Maneuver and Dynamic-Parameter Estimation

Impulsive and finite-burn estimation, burn timing, thrust magnitude and direction, empirical accelerations, atmospheric-drag parameters, solar-radiation-pressure parameters, observability, covariance, and physically meaningful maneuver reconstruction.

4.3.15 Maneuver and Dynamic-Parameter Estimation

Maneuver and dynamic-parameter estimation extends orbit determination beyond estimating only position and velocity. In many spacecraft-navigation problems, the trajectory depends on quantities that are uncertain, imperfectly known, or not known at all.

Examples include:

Instead of forcing the orbit state to absorb all these effects, the estimator can explicitly solve for the uncertain maneuver or dynamic parameters.

The central idea is

\[ \boxed{ \text{Trajectory Measurements} + \text{Dynamic Model} \rightarrow \text{State Estimate} + \text{Maneuver / Dynamic-Parameter Estimate} } \]

This section focuses on six major areas:

4.3.15.1 Why Maneuver Estimation Is Needed

Suppose a commanded impulsive maneuver is

\[ \Delta \mathbf v_{\rm cmd}. \]

The actual maneuver may be

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

where

\[ \delta\Delta \mathbf v \]

represents execution error.

Possible causes include:

If the OD system assumes the commanded maneuver was executed perfectly, the post-maneuver trajectory prediction may become biased.

4.3.15.2 Maneuver Error as a Dynamic Discontinuity

For an ideal impulsive burn at time t_m,

\[ \boxed{ \mathbf r^+ = \mathbf r^- } \]

while velocity changes instantaneously:

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

The superscripts - and + denote the states immediately before and after the maneuver.

Thus the maneuver introduces a discontinuity in velocity but not position.

4.3.15.3 Impulsive Maneuver Model

For state

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

an impulsive maneuver can be written

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

with

\[ \boxed{ \mathbf B = \begin{bmatrix} \mathbf 0_{3\times3}\\ \mathbf I_{3\times3} \end{bmatrix}. } \]

This is the simplest mathematical representation of a maneuver in OD.

4.3.15.4 Known versus Unknown Maneuver

If the maneuver is perfectly known,

\[ \Delta \mathbf v = \Delta \mathbf v_{\rm known}, \]

it can simply be inserted into the propagator.

If it is uncertain, write

\[ \boxed{ \Delta \mathbf v = \Delta \mathbf v_{\rm nominal} + \delta\Delta \mathbf v. } \]

Then

\[ \delta\Delta \mathbf v \]

can be estimated.

4.3.15.5 Maneuver Correction as an Estimated Parameter

The augmented unknown vector may be

\[ \boxed{ \mathbf y = \begin{bmatrix} \mathbf x_0\\ \delta\Delta\mathbf v \end{bmatrix}. } \]

Measurements before and after the maneuver constrain both the reference state and the maneuver correction.

This is especially powerful in batch OD because the complete observation arc can be processed simultaneously.

4.3.15.6 Maneuver Sensitivity

For a measurement after the maneuver,

\[ \mathbf z_k = h_k( \mathbf x_k ), \]

the measurement sensitivity to the maneuver is

\[ \boxed{ \frac{\partial\mathbf z_k} {\partial\Delta\mathbf v} = \mathbf H_k \boldsymbol\Phi( t_k,t_m^+ ) \mathbf B. } \]

This equation shows how an impulsive velocity error propagates forward and changes later measurements.

4.3.15.7 Measurements Before the Maneuver

Measurements collected before t_m generally have

\[ \boxed{ \frac{\partial\mathbf z_k} {\partial\Delta\mathbf v} = \mathbf0 \qquad t_k<t_m } \]

because the maneuver has not yet occurred.

Measurements after the burn contain information about the maneuver.

4.3.15.8 Why Pre- and Post-Maneuver Measurements Help

Measurements before the maneuver constrain the pre-burn trajectory.

Measurements after the maneuver constrain the post-burn trajectory.

The difference between the dynamically consistent pre- and post-burn trajectories provides information about the delivered maneuver.

Conceptually,

\[ \boxed{ \text{Pre-Burn Orbit} + \text{Post-Burn Orbit} \rightarrow \text{Estimated Maneuver} } \]

4.3.15.9 Maneuver Covariance

Suppose maneuver execution error has covariance

\[ \mathbf P_{\Delta v}. \]

Then after the maneuver,

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

where \mathbf M represents the deterministic state-event mapping.

For an ideal position-continuous impulsive maneuver,

\[ \mathbf M \]

may be the identity for the basic state.

4.3.15.10 Maneuver Covariance Interpretation

The term

\[ \mathbf B \mathbf P_{\Delta v} \mathbf B^T \]

injects uncertainty into the velocity state.

That velocity uncertainty later propagates into position uncertainty.

Thus,

\[ \boxed{ \text{Maneuver Uncertainty} \rightarrow \text{Velocity Uncertainty} \rightarrow \text{Future Position Uncertainty}. } \]

4.3.15.11 Components of Impulsive Maneuver Error

The maneuver error can be expressed in Cartesian form:

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

Alternatively, it can be expressed in a physically meaningful local orbital frame such as radial-transverse-normal:

\[ \boxed{ \delta\Delta\mathbf v_{\rm RTN} = \begin{bmatrix} \delta\Delta v_R\\ \delta\Delta v_T\\ \delta\Delta v_N \end{bmatrix}. } \]

The most suitable frame depends on the maneuver geometry.

4.3.15.12 Maneuver Magnitude Error

Let the commanded maneuver magnitude be

\[ \Delta v_c. \]

If the actual magnitude differs by scale factor \epsilon_m,

\[ \boxed{ \Delta v_{\rm actual} = (1+\epsilon_m) \Delta v_c. } \]

Then \epsilon_m can be estimated instead of estimating three independent Cartesian components.

This may improve conditioning when the commanded direction is known accurately.

4.3.15.13 Maneuver Direction Error

Let the nominal burn direction be

\[ \hat{\mathbf u}. \]

The commanded maneuver is

\[ \Delta\mathbf v_c = \Delta v_c \hat{\mathbf u}. \]

A small direction error can be represented by a small rotation vector

\[ \delta\boldsymbol\theta. \]

Then approximately,

\[ \boxed{ \delta\hat{\mathbf u} \approx - [ \hat{\mathbf u}\times ] \delta\boldsymbol\theta. } \]

This converts small angular misalignment into a velocity-vector error.

4.3.15.14 Combined Magnitude and Direction Error

A useful maneuver parameterization is

\[ \boxed{ \Delta\mathbf v = (1+\epsilon_m) \Delta v_c \hat{\mathbf u}_{\rm actual}. } \]

For small directional errors,

\[ \hat{\mathbf u}_{\rm actual} \approx \hat{\mathbf u} - [ \hat{\mathbf u}\times ] \delta\boldsymbol\theta. \]

Thus,

\[ \boxed{ \delta\Delta\mathbf v \approx \epsilon_m \Delta v_c \hat{\mathbf u} - \Delta v_c [ \hat{\mathbf u}\times ] \delta\boldsymbol\theta. } \]

This separates thrust-magnitude error from thrust-direction error.

4.3.15.15 Why Parameterization Matters

Estimating

\[ \Delta v_x,\Delta v_y,\Delta v_z \]

is very flexible.

Estimating

may be more physically meaningful.

The best parameterization should:

Thus,

\[ \boxed{ \text{Parameterization} \text{ should reflect what the measurements can actually identify.} } \]

4.3.15.16 Burn-Time Estimation

The burn epoch itself may be uncertain.

Let the nominal burn time be

\[ t_m. \]

The actual time may be

\[ \boxed{ t_{m,\rm actual} = t_m + \delta t_m. } \]

A timing error shifts where along the trajectory the velocity change is applied.

4.3.15.17 Why Burn Timing Matters

Suppose the spacecraft travels with velocity

\[ \mathbf v. \]

A burn occurring \delta t early or late changes both

Even if the delivered \Delta v is identical, a timing shift produces a different future trajectory.

Thus,

\[ \boxed{ \text{Same }\Delta\mathbf v + \text{Different Burn Time} \rightarrow \text{Different Orbit}. } \]

4.3.15.18 Burn-Time Sensitivity

For a small timing error,

\[ \delta t_m, \]

the state effect may be approximated locally using the dynamics immediately before and after the event.

A useful first-order event-time sensitivity is

\[ \boxed{ \frac{\partial\mathbf x^+} {\partial t_m} \approx \mathbf f^- - \mathbf f^+ } \]

for an ideal event mapping under appropriate formulation.

The exact expression depends on how the maneuver event is modeled.

4.3.15.19 Burn Timing and Position Error

For a simplified case with approximately constant pre-burn velocity,

\[ \delta\mathbf r \approx \mathbf v \delta t. \]

Thus even modest timing uncertainty can produce a meaningful spatial shift at the maneuver event.

This is especially important for high-speed trajectories.

4.3.15.20 Burn Timing and Estimability

Burn timing is estimable only if the available measurements are sensitive to it.

Timing may be difficult to separate from

Therefore,

\[ \boxed{ \text{Burn-Time Estimation} \text{ requires sufficient geometry and timing information.} } \]

4.3.15.21 Burn Start and Burn Stop Time

For a finite-duration burn, there are two important event times:

\[ \boxed{ t_{\rm start} } \]

and

\[ \boxed{ t_{\rm stop}. } \]

Burn duration is

\[ \boxed{ T_b = t_{\rm stop} - t_{\rm start}. } \]

Uncertainty may exist in one or both times.

4.3.15.22 Finite-Burn Dynamics

A finite burn cannot always be approximated as an instantaneous \Delta v.

During the burn,

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

and

\[ \boxed{ \dot{\mathbf v} = \mathbf a_{\rm natural} + \mathbf a_T. } \]

The thrust acceleration is

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

4.3.15.23 Variable-Mass Dynamics

If propellant consumption is significant,

\[ m=m(t). \]

Then

\[ \boxed{ \mathbf a_T(t) = \frac{\mathbf F_T(t)} {m(t)}. } \]

The mass evolution may follow

\[ \boxed{ \dot m = -\dot m_p. } \]

Thus finite-burn orbit determination may require propagation of mass as an additional state.

4.3.15.24 Thrust Magnitude

Let the commanded thrust magnitude be

\[ F_c(t). \]

A simple scale-factor model is

\[ \boxed{ F_{\rm actual}(t) = (1+s_F) F_c(t). } \]

The parameter

\[ s_F \]

can be estimated.

4.3.15.25 Thrust Bias

An additive thrust bias model may be

\[ \boxed{ F_{\rm actual} = F_c+b_F. } \]

Whether an additive or multiplicative representation is better depends on the actuator physics.

For many propulsion systems, percentage scale error is more physically intuitive than a fixed additive error.

4.3.15.26 Thrust Vector

The thrust force is

\[ \boxed{ \mathbf F_T = F_T \hat{\mathbf u}_T. } \]

Therefore thrust uncertainty can arise from two separate sources:

\[ \boxed{ \text{Thrust Magnitude Error} } \]

and

\[ \boxed{ \text{Thrust Direction Error}. } \]

These should not automatically be combined into one scalar uncertainty.

4.3.15.27 Thrust-Direction Parameterization

One representation uses two small angular deviations:

\[ \boxed{ \delta\boldsymbol\theta_T = \begin{bmatrix} \delta\theta_1\\ \delta\theta_2 \end{bmatrix}. } \]

A two-angle description is usually sufficient because a unit direction vector has only two independent degrees of freedom.

4.3.15.28 Misalignment Matrix

A small thrust-direction error can be represented as

\[ \boxed{ \mathbf C_{\rm actual} \approx \mathbf I - [ \delta\boldsymbol\theta\times ]. } \]

Then

\[ \boxed{ \mathbf F_{\rm actual} \approx ( \mathbf I - [ \delta\boldsymbol\theta\times ] ) \mathbf F_{\rm nominal}. } \]

This form is useful for estimating small propulsion misalignment.

4.3.15.29 Body-to-Inertial Thrust Direction

If thrust direction is defined in spacecraft body coordinates,

\[ \mathbf F_B, \]

the inertial thrust is

\[ \boxed{ \mathbf F_I = \mathbf C_{IB} \mathbf F_B. } \]

Therefore attitude error can also produce apparent thrust-direction error.

This creates correlation between

4.3.15.30 Attitude Error during Finite Burn

During a long burn, spacecraft attitude may evolve.

Therefore

\[ \hat{\mathbf u}_T = \hat{\mathbf u}_T(t). \]

The total delivered impulse is

\[ \boxed{ \mathbf J = \int_{t_{\rm start}}^{t_{\rm stop}} \mathbf F_T(t) \,dt. } \]

If attitude changes during the burn, the inertial direction of this impulse changes continuously.

4.3.15.31 Effective Delta-V from a Finite Burn

Ignoring gravity losses and assuming constant mass approximately,

\[ \boxed{ \Delta\mathbf v_T \approx \frac{1}{m} \int_{t_s}^{t_e} \mathbf F_T(t)\,dt. } \]

But an accurate finite-burn OD model integrates the thrust acceleration together with the natural dynamics.

4.3.15.32 Why Finite Burn Cannot Always Be Reduced to Impulsive Burn

An impulsive approximation assumes all velocity change occurs at one instant.

This may be inadequate when

Thus,

\[ \boxed{ \text{Long / Precision Burn} \rightarrow \text{Finite-Burn Modeling Preferred}. } \]

4.3.15.33 Finite-Burn Parameter Vector

A possible estimated burn-parameter vector is

\[ \boxed{ \boldsymbol\theta_b = \begin{bmatrix} s_F\\ \delta\theta_1\\ \delta\theta_2\\ \delta t_{\rm start}\\ \delta t_{\rm stop} \end{bmatrix}. } \]

More complex models may include

4.3.15.34 Finite-Burn Sensitivity Equations

If

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

the sensitivity satisfies

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

This provides the state sensitivity to burn parameters.

4.3.15.35 Measurement Sensitivity to Burn Parameters

At measurement epoch t_k,

\[ \boxed{ \frac{\partial\mathbf z_k} {\partial\boldsymbol\theta_b} = \mathbf H_k \mathbf S_{\theta_b,k} + \mathbf H_{\theta_b,k}. } \]

Usually the burn parameters influence measurements indirectly through the propagated state.

4.3.15.36 Maneuver Parameter Correlation

Burn parameters can be strongly correlated.

For example,

\[ \boxed{ \text{Thrust Magnitude} \leftrightarrow \text{Burn Duration} } \]

because both affect total impulse.

Similarly,

\[ \boxed{ \text{Burn Timing} \leftrightarrow \text{Initial State} } \]

and

\[ \boxed{ \text{Thrust Direction} \leftrightarrow \text{Attitude / Alignment}. } \]

These correlations should be examined through the covariance matrix.

4.3.15.37 Magnitude-Duration Ambiguity

For a constant thrust,

\[ \boxed{ J=FT_b. } \]

Therefore approximately the same impulse can be produced by

\[ F\uparrow,\quad T_b\downarrow \]

or

\[ F\downarrow,\quad T_b\uparrow. \]

If measurements primarily observe only the total integrated effect, thrust magnitude and duration may be difficult to estimate independently.

4.3.15.38 Improving Burn-Parameter Observability

Parameter separation improves when additional information is available, such as

Thus external telemetry can significantly improve maneuver estimation.

4.3.15.39 Telemetry-Aided Maneuver Estimation

Propulsion telemetry may provide

These can serve as

This reduces the number of completely free maneuver parameters.

4.3.15.40 Unknown Maneuver Detection

Sometimes the estimator does not know that a maneuver occurred.

A sudden change in innovation statistics may indicate an unmodeled acceleration event.

Conceptually,

\[ \boxed{ \text{Persistent Innovation Change} \rightarrow \text{Possible Unknown Maneuver}. } \]

However, sensor faults can produce similar behavior.

4.3.15.41 Distinguishing Maneuver from Measurement Fault

A maneuver affects the propagated physical trajectory and should generally influence multiple independent sensors consistently.

A measurement fault may affect only one sensor.

Thus multi-sensor consistency can help distinguish

\[ \boxed{ \text{Dynamic Event} } \]

from

\[ \boxed{ \text{Sensor Event}. } \]

4.3.15.42 Maneuver Epoch Search

If maneuver time is unknown, one approach is to test candidate burn times.

For each candidate t_m,

\[ J(t_m) \]

is evaluated.

The selected burn time may be the one that produces the best statistically valid residual fit.

This converts maneuver timing into a nonlinear search problem.

4.3.15.43 Maneuver Detection Using Multiple Hypotheses

One can consider hypotheses such as

\[ H_0: \text{No Maneuver} \]

and

\[ H_1: \text{Maneuver Occurred}. \]

The measurement likelihood under each hypothesis can be compared.

This is more rigorous than declaring a maneuver solely from one large residual.

4.3.15.44 Empirical Accelerations

Not all unmodeled forces need to be represented through detailed physical parameters.

Small empirical accelerations may be estimated:

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

where the components may be expressed in

4.3.15.45 Why Use Empirical Accelerations?

They can represent residual effects from

They are especially useful when a precise physical model is unavailable.

4.3.15.46 Constant Empirical Acceleration

Over a short arc,

\[ \boxed{ \dot{\mathbf a}_{\rm emp} = \mathbf0 } \]

may be assumed.

The estimator then solves for a constant acceleration over that interval.

This is simple but may become unrealistic over long arcs.

4.3.15.47 Random-Walk Empirical Acceleration

A more flexible model is

\[ \boxed{ \mathbf a_{{\rm emp},k+1} = \mathbf a_{{\rm emp},k} + \mathbf w_{a,k}. } \]

This allows slow changes over time.

4.3.15.48 Gauss-Markov Empirical Acceleration

A first-order model is

\[ \boxed{ \dot{\mathbf a}_{\rm emp} = -\frac{1}{\tau_a} \mathbf a_{\rm emp} + \mathbf w_a. } \]

Here,

\[ \tau_a \]

controls how long acceleration errors remain correlated.

4.3.15.49 Empirical Acceleration in the State Vector

The augmented state may be

\[ \boxed{ \mathbf x_a = \begin{bmatrix} \mathbf r\\ \mathbf v\\ \mathbf a_{\rm emp} \end{bmatrix}. } \]

Then

\[ \boxed{ \dot{\mathbf v} = \mathbf a_{\rm model} + \mathbf a_{\rm emp}. } \]

Measurements continuously update both trajectory and empirical acceleration.

4.3.15.50 Danger of Excessive Empirical Freedom

Empirical accelerations can improve residuals even when the physical model is wrong.

For example, they may absorb

Therefore,

\[ \boxed{ \text{Small Residuals with Empirical Terms} \neq \text{Correct Physics}. } \]

4.3.15.51 Empirical Acceleration versus Process Noise

These approaches are related but not identical.

Explicit empirical acceleration:

\[ \boxed{ \text{Estimate a State Representing the Disturbance} } \]

Process noise:

\[ \boxed{ \text{Represent Disturbance Statistically without Explicitly Estimating It}. } \]

Explicit states are useful when the disturbance has persistent structure.

4.3.15.52 Piecewise-Constant Acceleration

Another approach is to divide the trajectory into intervals and estimate a constant acceleration in each interval:

\[ \boxed{ \mathbf a(t) = \mathbf a_j, \qquad t_j\leq t<t_{j+1}. } \]

This provides greater flexibility but increases parameter count.

4.3.15.53 Correlation of Empirical Accelerations with Orbit State

A constant along-track acceleration can resemble

Thus empirical acceleration states often become correlated with the orbital state.

Longer or geometrically diverse measurement arcs can help separate them.

4.3.15.54 Atmospheric Drag Parameter Estimation

Atmospheric drag is commonly modeled as

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

The uncertain factor may be represented as

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

or by its inverse ballistic coefficient

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

The convention must always be stated.

4.3.15.55 Drag Scale Factor

Instead of estimating the complete physical parameter, one may estimate a scale factor:

\[ \boxed{ \mathbf a_D^{\rm actual} = s_D \mathbf a_D^{\rm model}. } \]

Nominally,

\[ s_D=1. \]

The estimate indicates how much the modeled drag must be scaled to fit the observations.

4.3.15.56 Drag-Coefficient Estimation

If area and mass are considered known,

\[ \boxed{ C_D } \]

may be estimated directly.

But the estimated value can absorb uncertainties in

Therefore the fitted C_D may be an effective parameter.

4.3.15.57 Ballistic-Coefficient Estimation

If

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

then drag acceleration becomes

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

One may augment

\[ B \]

into the navigation state.

4.3.15.58 Drag Parameter Sensitivity

For

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

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

Therefore,

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

This sensitivity enters the variational equations.

4.3.15.59 Ballistic-Coefficient Sensitivity

If

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

then

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

The sign is positive because increasing B reduces drag magnitude.

4.3.15.60 Density-Drag Correlation

The drag acceleration depends on approximately

\[ \rho C_DA/m. \]

Therefore atmospheric-density scale and ballistic parameter can be highly correlated.

A trajectory may not be able to determine whether excess drag came from

Thus,

\[ \boxed{ \text{Density Error} \leftrightarrow \text{Drag-Parameter Error}. } \]

4.3.15.61 Atmospheric Density Scale Factor

One option is to estimate

\[ \boxed{ \rho_{\rm actual} = s_\rho \rho_{\rm model}. } \]

Then s_\rho becomes an augmented parameter.

This may be useful when the atmosphere model is the dominant uncertainty source.

4.3.15.62 Drag-Parameter Dynamics

Over short arcs,

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

may be adequate.

For longer arcs,

\[ \boxed{ \beta_{k+1} = \beta_k+w_{\beta,k} } \]

may better represent slow variation.

4.3.15.63 Why Drag Parameters May Vary

The effective drag parameter can change because of

Thus a slowly varying effective parameter may represent the real trajectory better than one fixed constant.

4.3.15.64 Drag Observability

Drag is most observable when its trajectory effect becomes large enough relative to measurement noise.

Observability generally improves with

However, increasing the arc length also increases exposure to changing atmospheric conditions.

4.3.15.65 Solar Radiation Pressure

Solar radiation pressure may be approximated as

\[ \boxed{ \mathbf a_{\rm SRP} = P_\odot C_R \frac{A}{m} \hat{\mathbf s} } \]

with additional geometry, distance scaling, and eclipse factors as required.

Here,

4.3.15.66 SRP Coefficient Estimation

The coefficient

\[ \boxed{ C_R } \]

can be estimated as an augmented parameter.

Alternatively, estimate a scale factor

\[ \boxed{ \mathbf a_{\rm SRP}^{\rm actual} = s_{\rm SRP} \mathbf a_{\rm SRP}^{\rm model}. } \]

4.3.15.67 SRP Parameter Sensitivity

If

\[ \mathbf a_{\rm SRP} = C_R \mathbf a_{\rm SRP,unit}, \]

then

\[ \boxed{ \frac{\partial\mathbf a_{\rm SRP}} {\partial C_R} = \mathbf a_{\rm SRP,unit}. } \]

This sensitivity drives how measurements can identify C_R.

4.3.15.68 Area-to-Mass Ratio and SRP

SRP depends on

\[ \frac{A}{m}. \]

Thus uncertainty in

may produce similar trajectory effects.

As with drag,

\[ \boxed{ \text{Estimated SRP Parameter} \text{ may represent an effective combined quantity.} } \]

4.3.15.69 Eclipse Modeling

SRP changes when the spacecraft enters shadow.

A shadow factor

\[ \nu(t) \]

may be introduced:

\[ \boxed{ \mathbf a_{\rm SRP} = \nu(t) P_\odot C_R \frac{A}{m} \hat{\mathbf s}. } \]

Here,

\[ 0\leq\nu\leq1. \]

Accurate eclipse modeling can be important when estimating SRP parameters.

4.3.15.70 SRP Observability

SRP parameters become observable through accumulated trajectory deviations.

Observability depends on

4.3.15.71 Drag versus SRP Estimation

Drag and SRP act differently.

Drag is approximately opposite relative atmospheric velocity.

SRP is related to the Sun-spacecraft direction.

Therefore changing orbital and solar geometry can help separate them.

Still, poorly designed arcs may create significant parameter correlation.

4.3.15.72 Dynamic-Parameter State Vector

A general augmented OD state may be

\[ \boxed{ \mathbf x_a = \begin{bmatrix} \mathbf r\\ \mathbf v\\ B\\ C_R\\ \mathbf a_{\rm emp}\\ \boldsymbol\theta_{\rm maneuver} \end{bmatrix}. } \]

This is mathematically possible, but not all parameters should automatically be estimated simultaneously.

4.3.15.73 Parameter Selection

A dynamic parameter should generally be estimated when it is

\[ \boxed{ \text{Uncertain} + \text{Influential} + \text{Observable}. } \]

If one of these conditions is absent, explicit estimation may not be useful.

4.3.15.74 Dynamic Parameters versus Process Noise

A persistent, structured uncertainty may be a good candidate for explicit parameter estimation.

A rapidly varying or poorly modeled disturbance may be better represented through process noise.

Thus,

\[ \boxed{ \text{Persistent Structured Error} \rightarrow \text{Estimate Parameter} } \]

whereas

\[ \boxed{ \text{Random / Unstructured Error} \rightarrow \text{Process Noise}. } \]

4.3.15.75 Dynamic Parameters versus Empirical Acceleration

A physical parameter such as C_D or C_R has physical interpretation.

An empirical acceleration does not necessarily.

Thus,

\[ \boxed{ \text{Physical Parameter Model} \rightarrow \text{Interpretability} } \]

while

\[ \boxed{ \text{Empirical Acceleration} \rightarrow \text{Flexibility}. } \]

A good OD architecture balances both.

4.3.15.76 Parameter Priors

Ground testing or previous orbit solutions may provide prior information:

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

Examples include

A prior can stabilize weakly observable parameters.

4.3.15.77 Parameter Covariance

The augmented covariance is

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

The parameter covariance

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

quantifies parameter uncertainty.

The cross-covariance

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

quantifies coupling between trajectory and parameter uncertainty.

4.3.15.78 Parameter Correlation

For two parameters,

\[ \theta_i,\theta_j, \]

the correlation coefficient is

\[ \boxed{ \rho_{ij} = \frac{ P_{\theta_i\theta_j} }{ \sigma_{\theta_i}\sigma_{\theta_j} }. } \]

If

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

the two parameters are difficult to separate.

4.3.15.79 Typical Strong Correlations

Examples include

\[ \boxed{ C_D \leftrightarrow \rho_{\rm atmosphere} } \]

\[ \boxed{ F \leftrightarrow T_b } \]

\[ \boxed{ \text{Burn Time} \leftrightarrow \text{Initial State} } \]

\[ \boxed{ \text{Thrust Alignment} \leftrightarrow \text{Attitude Error}. } \]

These should be examined before trusting parameter estimates.

4.3.15.80 Identifiability

A parameter is identifiable only if the measurements distinguish its effect from other parameters.

If two sensitivity columns satisfy approximately

\[ \boxed{ \mathbf g_i \approx c\mathbf g_j, } \]

the corresponding parameters are difficult to identify independently.

This is a central issue in maneuver and dynamic-parameter estimation.

4.3.15.81 Parameter Information Matrix

For parameter sensitivity matrix

\[ \mathbf G_\theta, \]

the information matrix is

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

Small eigenvalues correspond to weakly observable parameter combinations.

4.3.15.82 Parameter Scaling

Dynamic parameters may have very different magnitudes.

For example,

\[ \Delta v\sim10^{-2}\ {\rm m/s}, \]

while a timing correction may be

\[ 10^{-3}\ {\rm s}. \]

Poor scaling can harm numerical conditioning.

Use scaled parameters where appropriate.

4.3.15.83 Physical Constraints

Many estimated parameters have physical limits:

\[ \boxed{ F>0 } \]

\[ \boxed{ T_b>0 } \]

\[ \boxed{ C_D>0 } \]

\[ \boxed{ C_R>0. } \]

These constraints may be enforced explicitly or through transformed parameterizations.

4.3.15.84 Logarithmic Parameterization

For a positive parameter \theta,

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

Then

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

This can improve numerical optimization while automatically satisfying positivity.

4.3.15.85 Sequential Dynamic-Parameter Estimation

In sequential OD, uncertain parameters are included in the state.

For example,

\[ \boxed{ \mathbf x_a = \begin{bmatrix} \mathbf r\\ \mathbf v\\ B\\ C_R \end{bmatrix}. } \]

Measurements continuously update both the orbit and the parameters.

4.3.15.86 Batch Dynamic-Parameter Estimation

In batch OD, the unknown vector might be

\[ \boxed{ \mathbf y = \begin{bmatrix} \mathbf x_0\\ B\\ C_R\\ \Delta\mathbf v \end{bmatrix}. } \]

The complete observation arc simultaneously constrains the state and the parameters.

Batch estimation can be particularly effective for maneuver reconstruction.

4.3.15.87 Batch versus Sequential Maneuver Estimation

Batch estimation is useful when

Sequential estimation is useful when

4.3.15.88 Maneuver Estimation in General Relative Navigation

In a general relative-navigation problem, a maneuver applied to one spacecraft changes the relative state.

For relative state

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

an impulsive maneuver produces

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

The same estimation principles apply to formation flying and rendezvous/proximity operations in general.

4.3.15.89 Finite Burn in Relative Navigation

For a finite thrust event,

\[ \boxed{ \dot{\mathbf v}_{\rm rel} = \mathbf a_{\rm rel,natural} + \mathbf a_{\rm control}. } \]

Uncertainty in the control acceleration affects predicted relative motion.

Range, LOS, or other relative measurements can then help estimate the delivered maneuver.

4.3.15.90 Validation of Maneuver Estimates

Estimated maneuver parameters should be checked against

An estimated maneuver that fits tracking data but is physically impossible should not be accepted blindly.

4.3.15.91 Post-Burn Prediction Check

After estimating the maneuver, propagate the corrected state forward.

If future measurements are predicted substantially better, this supports the maneuver estimate.

Thus,

\[ \boxed{ \text{Good Parameter Estimate} \rightarrow \text{Improved Out-of-Sample Prediction}. } \]

4.3.15.92 Residual Diagnostics

Before estimating the maneuver, residuals may show a systematic discontinuity.

After estimation, the residual structure should reduce.

For example,

\[ \boxed{ \text{Step-Like Residual} \rightarrow \text{Possible Maneuver Magnitude Error} } \]

while gradual drift may indicate

\[ \boxed{ \text{Force-Model Error}. } \]

4.3.15.93 Dynamic-Parameter Validation

Estimated C_D, C_R, empirical accelerations, or thrust parameters should be checked for

A parameter that changes wildly may indicate weak observability.

4.3.15.94 False Parameter Convergence

A parameter estimate may appear stable because

Therefore,

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

4.3.15.95 Over-Parameterization

Estimating simultaneously

may create severe correlation.

Therefore parameter sets should be introduced gradually and justified by observability.

4.3.15.96 Common Mistakes

Common mistakes include:

  1. Assuming commanded \Delta v equals delivered \Delta v.
  2. Ignoring maneuver execution covariance.
  3. Modeling a long finite burn as impulsive without checking validity.
  4. Estimating magnitude and duration separately when only total impulse is observable.
  5. Ignoring burn timing uncertainty.
  6. Ignoring attitude dependence of thrust direction.
  7. Estimating too many correlated parameters.
  8. Treating fitted C_D or C_R as exact physical constants.
  9. Allowing empirical accelerations to hide model errors.
  10. Using zero process noise for a genuinely varying parameter.
  11. Forgetting parameter-state cross-covariances.
  12. Ignoring telemetry that could improve observability.
  13. Failing to distinguish sensor faults from unknown maneuvers.
  14. Ignoring physical bounds.
  15. Declaring parameter convergence without checking covariance or correlation.
  16. Using overly long arcs when parameters are not constant.
  17. Using overly short arcs with insufficient parameter information.
  18. Ignoring density-drag correlation.
  19. Ignoring thrust-attitude coupling.
  20. Validating only against fitted residuals rather than future prediction.

4.3.15.97 Practical Maneuver-Estimation Workflow

A disciplined workflow is

\[ \boxed{ \begin{array}{c} \text{Identify Maneuver / Dynamic Uncertainty}\\ \downarrow\\ \text{Choose Parameterization}\\ \downarrow\\ \text{Define Prior Values and Covariance}\\ \downarrow\\ \text{Define Impulsive or Finite-Burn Model}\\ \downarrow\\ \text{Include Burn Timing if Required}\\ \downarrow\\ \text{Propagate State + Sensitivities}\\ \downarrow\\ \text{Predict Measurements}\\ \downarrow\\ \text{Compute Residuals}\\ \downarrow\\ \text{Estimate State + Parameters}\\ \downarrow\\ \text{Examine Parameter Correlations}\\ \downarrow\\ \text{Check Residual Structure}\\ \downarrow\\ \text{Check Physical Plausibility}\\ \downarrow\\ \text{Validate against Telemetry / Independent Data}\\ \downarrow\\ \text{Assess Prediction Improvement} \end{array} } \]

4.3.15.98 General Dynamic-Parameter Architecture

The complete structure can be written as

\[ \boxed{ \begin{array}{c} \text{Reference State}\\ \mathbf x_0 \\ + \\ \text{Maneuver Parameters}\\ \boldsymbol\theta_m \\ + \\ \text{Dynamic Parameters}\\ \boldsymbol\theta_d \\ \downarrow\\ \text{Propagate Dynamics} \\ \mathbf x(t) \\ \downarrow\\ \text{Predict Measurements} \\ h(\mathbf x) \\ \downarrow\\ \text{Residuals} \\ \downarrow\\ \text{Estimator} \\ \downarrow\\ \hat{\mathbf x}, \hat{\boldsymbol\theta}_m, \hat{\boldsymbol\theta}_d, \mathbf P \end{array} } \]

4.3.15.99 Core Equations

Impulsive maneuver

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

Maneuver correction

\[ \boxed{ \Delta\mathbf v = \Delta\mathbf v_{\rm nominal} + \delta\Delta\mathbf v } \]

Maneuver covariance injection

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

Finite-burn dynamics

\[ \boxed{ \dot{\mathbf v} = \mathbf a_{\rm natural} + \frac{\mathbf F_T}{m} } \]

Delivered impulse

\[ \boxed{ \mathbf J = \int_{t_s}^{t_e} \mathbf F_T(t)\,dt } \]

Burn duration

\[ \boxed{ T_b = t_{\rm stop} - t_{\rm start} } \]

Thrust scale factor

\[ \boxed{ F_{\rm actual} = (1+s_F) F_{\rm nominal} } \]

Empirical acceleration

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

Drag

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

Ballistic coefficient

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

Solar radiation pressure

\[ \boxed{ \mathbf a_{\rm SRP} = P_\odot C_R \frac{A}{m} \hat{\mathbf s} } \]

Parameter sensitivity

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

Measurement sensitivity to a dynamic parameter

\[ \boxed{ \frac{\partial\mathbf z_k} {\partial\boldsymbol\theta} = \mathbf H_k \mathbf S_{\theta,k} + \mathbf H_{\theta,k} } \]

4.3.15.100 Key Takeaways

Maneuver and dynamic-parameter estimation allows orbit determination to solve not only for where the spacecraft is, but also for why its motion differs from the nominal prediction.

For impulsive maneuvers,

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

provides the basic event model.

When the delivered maneuver is uncertain,

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

and the correction can be estimated from tracking data.

For finite-duration burns, the thrust must be integrated through the equations of motion:

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

Important burn parameters can include:

The delivered impulse is

\[ \boxed{ \mathbf J = \int \mathbf F(t)\,dt. } \]

This immediately creates potential correlation between thrust magnitude and burn duration.

Empirical accelerations provide a flexible way to capture unmodeled dynamics:

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

But they should not be allowed to hide identifiable physical modeling errors.

Atmospheric drag estimation commonly involves

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

and therefore creates strong coupling between atmospheric density and drag-related spacecraft parameters.

Solar-radiation-pressure estimation commonly involves

\[ \boxed{ \mathbf a_{\rm SRP} = P_\odot C_R \frac{A}{m} \hat{\mathbf s}. } \]

The estimator may solve for C_R, a scale factor, or another effective SRP parameter.

The most important requirement is estimability:

\[ \boxed{ \text{Adding a Maneuver or Dynamic Parameter} \neq \text{Making It Observable}. } \]

A parameter should generally be estimated only when it is

\[ \boxed{ \text{Uncertain} + \text{Dynamically Important} + \text{Observable}. } \]

The quality of the final maneuver or dynamic-parameter solution should therefore be judged using:

The overall principle is

\[ \boxed{ \text{Orbit Estimation} + \text{Maneuver Estimation} + \text{Dynamic-Parameter Estimation} \rightarrow \text{More Physically Complete Navigation Solution}. } \]

Next: 4.3.16 Geometry, Identifiability and Estimability

Maneuver and dynamic-parameter estimation depends on whether the available measurements can actually distinguish the parameters being solved for. The next section develops the geometry, rank, null-space, correlation, and weak-direction concepts that determine whether states and parameters are estimable.

4.3.16 Geometry, Identifiability and Estimability

Measurement geometry, rank, null spaces, parameter correlations, weak estimation directions, and practical estimability diagnostics.

Continue to Geometry, Identifiability and Estimability →