at Final Epoch} \\ \hat{\mathbf x}_{N|N}^{s} = \hat{\mathbf x}_{N|N} \\ \downarrow\\ \text{Backward Recursion} \\ \mathbf G_k \\ \downarrow\\ \hat{\mathbf x}_{k|N} = \hat{\mathbf x}_{k|k} + \mathbf G_k ( \hat{\mathbf x}_{k+1|N} - \hat{\mathbf x}_{k+1|k} ) \\ \downarrow\\ \mathbf P_{k|N} \\ \downarrow\\ \text{Smoothed State History} \\ \downarrow\\ \text{Trajectory Reconstruction and Validation} \end{array} } \]

4.3.7.160 Practical Fixed-Lag Workflow

For a fixed lag L:

1. Run the real-time forward filter.
2. Maintain a rolling buffer of the last L states.
3. Store corresponding covariances and transition quantities.
4. When a new measurement arrives, update the current state.
5. Run a backward smoothing pass through the stored window.
6. Output

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

1. Remove the oldest state from the buffer.
2. Advance the window.

Thus,

\[ \boxed{ \text{Streaming Measurements} \rightarrow \text{Rolling Smoothing Window}. } \]

4.3.7.161 Key Equations

Filtered estimate

\[ \boxed{ \hat{\mathbf x}_{k|k} = E[ \mathbf x_k | \mathbf z_{1:k} ] } \]

Smoothed estimate

\[ \boxed{ \hat{\mathbf x}_{k|N} = E[ \mathbf x_k | \mathbf z_{1:N} ], \qquad N>k } \]

Fixed-lag estimate

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

Linear state prediction

\[ \boxed{ \hat{\mathbf x}_{k+1|k} = \mathbf F_k \hat{\mathbf x}_{k|k} } \]

Predicted covariance

\[ \boxed{ \mathbf P_{k+1|k} = \mathbf F_k \mathbf P_{k|k} \mathbf F_k^T + \mathbf Q_k } \]

RTS smoother gain

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

RTS smoothed-state update

\[ \boxed{ \hat{\mathbf x}_{k|N} = \hat{\mathbf x}_{k|k} + \mathbf G_k \left( \hat{\mathbf x}_{k+1|N} - \hat{\mathbf x}_{k+1|k} \right) } \]

RTS smoothed-covariance update

\[ \boxed{ \mathbf P_{k|N} = \mathbf P_{k|k} + \mathbf G_k \left( \mathbf P_{k+1|N} - \mathbf P_{k+1|k} \right) \mathbf G_k^T } \]

Final condition

\[ \boxed{ \hat{\mathbf x}_{N|N}^{s} = \hat{\mathbf x}_{N|N} } \]

\[ \boxed{ \mathbf P_{N|N}^{s} = \mathbf P_{N|N} } \]

Dynamic residual in batch smoothing

\[ \boxed{ \mathbf r_{f,k} = \mathbf x_{k+1} - f_k( \mathbf x_k ) } \]

Measurement residual

\[ \boxed{ \mathbf r_{z,k} = \mathbf z_k - h_k( \mathbf x_k ) } \]

Gaussian trajectory-MAP cost

\[ \boxed{ \begin{aligned} J ={}& \| \mathbf x_0-\bar{\mathbf x}_0 \|_{\mathbf P_0^{-1}}^2 \\ &+ \sum_{k=0}^{N-1} \| \mathbf x_{k+1} - f_k( \mathbf x_k ) \|_{\mathbf Q_k^{-1}}^2 \\ &+ \sum_{k=1}^{N} \| \mathbf z_k - h_k( \mathbf x_k ) \|_{\mathbf R_k^{-1}}^2 \end{aligned} } \]

Smoothed-state correction relative to filtering

\[ \boxed{ \Delta\mathbf x_{s,k} = \hat{\mathbf x}_{k|N} - \hat{\mathbf x}_{k|k} } \]

Covariance improvement

\[ \boxed{ \Delta\mathbf P_k = \mathbf P_{k|k} - \mathbf P_{k|N} } \]

4.3.7.162 Key Takeaways

Filtering uses only measurements available through the current epoch:

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

Smoothing estimates a past state using later measurements:

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

Therefore,

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

while

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

The fundamental reason smoothing works is that spacecraft states are dynamically correlated across time.

Thus,

\[ \boxed{ \text{Future Measurement} \rightarrow \text{Future-State Improvement} \rightarrow \text{Past-State Improvement}. } \]

Three important smoothing forms are:

\[ \boxed{ \text{Fixed Interval}, \quad \text{Fixed Lag}, \quad \text{Fixed Point}. } \]

The Rauch–Tung–Striebel smoother uses a forward filtering pass followed by a backward recursion.

Its key gain is

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

The smoothed state is

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

The interpretation is straightforward:

\[ \boxed{ \text{Past Smoothed State} = \text{Past Filtered State} + \text{Back-Propagated Future Improvement}. } \]

Smoothing commonly reduces covariance:

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

But reduced covariance is meaningful only if the model and uncertainty assumptions are correct.

The benefit of smoothing depends strongly on:

• dynamic coupling,
• process noise,
• measurement noise,
• future measurement geometry,
• observability,
• sensor availability.

Smoothing is particularly useful for:

• trajectory reconstruction,
• orbit reconstruction,
• maneuver reconstruction,
• general relative-navigation analysis,
• calibration,
• fault analysis,
• post-flight navigation validation.

However,

\[ \boxed{ \text{Smoothing Does Not Create Information}. } \]

It redistributes information already present in the measurement history and dynamic model.

Also,

\[ \boxed{ \text{Smoothing Does Not Repair Incorrect Models Automatically}. } \]

A wrong frame, timing model, sensor model, maneuver model, or covariance assumption can still produce a precise-looking but incorrect reconstructed trajectory.

Finally,

\[ \boxed{ \text{Batch Estimation} \leftrightarrow \text{Trajectory MAP Estimation} \leftrightarrow \text{Smoothing} } \]

are closely related mathematical views of the same fundamental problem:

How can all available information over a time interval be used to obtain the best estimate of the spacecraft state history?

4.3.7.163 What Comes Next

Smoothing relies fundamentally on relationships between states at different times.

For example, the RTS gain contains the transition matrix

\[ \mathbf F_k, \]

and dynamic trajectory estimation repeatedly uses

\[ \boldsymbol{\Phi}(t_k,t_0). \]

Likewise, parameter estimation requires quantities such as

\[ \frac{\partial\mathbf x(t)} {\partial\theta}. \]

This raises the next fundamental question:

\[ \boxed{ \text{How exactly do small perturbations in an initial state or model parameter propagate through spacecraft dynamics?} } \]

The answer is provided by State Transition Matrices and sensitivity matrices.

That leads directly to

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

where the next section should develop:

• the formal STM definition,
• state perturbations,
• local linearization of dynamics,
• variational equations,
• STM differential equations,
• STM composition and inverse properties,
• covariance propagation,
• parameter sensitivity matrices,
• sensitivity differential equations,
• measurement sensitivities,
• dynamic design matrices,
• trajectory sensitivity,
• control-input sensitivity,
• maneuver sensitivity,
• numerical and finite-difference STM computation,
• STM verification,
• conditioning,
• and the role of sensitivity matrices in orbit determination, batch estimation, filtering, smoothing, guidance, and general RPO analysis.

Next: 4.3.8 State Transition and Sensitivity Matrices

Smoothing depends on how perturbations and uncertainty are connected between states at different epochs. The next section develops the state transition matrix, variational equations, parameter and control sensitivities, covariance mapping, dynamic design matrices, observability links, numerical STM computation, and verification.