Orbit-Determination Measurement Models

4.3.12 Orbit-Determination Measurement Models

Orbit determination depends on two major ingredients:

\[ \boxed{ \text{Dynamic Model} + \text{Measurement Model} } \]

The dynamic model predicts how the spacecraft state evolves with time. The measurement model predicts what a sensor should observe for a given spacecraft state.

If the spacecraft state is

\[ \mathbf x_k \]

and the measurement at time t_k is

\[ \mathbf z_k, \]

the general measurement equation is

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

where

• h_k(\cdot) is the measurement function,
• \mathbf v_k is measurement noise.

For many OD problems, the state may be

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

or an augmented state containing additional quantities such as biases, clock states, calibration parameters, and dynamic coefficients.

The purpose of a measurement model is therefore:

\[ \boxed{ \text{Spacecraft State} \rightarrow \text{Predicted Sensor Observation} } \]

and the difference between the actual and predicted measurement provides the information used by the estimator.

This section covers the major measurement types used in orbit determination:

• range,
• range rate,
• Doppler,
• azimuth/elevation,
• right ascension/declination,
• line of sight,
• GNSS measurements,
• optical observations,
• measurement partial derivatives,
• measurement covariance and residual construction,
• practical corrections required for precise tracking.

4.3.12.1 General Measurement Equation

The generic nonlinear measurement model is

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

where

\[ \boldsymbol\theta_k \]

may contain measurement-related parameters such as

• sensor biases,
• station coordinates,
• timing offsets,
• clock parameters,
• calibration coefficients.

The noise is commonly modeled as

\[ \boxed{ \mathbf v_k \sim \mathcal N( \mathbf 0, \mathbf R_k ) } \]

where

\[ \mathbf R_k \]

is the measurement-noise covariance matrix.

4.3.12.2 Predicted Measurement

Given a predicted spacecraft state

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

the expected measurement is

\[ \boxed{ \hat{\mathbf z}_k = h_k( \hat{\mathbf x}_k^- ) } \]

The estimator then compares this predicted value with the actual observation.

The residual or innovation is

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

The residual drives the state correction.

4.3.12.3 Measurement Residual Interpretation

A nonzero residual may result from

• state-estimation error,
• sensor noise,
• sensor bias,
• timing error,
• calibration error,
• force-model error,
• station-coordinate error,
• maneuver mismatch.

Therefore,

\[ \boxed{ \text{Residual} \neq \text{Pure Position or Velocity Error} } \]

A physically meaningful OD solution requires correct interpretation of residuals.

4.3.12.4 Observer and Spacecraft Geometry

Let

\[ \mathbf r_{\rm sc} \]

be spacecraft position and

\[ \mathbf r_{\rm obs} \]

be observer position.

Define relative position:

\[ \boxed{ \boldsymbol\rho = \mathbf r_{\rm sc} - \mathbf r_{\rm obs} } \]

and relative velocity:

\[ \boxed{ \dot{\boldsymbol\rho} = \mathbf v_{\rm sc} - \mathbf v_{\rm obs} } \]

The geometric range is

\[ \boxed{ \rho = \|\boldsymbol\rho\| } \]

and the line-of-sight unit vector is

\[ \boxed{ \hat{\boldsymbol\rho} = \frac{\boldsymbol\rho}{\rho} } \]

These quantities form the basis of many OD measurement models.

4.3.12.5 One-Way Range

The simplest geometric range measurement is

\[ \boxed{ \rho = \| \mathbf r_{\rm sc} - \mathbf r_{\rm obs} \| } \]

The measurement may be modeled as

\[ \boxed{ z_\rho = \rho + b_\rho + v_\rho } \]

where

• b_\rho is range bias,
• v_\rho is random noise.

Range directly measures distance along the line of sight.

4.3.12.6 Physical Interpretation of Range

Range strongly constrains the spacecraft position along the observer-to-spacecraft direction.

However, one range measurement generally does not determine the full 3-D position.

The measurement contains much weaker information about transverse directions.

Thus,

\[ \boxed{ \text{Range} \rightarrow \text{Strong LOS Distance Information} } \]

while changing geometry over time is needed to obtain full position observability.

4.3.12.7 Two-Way Range

Ground-based tracking often uses two-way ranging.

A signal is transmitted from the station, received or transponded by the spacecraft, and returned to the station.

Approximately,

\[ \boxed{ \rho_{\rm 2way} \approx \frac{c\Delta t}{2} } \]

where

• c is the speed of light,
• \Delta t is round-trip signal time.

For precision OD, the spacecraft and station move during signal propagation, so full light-time modeling is required.

4.3.12.8 Range Bias

A practical range model may include

\[ \boxed{ z_\rho = \rho + b_{\rm station} + b_{\rm transponder} + b_{\rm media} + v_\rho } \]

Bias may come from

• instrumentation,
• transponder delay,
• calibration offset,
• atmosphere,
• clock errors.

If significant, these biases may need to be calibrated or estimated.

4.3.12.9 Range Partial Derivative

For

\[ \rho = \|\boldsymbol\rho\|, \]

the derivative with respect to spacecraft position is

\[ \boxed{ \frac{\partial\rho} {\partial\mathbf r_{\rm sc}} = \hat{\boldsymbol\rho}^T } \]

assuming observer position is fixed relative to the differentiation variable.

For the basic geometric range model,

\[ \boxed{ \frac{\partial\rho} {\partial\mathbf v_{\rm sc}} = \mathbf0^T } \]

instantaneously.

Therefore the local measurement Jacobian is

\[ \boxed{ \mathbf H_\rho = \begin{bmatrix} \hat{\boldsymbol\rho}^T & \mathbf0^T \end{bmatrix} } \]

for a position-velocity state.

4.3.12.10 Range Rate

Range rate is the time derivative of range.

Starting from

\[ \rho = \|\boldsymbol\rho\|, \]

the derivative is

\[ \boxed{ \dot\rho = \frac{ \boldsymbol\rho^T \dot{\boldsymbol\rho} } {\rho} } \]

or equivalently

\[ \boxed{ \dot\rho = \hat{\boldsymbol\rho}^T \dot{\boldsymbol\rho} } \]

Thus range rate measures relative velocity projected along the line of sight.

4.3.12.11 Range-Rate Measurement Model

A simple model is

\[ \boxed{ z_{\dot\rho} = \dot\rho + b_{\dot\rho} + v_{\dot\rho} } \]

where

• b_{\dot\rho} is range-rate bias,
• v_{\dot\rho} is noise.

Range and range rate are complementary because range constrains relative position while range rate constrains relative velocity along the LOS.

4.3.12.12 Physical Meaning of Range Rate

If

\[ \dot\rho>0, \]

the objects are separating along the line of sight.

If

\[ \dot\rho<0, \]

they are closing.

If

\[ \dot\rho=0, \]

the instantaneous relative motion is transverse to the LOS or momentarily stationary in range.

4.3.12.13 Range-Rate Sensitivity to Velocity

Because

\[ \dot\rho = \hat{\boldsymbol\rho}^T \mathbf v_{\rm rel}, \]

where

\[ \mathbf v_{\rm rel} = \mathbf v_{\rm sc} - \mathbf v_{\rm obs}, \]

the velocity derivative is

\[ \boxed{ \frac{\partial\dot\rho} {\partial\mathbf v_{\rm sc}} = \hat{\boldsymbol\rho}^T } \]

for the basic geometric model.

4.3.12.14 Range-Rate Sensitivity to Position

Range rate also depends on position because the LOS unit vector depends on position.

Let

\[ \mathbf P_\perp = \mathbf I - \hat{\boldsymbol\rho} \hat{\boldsymbol\rho}^T. \]

Then

\[ \boxed{ \frac{\partial\dot\rho} {\partial\mathbf r_{\rm sc}} = \frac{ \mathbf v_{\rm rel}^T \mathbf P_\perp }{\rho} } \]

Therefore,

\[ \boxed{ \mathbf H_{\dot\rho} = \begin{bmatrix} \dfrac{ \mathbf v_{\rm rel}^T (\mathbf I-\hat{\boldsymbol\rho}\hat{\boldsymbol\rho}^T) }{\rho} & \hat{\boldsymbol\rho}^T \end{bmatrix} } \]

for the simplified model.

4.3.12.15 Doppler Measurement

Doppler measures frequency shift caused by relative motion.

For a one-way signal, a simplified relation is

\[ \boxed{ \Delta f \approx -\frac{f_0}{c}\dot\rho } \]

where

• f_0 is transmitted frequency,
• c is the speed of light.

Thus Doppler is fundamentally related to line-of-sight relative velocity.

4.3.12.16 Doppler versus Range Rate

Doppler and range rate are closely related but are not always numerically identical measurement types.

Range rate is expressed in

\[ \mathrm{m/s} \]

whereas Doppler is measured in

\[ \mathrm{Hz}. \]

Their simplified relationship is

\[ \boxed{ \dot\rho \approx -\frac{c}{f_0}\Delta f. } \]

Real tracking systems may use coherent transponders, turnaround ratios, accumulated Doppler counts, and finite integration intervals.

4.3.12.17 Integrated Doppler

A Doppler observable may represent accumulated phase or frequency over an interval.

Conceptually,

\[ \boxed{ N_D \propto \rho(t_2)-\rho(t_1) } \]

over a counting interval.

This makes Doppler highly sensitive to relative radial motion.

For accurate modeling, the finite measurement integration period should be represented.

4.3.12.18 Two-Way Doppler

In two-way tracking, frequency shift occurs on both the uplink and downlink.

The full model depends on

• uplink frequency,
• transponder turnaround ratio,
• spacecraft motion,
• station motion,
• light-time,
• relativistic effects.

The simplified proportionality to range rate remains useful for understanding the geometry.

4.3.12.19 Doppler Measurement Noise

Doppler precision depends on

• oscillator stability,
• signal-to-noise ratio,
• integration time,
• receiver tracking,
• propagation effects.

Longer integration times can reduce random frequency noise but may reduce temporal resolution.

Thus measurement cadence and precision are coupled.

4.3.12.20 Azimuth and Elevation Measurements

Ground tracking systems frequently measure spacecraft direction in a local topocentric frame.

Let the line-of-sight vector in a local East-North-Up frame be

\[ \boldsymbol\rho_{\rm ENU} = \begin{bmatrix} \rho_E\\ \rho_N\\ \rho_U \end{bmatrix}. \]

Then azimuth can be defined as

\[ \boxed{ A = \operatorname{atan2}( \rho_E,\rho_N ) } \]

and elevation as

\[ \boxed{ E = \arctan \left( \frac{\rho_U} {\sqrt{\rho_E^2+\rho_N^2}} \right) } \]

subject to the chosen convention.

4.3.12.21 Azimuth Convention

Azimuth conventions must be stated explicitly.

A common convention is

• 0^\circ: North,
• 90^\circ: East,
• increasing clockwise when viewed from above.

Other conventions may exist.

Therefore,

\[ \boxed{ \text{Angle Convention Must Be Defined} } \]

before implementing the measurement equation.

4.3.12.22 Elevation

Elevation represents the angle above the local horizon.

If

\[ E=0^\circ, \]

the spacecraft lies on the local horizon.

If

\[ E=90^\circ, \]

it is at the local zenith.

Low-elevation measurements may suffer increased atmospheric effects and poorer geometry.

4.3.12.23 Topocentric Transformation

The relative vector is often initially known in an inertial or Earth-fixed frame.

It must be transformed to the station-local frame:

\[ \boxed{ \boldsymbol\rho_{\rm ENU} = \mathbf C_{\rm ENU/ECEF} \boldsymbol\rho_{\rm ECEF} } \]

or the appropriate local frame.

Therefore azimuth/elevation prediction requires correct

• Earth orientation,
• station coordinates,
• Earth rotation,
• frame transformations.

4.3.12.24 Azimuth Measurement Model

The measured azimuth may be written

\[ \boxed{ z_A = A + b_A + v_A } \]

where

• b_A is angular bias,
• v_A is random measurement noise.

Azimuth constrains the horizontal LOS direction.

4.3.12.25 Elevation Measurement Model

Similarly,

\[ \boxed{ z_E = E + b_E + v_E } \]

Elevation constrains vertical LOS geometry.

Azimuth and elevation together provide a 2-D angular direction but no direct range.

4.3.12.26 Angular Singularities

Azimuth becomes poorly defined near zenith because horizontal LOS components approach zero.

This is a coordinate singularity, not a physical singularity.

Alternative representations such as LOS unit vectors can avoid this issue.

4.3.12.27 Right Ascension and Declination

Astronomical optical measurements are often expressed using right ascension and declination.

Let an inertial LOS vector be

\[ \boldsymbol\rho = \begin{bmatrix} \rho_x\\ \rho_y\\ \rho_z \end{bmatrix}. \]

Then right ascension is

\[ \boxed{ \alpha = \operatorname{atan2}( \rho_y,\rho_x ) } \]

and declination is

\[ \boxed{ \delta = \arcsin \left( \frac{\rho_z}{\rho} \right) } \]

where

\[ \rho=\|\boldsymbol\rho\|. \]

4.3.12.28 Right Ascension

Right ascension describes angular direction in an equatorial inertial coordinate system.

It is analogous to longitude on the celestial sphere.

Because it is periodic, care is required near the 0/2\pi wrap boundary.

Residuals should be wrapped consistently.

4.3.12.29 Declination

Declination describes angular distance above or below the equatorial plane.

Its range is typically

\[ -\frac{\pi}{2} \le \delta \le \frac{\pi}{2}. \]

Together, RA and Dec specify LOS direction but do not directly provide distance.

4.3.12.30 Angle-Only Orbit Determination

When only angular measurements are available, the range to the spacecraft is initially weakly constrained.

The estimator relies heavily on

• changing observation geometry,
• dynamic propagation,
• multiple observation times,
• possibly multiple observers.

Thus,

\[ \boxed{ \text{Angle Measurements} + \text{Dynamics} + \text{Geometry Change} \rightarrow \text{Range Information Over Time} } \]

4.3.12.31 Line-of-Sight Measurement

Instead of representing angular direction using two angles, one may directly use a unit LOS vector:

\[ \boxed{ \hat{\boldsymbol\rho} = \frac{ \mathbf r_{\rm target} - \mathbf r_{\rm observer} } { \| \mathbf r_{\rm target} - \mathbf r_{\rm observer} \| } } \]

This avoids angle wrapping and some coordinate singularities.

4.3.12.32 LOS Vector Measurement Model

A measured LOS can be modeled as

\[ \boxed{ \mathbf z_{\rm LOS} = \hat{\boldsymbol\rho} + \mathbf v_{\rm LOS} } \]

with appropriate normalization and covariance representation.

Because a unit vector has only two independent degrees of freedom, its three Cartesian components are constrained:

\[ \boxed{ \hat{\boldsymbol\rho}^T \hat{\boldsymbol\rho} = 1. } \]

This must be considered when defining measurement covariance.

4.3.12.33 LOS Jacobian

For

\[ \hat{\boldsymbol\rho} = \frac{\boldsymbol\rho}{\rho}, \]

the derivative with respect to relative position is

\[ \boxed{ \frac{\partial\hat{\boldsymbol\rho}} {\partial\boldsymbol\rho} = \frac{1}{\rho} \left( \mathbf I - \hat{\boldsymbol\rho} \hat{\boldsymbol\rho}^T \right) } \]

This matrix projects perturbations into directions perpendicular to the LOS.

Therefore LOS measurements contain almost no first-order sensitivity to motion directly along the LOS.

4.3.12.34 Physical Meaning of LOS Sensitivity

The matrix

\[ \mathbf I - \hat{\boldsymbol\rho} \hat{\boldsymbol\rho}^T \]

removes the radial component.

Therefore,

\[ \boxed{ \text{LOS Measurement} \rightarrow \text{Strong Transverse Information} } \]

whereas

\[ \boxed{ \text{Range Measurement} \rightarrow \text{Strong Radial Information} } \]

This illustrates why combining range and LOS measurements can be very powerful.

4.3.12.35 Optical Measurements

Optical sensors may produce measurements such as

• pixel coordinates,
• centroid locations,
• bearings,
• LOS vectors,
• angular positions,
• apparent object size,
• landmark directions,
• star directions.

The raw camera image is not itself the OD measurement until it has been converted into a feature or geometric observable.

4.3.12.36 Pinhole Camera Model

For a point expressed in camera coordinates

\[ \mathbf p_C = \begin{bmatrix} X_C\\ Y_C\\ Z_C \end{bmatrix}, \]

the ideal pinhole projection is

\[ \boxed{ u = f_x\frac{X_C}{Z_C} + c_x } \]

\[ \boxed{ v = f_y\frac{Y_C}{Z_C} + c_y } \]

where

• f_x,f_y are effective focal lengths,
• c_x,c_y are principal-point coordinates.

4.3.12.37 Camera LOS from Pixel Coordinates

Given calibrated image coordinates,

\[ \tilde u = \frac{u-c_x}{f_x}, \qquad \tilde v = \frac{v-c_y}{f_y}, \]

the unnormalized camera-frame LOS is

\[ \begin{bmatrix} \tilde u\\ \tilde v\\ 1 \end{bmatrix}. \]

Normalize it:

\[ \boxed{ \hat{\boldsymbol\rho}_C = \frac{ \begin{bmatrix} \tilde u\\ \tilde v\\ 1 \end{bmatrix} }{ \left\| \begin{bmatrix} \tilde u\\ \tilde v\\ 1 \end{bmatrix} \right\| } } \]

4.3.12.38 Optical Frame Transformation

The LOS may need transformation from camera frame to body frame and then to inertial or navigation frame:

\[ \boxed{ \hat{\boldsymbol\rho}_I = \mathbf C_{IB} \mathbf C_{BC} \hat{\boldsymbol\rho}_C } \]

where

• \mathbf C_{BC}: camera-to-body rotation,
• \mathbf C_{IB}: body-to-inertial rotation.

Thus optical OD depends on both orbital and attitude knowledge.

4.3.12.39 Camera Alignment Error

Camera alignment uncertainty introduces LOS error.

A small alignment error

\[ \delta\boldsymbol\theta_C \]

produces approximately

\[ \boxed{ \delta\hat{\boldsymbol\rho} \approx - [\hat{\boldsymbol\rho}\times] \delta\boldsymbol\theta_C. } \]

Therefore optical measurement accuracy may depend strongly on camera-to-body calibration.

4.3.12.40 Optical Centroid Error

An image-processing algorithm estimates target centroid:

\[ \boxed{ \mathbf z_{\rm pix} = \begin{bmatrix} u\\ v \end{bmatrix} } \]

with covariance

\[ \boxed{ \mathbf R_{\rm pix} = \begin{bmatrix} \sigma_u^2 & \sigma_{uv}\\ \sigma_{uv} & \sigma_v^2 \end{bmatrix}. } \]

Pixel uncertainty can be transformed into angular uncertainty using the camera model Jacobian.

4.3.12.41 Pixel-to-Angle Uncertainty

For small angles,

\[ \theta_x \approx \frac{u-c_x}{f_x}. \]

Therefore approximately,

\[ \boxed{ \sigma_{\theta_x} \approx \frac{\sigma_u}{f_x}. } \]

Likewise,

\[ \boxed{ \sigma_{\theta_y} \approx \frac{\sigma_v}{f_y}. } \]

This links image-processing precision directly to navigation performance.

4.3.12.42 Apparent Angular Size

If an object of known physical size D has angular diameter \theta,

\[ \boxed{ \theta \approx \frac{D}{\rho} } \]

for small angles.

Thus apparent size can provide approximate range information if physical size and geometry are known.

However, such range estimates may be sensitive to

• target attitude,
• shape,
• illumination,
• segmentation accuracy.

4.3.12.43 Feature-Based Optical Navigation

Instead of using the object centroid, an optical system may track features.

A feature measurement may be

\[ \boxed{ \mathbf z_i = \begin{bmatrix} u_i\\ v_i \end{bmatrix} } \]

for feature i.

If feature locations are known in the target/body frame, the camera model predicts their image positions.

This can support simultaneous estimation of

• relative position,
• relative attitude,
• calibration parameters.

4.3.12.44 Optical Measurement Bias Sources

Possible optical biases include

• focal-length error,
• principal-point error,
• lens distortion,
• camera alignment error,
• timing offset,
• thermal deformation,
• centroid bias,
• feature-location uncertainty.

These may need explicit calibration or augmented-state estimation.

4.3.12.45 GNSS Measurements

GNSS-based orbit determination uses measurements from navigation satellites.

Common observables include

• pseudorange,
• carrier phase,
• Doppler,
• navigation-message ephemerides.

The receiver position and clock states are inferred from simultaneous observations to multiple GNSS satellites.

4.3.12.46 GNSS Pseudorange

For GNSS satellite i, a simplified pseudorange model is

\[ \boxed{ P_i = \rho_i + c( \delta t_r-\delta t_{s,i} ) + I_i + T_i + b_i + \epsilon_i } \]

where

• \rho_i: geometric range,
• \delta t_r: receiver clock bias,
• \delta t_{s,i}: satellite clock bias,
• I_i: ionospheric delay,
• T_i: tropospheric delay,
• b_i: hardware or other bias,
• \epsilon_i: noise.

4.3.12.47 GNSS Geometric Range

The geometric range is

\[ \boxed{ \rho_i = \| \mathbf r_r - \mathbf r_{s,i} \| } \]

where

• \mathbf r_r: receiver spacecraft position,
• \mathbf r_{s,i}: GNSS satellite position.

Because multiple GNSS satellites have different LOS directions, simultaneous pseudoranges provide 3-D geometry.

4.3.12.48 GNSS Clock Bias

Receiver clock bias contributes directly to all pseudorange measurements.

Therefore the GNSS navigation state commonly includes

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

The clock states are essential because GNSS receiver clocks are generally not synchronized perfectly with GNSS system time.

4.3.12.49 GNSS Doppler

GNSS Doppler measures relative radial velocity between receiver and navigation satellite.

A simplified model is

\[ \boxed{ \dot\rho_i = \hat{\boldsymbol\rho}_i^T ( \mathbf v_r-\mathbf v_{s,i} ) } \]

plus clock-drift and measurement-error terms.

It provides strong information about receiver velocity.

4.3.12.50 GNSS Carrier Phase

Carrier-phase measurements provide very high precision.

A simplified model is

\[ \boxed{ \Phi_i = \rho_i + c( \delta t_r-\delta t_{s,i} ) + \lambda N_i + \epsilon_{\Phi,i} +\cdots } \]

where

• \lambda is carrier wavelength,
• N_i is integer ambiguity.

Carrier phase is highly precise but requires ambiguity handling.

4.3.12.51 GNSS Measurement Geometry

GNSS navigation quality depends on satellite geometry.

Poor satellite geometry can produce large position uncertainty even when measurement noise is small.

This is related to dilution of precision.

Conceptually,

\[ \boxed{ \text{GNSS Accuracy} = f( \text{Measurement Noise}, \text{Satellite Geometry} ) } \]

4.3.12.52 GNSS in Space Applications

For spacecraft, GNSS use depends on

• altitude,
• antenna orientation,
• receiver sensitivity,
• GNSS constellation visibility,
• side-lobe reception,
• signal strength.

In LEO, GNSS can provide frequent high-quality navigation updates.

At higher altitudes, signal geometry and availability can become more challenging.

4.3.12.53 Inter-Satellite Range

For two spacecraft,

\[ \boxed{ \rho_{12} = \| \mathbf r_2-\mathbf r_1 \| } \]

An inter-satellite range measurement can be written

\[ \boxed{ z_{\rho,12} = \rho_{12} + b_{\rho,12} + v_{\rho,12}. } \]

This supports relative orbit determination and formation navigation.

4.3.12.54 Inter-Satellite Range Rate

Similarly,

\[ \boxed{ \dot\rho_{12} = \frac{ ( \mathbf r_2-\mathbf r_1 )^T ( \mathbf v_2-\mathbf v_1 ) }{ \rho_{12} } } \]

This provides information about relative closing or separation velocity.

4.3.12.55 Relative Measurements in General RPO

In general rendezvous and proximity operations, relative navigation may use

• range,
• range rate,
• bearing angles,
• LOS vectors,
• camera features,
• LiDAR,
• radar,
• inter-satellite GNSS.

The relative state is commonly

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

The same measurement-model principles used for absolute OD apply to relative OD.

4.3.12.56 Relative Range

For relative position

\[ \mathbf r_{\rm rel}, \]

the range is

\[ \boxed{ \rho_{\rm rel} = \| \mathbf r_{\rm rel} \| } \]

and the measurement model is

\[ \boxed{ z_\rho = \rho_{\rm rel} + v_\rho. } \]

This provides radial separation but not complete 3-D position.

4.3.12.57 Relative LOS

The relative LOS is

\[ \boxed{ \hat{\boldsymbol\rho}_{\rm rel} = \frac{ \mathbf r_{\rm rel} }{ \|\mathbf r_{\rm rel}\| } } \]

which provides direction to the other object.

Combining relative range and LOS can reconstruct relative position geometrically:

\[ \boxed{ \mathbf r_{\rm rel} = \rho_{\rm rel} \hat{\boldsymbol\rho}_{\rm rel}. } \]

4.3.12.58 LiDAR Measurements

LiDAR may provide

• range,
• point clouds,
• relative pose,
• surface geometry.

A basic LiDAR range model is

\[ \boxed{ z_L = \rho + b_L + v_L. } \]

More advanced LiDAR systems may directly estimate 3-D relative position and orientation.

4.3.12.59 Radar Measurements

Radar tracking may provide combinations of

• range,
• range rate,
• azimuth,
• elevation.

A radar measurement vector can be written

\[ \boxed{ \mathbf z_{\rm radar} = \begin{bmatrix} \rho\\ \dot\rho\\ A\\ E \end{bmatrix} + \mathbf v. } \]

Radar therefore combines radial and angular information.

4.3.12.60 Measurement Coordinate Transformation

Suppose a sensor measurement is naturally expressed in sensor frame S, while the spacecraft state is expressed in inertial frame I.

The geometric prediction requires

\[ \boxed{ \mathbf r_S = \mathbf C_{SI} \mathbf r_I } \]

or the appropriate chain of transformations.

Incorrect frame transformations are a common source of systematic measurement residuals.

4.3.12.61 Sensor Lever Arm

A sensor may not be located at the spacecraft center of mass.

Let the sensor position relative to the spacecraft reference point be

\[ \mathbf r_{B/S}. \]

Then the sensor inertial position is

\[ \boxed{ \mathbf r_S^I = \mathbf r_B^I + \mathbf C_{IB} \mathbf r_{B/S}. } \]

This lever arm can matter for precise angular or range measurements.

4.3.12.62 Sensor Attitude Dependence

Some measurements depend explicitly on spacecraft attitude.

Optical LOS measured in a body-fixed camera frame is one example.

Thus the state may need to include attitude:

\[ \boxed{ \mathbf x = \begin{bmatrix} \mathbf r\\ \mathbf v\\ \mathbf q\\ \boldsymbol\omega \end{bmatrix} } \]

if the measurement model requires it.

4.3.12.63 Time Tagging

A measurement should correspond to the correct state time:

\[ \boxed{ \mathbf z_k = h( \mathbf x(t_k) ). } \]

If the time tag contains error

\[ \Delta t, \]

the actual state used should be approximately

\[ \mathbf x(t_k+\Delta t). \]

For a moving spacecraft, even small timing offsets can generate significant apparent measurement errors.

4.3.12.64 Timing-Error Approximation

For a small timing error,

\[ \mathbf x(t+\Delta t) \approx \mathbf x(t) + \dot{\mathbf x}(t)\Delta t. \]

Therefore measurement error due to timing is approximately

\[ \boxed{ \delta\mathbf z \approx \mathbf H \dot{\mathbf x} \Delta t. } \]

This equation shows why timing synchronization is part of the measurement model.

4.3.12.65 Light-Time Correction

Signals propagate at finite speed.

For a one-way signal,

\[ \boxed{ t_r-t_t = \frac{\rho}{c} } \]

where

• t_t: transmission time,
• t_r: reception time.

Because spacecraft and observer move during this interval, the geometric range must be evaluated at different epochs.

4.3.12.66 Iterative Light-Time Solution

The measurement geometry may require solving

\[ \boxed{ \rho = \left\| \mathbf r_{\rm sc}(t_t) - \mathbf r_{\rm obs}(t_r) \right\| } \]

with

\[ t_r-t_t = \frac{\rho}{c}. \]

Because \rho depends on t_t, an iterative solution may be required.

4.3.12.67 Atmospheric Delay

Ground-based radio measurements may be affected by

• tropospheric delay,
• ionospheric delay.

A practical range model may therefore be

\[ \boxed{ z_\rho = \rho + \Delta\rho_{\rm trop} + \Delta\rho_{\rm iono} + b + v. } \]

The importance depends on tracking frequency, elevation angle, and required precision.

4.3.12.68 Refraction in Angular Measurements

Atmospheric refraction can shift measured elevation.

The apparent elevation may differ from the geometric elevation:

\[ \boxed{ E_{\rm measured} = E_{\rm geometric} + \Delta E_{\rm refr}. } \]

This effect becomes stronger at low elevation angles.

4.3.12.69 Relativistic Measurement Corrections

For precision tracking, corrections may be required for

• gravitational time delay,
• relativistic clock effects,
• frequency shifts,
• signal propagation near massive bodies.

These are generally not needed for basic OD but can be important in high-precision GNSS and deep-space navigation.

4.3.12.70 Measurement Bias State

A sensor bias may be included as an augmented state:

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

The measurement becomes

\[ \boxed{ z = h(\mathbf x) + b + v. } \]

The bias may be modeled as constant or slowly varying.

4.3.12.71 Measurement Scale Factor

A measurement may contain multiplicative error:

\[ \boxed{ z_m = (1+s) h(\mathbf x) + v } \]

where s is scale-factor error.

This may need calibration or augmented-state estimation.

4.3.12.72 Measurement Covariance

For measurement vector

\[ \mathbf z = \begin{bmatrix} z_1\\ z_2\\ \vdots \end{bmatrix}, \]

the measurement covariance is

\[ \boxed{ \mathbf R = E[ \mathbf v \mathbf v^T ]. } \]

For independent measurements,

\[ \boxed{ \mathbf R = \operatorname{diag} ( \sigma_1^2, \sigma_2^2, \ldots ) } \]

but correlated measurements require off-diagonal terms.

4.3.12.73 Why Correlated Measurement Noise Matters

Suppose two measurement errors share a common clock bias or atmospheric error.

Then

\[ R_{12}\neq0. \]

Treating them as independent would count shared information more than once and make the estimator overconfident.

Therefore,

\[ \boxed{ \text{Measurement Correlation Must Be Modeled When Significant} } \]

4.3.12.74 Measurement Weighting

For batch OD,

\[ \boxed{ \mathbf W = \mathbf R^{-1}. } \]

Measurements with lower variance receive greater weight.

However, weighting should represent realistic statistical uncertainty, not simply nominal sensor resolution.

4.3.12.75 Resolution versus Accuracy

Sensor resolution is the smallest change represented by the instrument.

Accuracy describes closeness to the true value.

These are not equivalent.

A measurement may have excellent digital resolution but poor absolute accuracy due to bias or calibration error.

Thus OD measurement covariance should not be based on resolution alone.

4.3.12.76 Measurement Precision versus Bias

A highly precise sensor can repeatedly produce nearly identical but biased measurements.

Thus,

\[ \boxed{ \text{Precision} \neq \text{Accuracy} } \]

Bias should be calibrated, corrected, or estimated separately.

4.3.12.77 Measurement Linearization

For nonlinear measurement model

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

linearize around

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

Then

\[ \boxed{ h( \mathbf x ) \approx h( \hat{\mathbf x} ) + \mathbf H ( \mathbf x-\hat{\mathbf x} ) } \]

where

\[ \boxed{ \mathbf H = \frac{\partial h} {\partial\mathbf x} \bigg|_{\hat{\mathbf x}}. } \]

This matrix is the measurement Jacobian.

4.3.12.78 Role of the Measurement Jacobian

The Jacobian describes local measurement sensitivity.

If

\[ H_{ij} = \frac{\partial h_i}{\partial x_j}, \]

then H_{ij} indicates how strongly measurement i responds to state j.

Therefore,

\[ \boxed{ \mathbf H \rightarrow \text{Local Measurement Information Geometry} } \]

4.3.12.79 Measurement Jacobian in EKF

In an Extended Kalman Filter,

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

and

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

Thus the measurement Jacobian determines how state uncertainty projects into measurement space.

4.3.12.80 Measurement Jacobian in Batch OD

In batch OD, measurements depend on the state at each measurement time.

But the estimated unknown may be the reference-epoch state.

Using

\[ \delta\mathbf x_k = \boldsymbol\Phi(t_k,t_0) \delta\mathbf x_0, \]

we obtain

\[ \boxed{ \delta\mathbf z_k = \mathbf H_k \boldsymbol\Phi(t_k,t_0) \delta\mathbf x_0. } \]

Therefore the reference-epoch sensitivity is

\[ \boxed{ \mathbf G_k = \mathbf H_k \boldsymbol\Phi(t_k,t_0). } \]

4.3.12.81 Partial Derivatives with Respect to Parameters

If the measurement depends on estimated parameter \theta,

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

then

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

If the parameter influences the state dynamically, the total sensitivity also includes state sensitivity:

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

This is important in augmented-state OD.

4.3.12.82 Analytical Jacobians

Analytical derivatives are derived symbolically from measurement equations.

Advantages include

• speed,
• accuracy,
• deterministic behavior.

Disadvantages include

• derivation complexity,
• risk of implementation errors,
• maintenance burden.

They are often preferred for mature high-performance flight or OD software.

4.3.12.83 Finite-Difference Jacobians

A numerical derivative may be approximated as

\[ \boxed{ \frac{\partial h} {\partial x_i} \approx \frac{ h( \mathbf x+\epsilon\mathbf e_i ) - h( \mathbf x-\epsilon\mathbf e_i ) }{ 2\epsilon } } \]

This is useful for

• checking analytical derivatives,
• prototyping,
• complex models.

However, step-size selection is important.

4.3.12.84 Automatic Differentiation

Automatic differentiation computes derivatives by propagating derivatives through the numerical operations used by the model.

It can provide near-machine-precision derivatives without deriving every expression manually.

It is particularly useful for complex nonlinear measurement models.

4.3.12.85 Jacobian Verification

A good implementation practice is

\[ \boxed{ \text{Analytical Jacobian} \leftrightarrow \text{Finite-Difference Check} } \]

Small discrepancies are expected from numerical effects, but large differences can indicate

• sign errors,
• frame mistakes,
• missing terms,
• indexing errors.

4.3.12.86 Angle Residual Wrapping

Suppose angular residual is

\[ r_\theta = \theta_{\rm measured} - \theta_{\rm predicted}. \]

Near 0 and 2\pi, direct subtraction can give a false large residual.

Therefore angle residuals should be wrapped:

\[ \boxed{ r_\theta = \operatorname{wrapToPi} ( \theta_m-\theta_p ) } \]

or an equivalent convention.

4.3.12.87 Innovation Covariance

The predicted residual covariance is

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

This contains uncertainty from both

• predicted state,
• measurement noise.

It is therefore the correct scale for evaluating measurement consistency.

4.3.12.88 Normalized Innovation

For scalar measurement,

\[ \boxed{ \nu = \frac{y} {\sqrt S} } \]

provides a normalized residual.

For a vector measurement,

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

This is the normalized innovation squared.

4.3.12.89 Innovation Gating

Measurements may be accepted only if

\[ \boxed{ \mathbf y^T \mathbf S^{-1} \mathbf y < \gamma } \]

where \gamma is selected from an appropriate statistical threshold.

This is called innovation gating.

It helps reject outliers while accounting for predicted uncertainty.

4.3.12.90 Outlier Handling

Potential outlier causes include

• incorrect tracking association,
• sensor glitches,
• image-processing failure,
• multipath,
• poor signal lock,
• bad timing,
• corrupted packets.

Outlier logic should distinguish rare bad measurements from genuine spacecraft motion.

4.3.12.91 Measurement Availability

Different measurement types may not be available continuously.

Examples include

• ground-station visibility windows,
• camera lighting constraints,
• GNSS blockage,
• optical occultation,
• range-sensor duty cycles.

A robust estimator should handle asynchronous measurements.

4.3.12.92 Asynchronous Measurements

Suppose range arrives at time

\[ t_a \]

and optical LOS arrives at

\[ t_b. \]

The estimator should propagate the state to each actual measurement epoch before processing it.

Thus,

\[ \boxed{ \text{Measurement Update} \text{ should occur at the measurement time, not merely at a common nominal rate.} } \]

4.3.12.93 Multi-Rate Sensor Fusion

Different sensors may operate at different rates.

For example,

\[ f_1\neq f_2\neq f_3. \]

The estimator can propagate continuously or at a high internal rate while incorporating measurements whenever each sensor produces data.

Thus,

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

in general.

4.3.12.94 Measurement Latency

A sensor measurement may arrive after the physical observation time.

If the measurement corresponds to

\[ t_m \]

but arrives at

\[ t_m+\Delta t_{\rm latency}, \]

the estimator should distinguish

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

from

\[ \boxed{ \text{Data Arrival Time}. } \]

This is important in real-time navigation.

4.3.12.95 Delayed Measurement Processing

A delayed measurement can sometimes be handled through

• state buffering,
• out-of-sequence measurement processing,
• retrodiction,
• smoothing,
• state transition mapping.

The correct method depends on the estimator architecture.

4.3.12.96 Measurement Observability

A measurement is useful only if it is sensitive to the desired state direction.

The local sensitivity is given by

\[ \mathbf H. \]

Across time, the combination of

\[ \mathbf H_k \]

and

\[ \boldsymbol\Phi(t_k,t_0) \]

determines accumulated observability.

Thus,

\[ \boxed{ \text{Measurement Value} + \text{Measurement Geometry} + \text{Dynamics} \rightarrow \text{State Information} } \]

4.3.12.97 Complementary Measurements

Different sensors can constrain different state directions.

Examples:

\[ \boxed{ \text{Range} \rightarrow \text{Radial Position} } \]

\[ \boxed{ \text{Range Rate} \rightarrow \text{Radial Velocity} } \]

\[ \boxed{ \text{LOS Angles} \rightarrow \text{Transverse Direction} } \]

Therefore combining complementary measurements can greatly improve OD performance.

4.3.12.98 Range plus LOS

If both range and LOS are measured,

\[ \rho \]

and

\[ \hat{\boldsymbol\rho} \]

give

\[ \boxed{ \boldsymbol\rho = \rho \hat{\boldsymbol\rho}. } \]

This provides direct geometric 3-D relative-position information, subject to measurement uncertainty.

4.3.12.99 Range plus Range Rate

Range plus range rate gives

\[ \boxed{ \rho,\qquad\dot\rho } \]

which constrain radial position and radial velocity.

Transverse state components still rely on changing geometry, dynamics, or angular measurements.

4.3.12.100 Range plus Angles

A measurement set

\[ \boxed{ \mathbf z = \begin{bmatrix} \rho\\ A\\ E \end{bmatrix} } \]

or

\[ \boxed{ \mathbf z = \begin{bmatrix} \rho\\ \alpha\\ \delta \end{bmatrix} } \]

contains sufficient instantaneous geometric information to reconstruct a 3-D relative position under the chosen coordinate model.

Velocity still generally requires multiple measurements or velocity-sensitive observables.

4.3.12.101 Full Radar-Type Measurement

A common complete measurement vector is

\[ \boxed{ \mathbf z = \begin{bmatrix} \rho\\ \dot\rho\\ A\\ E \end{bmatrix} } \]

with covariance

\[ \boxed{ \mathbf R = E[ \mathbf v\mathbf v^T ]. } \]

Such a sensor provides both radial and angular information.

4.3.12.102 Camera Measurement Geometry

For a monocular camera, a single image typically provides angular direction but not absolute range directly.

Therefore,

\[ \boxed{ \text{Monocular Image} \rightarrow \text{Strong Bearing Information} + \text{Weak Instantaneous Range Information} } \]

Range may become observable through

• target size,
• known geometry,
• multiple features,
• spacecraft motion,
• multiple observations.

4.3.12.103 Stereo Optical Measurement

With two spatially separated cameras, depth may be obtained through triangulation.

If baseline is B and disparity is d,

\[ \boxed{ Z \approx \frac{fB}{d} } \]

for the simplified stereo model.

Stereo therefore provides direct geometric depth information but depends strongly on calibration and feature matching.

4.3.12.104 Optical Feature Uncertainty

The covariance of the image feature can propagate into LOS covariance:

\[ \boxed{ \mathbf P_{\rm LOS} \approx \mathbf J_{\rm pix\rightarrow LOS} \mathbf R_{\rm pix} \mathbf J_{\rm pix\rightarrow LOS}^T. } \]

Thus image processing and navigation estimation should not be treated as completely separate uncertainty problems.

4.3.12.105 GNSS Pseudorange Jacobian

For satellite i,

\[ \rho_i = \| \mathbf r-\mathbf r_i \|. \]

Therefore

\[ \boxed{ \frac{\partial \rho_i} {\partial\mathbf r} = \hat{\boldsymbol\rho}_i^T } \]

and with clock bias expressed as range-equivalent state b_c,

\[ \boxed{ \frac{\partial P_i} {\partial b_c} = 1. } \]

Hence a simplified pseudorange row is

\[ \boxed{ \mathbf H_i = \begin{bmatrix} \hat{\boldsymbol\rho}_i^T & 0\ 0\ 0 & 1 & 0 \end{bmatrix}. } \]

The exact sign depends on how the LOS vector is defined, so the convention must remain consistent.

4.3.12.106 GNSS Doppler Jacobian

GNSS Doppler depends strongly on receiver velocity and clock drift.

A simplified state sensitivity has the form

\[ \boxed{ \mathbf H_{\dot\rho,i} = \begin{bmatrix} \mathbf H_{r,i} & \hat{\boldsymbol\rho}_i^T & 0 & 1 \end{bmatrix} } \]

where \mathbf H_{r,i} contains position-dependent LOS terms.

4.3.12.107 Measurement Design Matrix

For measurements

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

the local sensitivities can be stacked:

\[ \boxed{ \mathbf H = \begin{bmatrix} \mathbf H_1\\ \mathbf H_2\\ \vdots\\ \mathbf H_N \end{bmatrix}. } \]

For reference-epoch OD,

\[ \boxed{ \mathbf G = \begin{bmatrix} \mathbf H_1\boldsymbol\Phi_1\\ \mathbf H_2\boldsymbol\Phi_2\\ \vdots\\ \mathbf H_N\boldsymbol\Phi_N \end{bmatrix}. } \]

This matrix is fundamental in batch orbit determination.

4.3.12.108 Information from Measurements

For linearized Gaussian measurements,

\[ \boxed{ \mathbf I = \mathbf H^T \mathbf R^{-1} \mathbf H } \]

is a local information matrix.

Large information corresponds to smaller uncertainty.

If \mathbf H is poorly conditioned or rank deficient, some state directions are weakly observed.

4.3.12.109 Measurement Geometry and Information

Two sensors with identical measurement noise may provide very different information depending on geometry.

Thus,

\[ \boxed{ \text{Measurement Quality} = \text{Precision} + \text{Geometry} + \text{Sensitivity} } \]

not precision alone.

4.3.12.110 Measurement Calibration

Before use in OD, a measurement system should ideally be characterized for

• bias,
• scale factor,
• alignment,
• nonlinearity,
• timing,
• latency,
• noise,
• environmental sensitivity.

These quantities determine how the raw sensor observation should be represented mathematically.

4.3.12.111 Ground Calibration versus In-Flight Calibration

Some calibration parameters can be determined before flight.

Others may change because of

• launch loads,
• thermal deformation,
• aging,
• deployment,
• radiation,
• mechanical shift.

A general strategy is

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

4.3.12.112 Measurement Model Fidelity

Increasing estimator complexity cannot compensate indefinitely for an incorrect measurement model.

For example, a sophisticated EKF using the wrong frame transformation can still produce biased estimates.

Therefore,

\[ \boxed{ \text{Good Estimator} + \text{Bad Measurement Model} \rightarrow \text{Bad Navigation} } \]

4.3.12.113 Measurement Model Validation

Measurement models should be tested through

• analytical checks,
• limiting cases,
• simulation,
• finite differences,
• synthetic truth data,
• hardware data when available.

For example, if spacecraft and observer positions are identical,

\[ \rho\rightarrow0 \]

and range-angle models may become singular.

Such cases should be handled explicitly.

4.3.12.114 Limiting-Case Checks

Useful model checks include:

• zero relative velocity \rightarrow \dot\rho=0,
• pure radial motion \rightarrow LOS direction unchanged,
• pure transverse motion \rightarrow first-order range rate may be zero,
• increasing range \rightarrow \dot\rho>0,
• identical position \rightarrow LOS undefined.

These tests catch many implementation errors.

4.3.12.115 Sign-Conventions Check

Range-rate and Doppler sign conventions frequently differ between systems.

For one implementation,

\[ \dot\rho>0 \]

may mean receding.

Another instrument may report positive Doppler for approaching motion.

Therefore the measurement-model sign must match the sensor-interface definition.

4.3.12.116 Units

Measurement-model units must remain explicit.

Typical units include

• range: m,
• range rate: m/s,
• Doppler: Hz,
• azimuth/elevation: rad or deg,
• RA/Dec: rad or deg,
• pixel coordinates: pixels,
• pseudorange: m,
• carrier phase: cycles or m.

Mixed angular units are a common source of errors.

4.3.12.117 Degrees versus Radians

Estimation software generally uses radians internally.

If a sensor reports degrees,

\[ \boxed{ \theta_{\rm rad} = \theta_{\rm deg} \frac{\pi}{180}. } \]

Jacobians must be consistent with the units of the state and measurement.

4.3.12.118 Measurement Frame Metadata

A measurement is incomplete unless its frame is known.

For example,

\[ \hat{\boldsymbol\rho} = \begin{bmatrix} 0.2\\0.4\\0.894 \end{bmatrix} \]

has no useful navigation meaning unless one knows whether it is expressed in

• camera frame,
• body frame,
• inertial frame,
• local orbital frame.

Thus,

\[ \boxed{ \text{Measurement Value} + \text{Frame} + \text{Time} + \text{Uncertainty} } \]

should be considered together.

4.3.12.119 Measurement Packet Concept

A useful navigation measurement record includes

\[ \boxed{ \{ t_k,\; \mathbf z_k,\; \mathbf R_k,\; \text{frame},\; \text{sensor ID},\; \text{quality flags} \}. } \]

This allows the estimator to interpret the measurement correctly.

4.3.12.120 Measurement Quality Flags

A sensor may provide flags indicating

• valid/invalid,
• degraded mode,
• low signal strength,
• poor feature tracking,
• saturation,
• insufficient satellites,
• calibration warning.

These flags can be used before measurement incorporation.

4.3.12.121 Raw Measurement versus Processed Measurement

An estimator may receive

• raw sensor measurements,
• preprocessed measurements,
• already estimated intermediate quantities.

For example,

\[ \text{Camera Image} \rightarrow \text{Pixel Features} \rightarrow \text{LOS} \rightarrow \text{Navigation Estimator}. \]

Each stage introduces uncertainty.

The covariance passed downstream should represent the uncertainty after that processing stage.

4.3.12.122 Avoiding Double Counting

Suppose a relative position estimate is generated from camera and range measurements.

If the main estimator then ingests both

• the raw camera/range measurements,
• the derived relative-position estimate,

without accounting for correlation, information may be double counted.

Therefore,

\[ \boxed{ \text{Derived Measurement Correlation Must Be Considered} } \]

4.3.12.123 Measurement Whitening

If measurement covariance is

\[ \mathbf R, \]

a whitened residual may be defined using a factor

\[ \mathbf R = \mathbf L\mathbf L^T. \]

Then

\[ \boxed{ \mathbf r_w = \mathbf L^{-1}\mathbf r. } \]

Whitened residuals are useful for diagnostics and least-squares implementations.

4.3.12.124 Pre-Fit Measurement Residual

Before state correction,

\[ \boxed{ \mathbf r_k^{\rm pre} = \mathbf z_k - h( \hat{\mathbf x}_k^- ). } \]

This indicates how well the propagated state predicts the measurement.

4.3.12.125 Post-Fit Measurement Residual

After correction,

\[ \boxed{ \mathbf r_k^{\rm post} = \mathbf z_k - h( \hat{\mathbf x}_k^+ ). } \]

Comparison of pre-fit and post-fit residuals can help evaluate estimator effectiveness.

4.3.12.126 Persistent Residual Bias

If residual mean is significantly nonzero,

\[ E[\mathbf r]\neq0, \]

possible causes include

• missing measurement bias,
• incorrect calibration,
• wrong force model,
• timing offset,
• frame error.

Persistent bias should not automatically be removed by increasing process noise.

4.3.12.127 Temporal Correlation in Residuals

Residuals should often be approximately white after correct modeling.

If

\[ E[ r_k r_{k+\ell} ] \neq0 \]

for significant lag \ell, the system may contain unmodeled temporal structure.

Possible causes include

• slowly varying bias,
• environmental error,
• correlated sensor noise,
• unmodeled dynamics.

4.3.12.128 Measurement Model in Batch OD

In batch OD,

\[ \boxed{ \mathbf z_k = h_k[ \phi( t_k,t_0,\mathbf x_0 )] + \mathbf v_k. } \]

The unknown is the reference-epoch state

\[ \mathbf x_0. \]

Thus every measurement is related back to the same state through orbit propagation.

4.3.12.129 Measurement Model in Sequential OD

In sequential OD,

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

The estimator propagates the state from the previous epoch and then processes the new measurement.

The physical measurement equations are the same as in batch OD; only the estimation architecture differs.

4.3.12.130 Measurement Models and UKF

In an Unscented Kalman Filter, explicit measurement Jacobians are not required for the basic UKF update.

Instead, sigma points are passed through

\[ \boxed{ \mathbf z^{(i)} = h( \mathbf x^{(i)} ). } \]

However, the measurement model itself must still be accurate.

Avoiding Jacobians does not remove the need for correct sensor geometry.

4.3.12.131 Measurement Models and Particle Filters

For particle i,

\[ \mathbf x_k^{(i)}, \]

the predicted measurement is

\[ \hat{\mathbf z}_k^{(i)} = h( \mathbf x_k^{(i)} ). \]

The measurement likelihood determines particle weight:

\[ \boxed{ w_k^{(i)} \propto p( \mathbf z_k | \mathbf x_k^{(i)} ). } \]

Thus the same measurement models are fundamental to nonlinear Bayesian estimators.

4.3.12.132 Measurement Likelihood

For Gaussian measurement noise,

\[ \boxed{ p( \mathbf z | \mathbf x ) \propto \exp \left[ -\frac12 ( \mathbf z-h(\mathbf x) )^T \mathbf R^{-1} ( \mathbf z-h(\mathbf x) ) \right]. } \]

This provides the probabilistic interpretation of the residual.

4.3.12.133 Measurement Model and Observability

Measurement equations determine what state combinations are directly visible.

The dynamics determine how information propagates from those observable combinations to other states.

Therefore,

\[ \boxed{ \text{OD Observability} = \text{Measurement Sensitivity} + \text{Dynamic Coupling} + \text{Geometry Evolution}. } \]

4.3.12.134 Measurement Model and Error Budget

Measurement uncertainty contributes to overall navigation uncertainty through

\[ \mathbf R. \]

But the final state error depends on the measurement sensitivity.

A sensor with small \sigma may still provide little benefit if it is insensitive to the dominant uncertain state direction.

Thus

\[ \boxed{ \text{Sensor Precision} \neq \text{Final Navigation Precision} } \]

4.3.12.135 Common Mistakes in OD Measurement Modeling

Common errors include:

1. Using incorrect frame transformations.
2. Mixing inertial and Earth-fixed quantities.
3. Ignoring observer motion.
4. Using the wrong range-rate sign convention.
5. Mixing degrees and radians.
6. Ignoring time-tag offsets.
7. Ignoring sensor latency.
8. Neglecting light-time where required.
9. Using sensor resolution as measurement covariance.
10. Ignoring measurement bias.
11. Treating correlated measurements as independent.
12. Using angle subtraction without wrapping.
13. Ignoring camera alignment.
14. Ignoring sensor lever arm.
15. Using an incorrect GNSS clock model.
16. Confusing Doppler frequency with geometric range rate.
17. Failing to validate analytical Jacobians.
18. Double counting derived measurements.
19. Using an inappropriate noise distribution.
20. Assuming more measurement rate always means more independent information.

4.3.12.136 Practical Measurement-Model Development Workflow

A disciplined workflow is

\[ \boxed{ \begin{array}{c} \text{Identify Physical Observable}\\ \downarrow\\ \text{Define Measurement Frame}\\ \downarrow\\ \text{Define Measurement Epoch}\\ \downarrow\\ \text{Define Observer/Sensor Geometry}\\ \downarrow\\ \text{Write Ideal Measurement Equation}\\ \downarrow\\ \text{Add Calibration Terms}\\ \downarrow\\ \text{Add Bias Terms}\\ \downarrow\\ \text{Define Noise Covariance}\\ \downarrow\\ \text{Derive Measurement Jacobian}\\ \downarrow\\ \text{Verify Jacobian}\\ \downarrow\\ \text{Implement Frame/Time Corrections}\\ \downarrow\\ \text{Generate Predicted Measurement}\\ \downarrow\\ \text{Compute Residual}\\ \downarrow\\ \text{Check Residual Statistics}\\ \downarrow\\ \text{Validate with Simulated or Real Data} \end{array} } \]

4.3.12.137 General Measurement Information Flow

The measurement-processing chain can be summarized as

\[ \boxed{ \begin{array}{c} \text{Spacecraft State}\\ \mathbf x \\ + \\ \text{Observer/Sensor Geometry} \\ + \\ \text{Frame and Time Information} \\ \downarrow\\ \text{Measurement Model} \\ h(\mathbf x) \\ \downarrow\\ \text{Predicted Measurement} \\ \hat{\mathbf z} \\ \downarrow\\ \text{Actual Measurement} - \text{Prediction} \\ \downarrow\\ \text{Residual} \\ \mathbf y \\ \downarrow\\ \text{Estimator Correction} \end{array} } \]

4.3.12.138 Major Measurement Types Summary

4.3.12.139 Key Equations

General measurement equation

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

Predicted measurement

\[ \boxed{ \hat{\mathbf z}_k = h_k( \hat{\mathbf x}_k^- ) } \]

Innovation

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

Relative position

\[ \boxed{ \boldsymbol\rho = \mathbf r_{\rm sc} - \mathbf r_{\rm obs} } \]

Range

\[ \boxed{ \rho = \|\boldsymbol\rho\| } \]

Range rate

\[ \boxed{ \dot\rho = \frac{ \boldsymbol\rho^T \dot{\boldsymbol\rho} }{ \rho } } \]

LOS vector

\[ \boxed{ \hat{\boldsymbol\rho} = \frac{\boldsymbol\rho}{\rho} } \]

Doppler approximation

\[ \boxed{ \Delta f \approx -\frac{f_0}{c} \dot\rho } \]

Azimuth

\[ \boxed{ A = \operatorname{atan2}( \rho_E,\rho_N ) } \]

Elevation

\[ \boxed{ E = \arctan \left( \frac{\rho_U} {\sqrt{\rho_E^2+\rho_N^2}} \right) } \]

Right ascension

\[ \boxed{ \alpha = \operatorname{atan2}( \rho_y,\rho_x ) } \]

Declination

\[ \boxed{ \delta = \arcsin \left( \frac{\rho_z}{\rho} \right) } \]

Measurement Jacobian

\[ \boxed{ \mathbf H_k = \frac{\partial h_k} {\partial\mathbf x_k} } \]

Reference-epoch sensitivity

\[ \boxed{ \mathbf G_k = \mathbf H_k \boldsymbol\Phi(t_k,t_0) } \]

Innovation covariance

\[ \boxed{ \mathbf S_k = \mathbf H_k \mathbf P_k^- \mathbf H_k^T + \mathbf R_k } \]

4.3.12.140 Key Takeaways

Orbit-determination measurement models mathematically connect the spacecraft state to what a sensor actually observes.

The fundamental relationship is

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

Different sensors observe different aspects of the state.

Range measures LOS distance:

\[ \boxed{ \rho = \| \mathbf r_{\rm sc} - \mathbf r_{\rm obs} \| } \]

Range rate measures LOS-relative velocity:

\[ \boxed{ \dot\rho = \hat{\boldsymbol\rho}^T \mathbf v_{\rm rel}. } \]

Doppler converts radial motion into frequency shift:

\[ \boxed{ \Delta f \approx -\frac{f_0}{c}\dot\rho. } \]

Angular measurements such as azimuth/elevation and RA/Dec provide LOS direction.

A direct LOS representation is

\[ \boxed{ \hat{\boldsymbol\rho} = \frac{\boldsymbol\rho}{\|\boldsymbol\rho\|}. } \]

GNSS measurements combine geometric range with clock and propagation effects, while optical measurements convert image geometry into bearing or LOS information.

The measurement Jacobian

\[ \boxed{ \mathbf H = \frac{\partial h}{\partial\mathbf x} } \]

describes how the measurement responds to changes in the state.

For reference-epoch orbit determination, the measurement sensitivity becomes

\[ \boxed{ \mathbf G_k = \mathbf H_k \boldsymbol\Phi(t_k,t_0). } \]

A complete measurement model must include not only geometry but also

• frame definitions,
• time tagging,
• sensor location,
• sensor alignment,
• light-time when required,
• bias,
• calibration,
• measurement covariance,
• latency,
• relevant propagation corrections.

The central engineering principle is

\[ \boxed{ \text{Accurate Orbit Determination Requires} \quad \text{Accurate Dynamics} + \text{Accurate Measurement Models}. } \]

Even the most sophisticated estimator cannot compensate reliably for a systematically incorrect measurement model.

Next: 4.3.13 Batch Orbit Determination

Measurement models define how each tracking observation depends on spacecraft state, sensor geometry, timing, calibration, and uncertainty. The next section combines an observation arc into a single differential-correction problem using STM-based design matrices, weighted residuals, iteration, convergence, covariance, and post-fit residual analysis.