Observability in Sensor Fusion

4.2.17 Observability in Sensor Fusion

A navigation filter can only estimate a state if the available measurements contain enough information about that state.

This leads to one of the most important ideas in spacecraft navigation and sensor fusion:

\[ \boxed{ \text{Observability} } \]

Observability answers the question:

\[ \boxed{ \text{Can the internal spacecraft state be reconstructed from the available measurements and known dynamics?} } \]

This is different from simply asking whether a sensor is available.

A spacecraft may have several healthy sensors and still be unable to estimate some states uniquely.

For example, a camera may provide only the direction toward a target. It may determine the line of sight very accurately but provide poor instantaneous information about range.

Similarly, an IMU may measure angular rate and specific force very accurately but cannot, by itself, prevent position and velocity error from drifting indefinitely.

Therefore,

\[ \boxed{ \text{Sensor Availability} \neq \text{State Observability}. } \]

Observability depends on

• the states being estimated,
• spacecraft dynamics,
• measurement equations,
• sensor geometry,
• spacecraft motion,
• target motion,
• mission phase,
• sensor combinations,
• and sometimes the duration over which measurements are collected.

A useful conceptual chain is

\[ \boxed{ \text{Spacecraft Dynamics} } \]

\[ + \]

\[ \boxed{ \text{Sensor Geometry} } \]

\[ + \]

\[ \boxed{ \text{Vehicle Motion} } \]

\[ \downarrow \]

\[ \boxed{ \text{Information About the State} } \]

\[ \downarrow \]

\[ \boxed{ \text{Observable / Weakly Observable / Unobservable States}. } \]

This becomes particularly important after the sensor faults discussed in Section 4.2.16.

Removing one sensor does not merely reduce accuracy.

It may remove the only measurement that made a particular state observable.

4.2.17.1 What Does Observable Mean?

Consider a state vector

\[ \boxed{ \mathbf x = \begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n \end{bmatrix}. } \]

A state is observable if its value can, in principle, be determined from the available measurements and the known system dynamics over a finite interval.

Conceptually:

\[ \boxed{ \text{Different States} \rightarrow \text{Different Measurement Histories}. } \]

If two different states always produce exactly the same measurement history, the estimator cannot distinguish them.

Those state differences are unobservable.

4.2.17.2 Observable Does Not Mean Directly Measured

A state does not have to be measured directly to be observable.

For example, consider

\[ \mathbf x = \begin{bmatrix} \theta\\ \omega \end{bmatrix} \]

while the sensor measures only attitude:

\[ \boxed{ y=\theta. } \]

Angular rate is not directly measured.

However, because

\[ \dot\theta=\omega, \]

the evolution of attitude over time contains information about angular rate.

Therefore \omega can be observable through the system dynamics.

This gives an important principle:

\[ \boxed{ \text{A State Can Be Observable Without Being Directly Measured}. } \]

4.2.17.3 Direct Measurement vs Dynamic Observability

Direct measurement means a state appears explicitly in the measurement equation.

For example,

\[ \boxed{ z=x+v. } \]

Dynamic observability means information about a state appears indirectly through how other measured quantities evolve.

For example,

\[ \dot x_1=x_2 \]

while only

\[ x_1 \]

is measured.

Over time, measurements of x_1 reveal information about x_2.

Thus:

\[ \boxed{ \text{Measurement Model} + \text{Dynamics} \rightarrow \text{Observability}. } \]

4.2.17.4 Unobservable State

A state component is unobservable when changes in that state cannot be distinguished from the available measurements.

Suppose

\[ \mathbf x= \begin{bmatrix} x_1\\ x_2 \end{bmatrix} \]

and the measurement is

\[ \boxed{ y=x_1. } \]

If the dynamics contain no coupling between x_1 and x_2, then changing x_2 does not affect y.

Therefore x_2 is unobservable.

The estimator may still contain an estimate for x_2, but that estimate is not being constrained by measurement information.

4.2.17.5 Why Unobservable States Are Dangerous

An estimator may numerically produce

\[ \hat x_2 \]

even when x_2 is unobservable.

This can create a false impression that the state is known.

The covariance should reveal the lack of information.

Ideally,

\[ \boxed{ P_{22} } \]

should remain large or grow.

If the filter instead reports very small covariance for an unobservable state, the estimator is inconsistent.

4.2.17.6 Weak Observability

Observability is not always practically binary.

Some state directions may technically be observable but only very weakly.

For example, a camera observing a distant target may contain some range information through apparent target size, but the sensitivity may be extremely small.

Thus:

\[ \boxed{ \text{Mathematically Observable} } \]

does not always imply

\[ \boxed{ \text{Practically Well Estimated}. } \]

This motivates the concept of weak observability.

4.2.17.7 Strong vs Weak Observability

A strongly observable state produces a significant change in measurements when the state changes.

A weakly observable state produces only a very small measurement change.

Conceptually:

\[ \boxed{ \left\| \frac{\partial h}{\partial x_i} \right\| \text{ large} \rightarrow \text{Strong Measurement Sensitivity}. } \]

Conversely,

\[ \boxed{ \left\| \frac{\partial h}{\partial x_i} \right\| \approx0 \rightarrow \text{Weak Measurement Sensitivity}. } \]

4.2.17.8 Linear State-Space System

Consider the linear system

\[ \boxed{ \dot{\mathbf x} = A\mathbf x } \]

and measurement equation

\[ \boxed{ \mathbf y = C\mathbf x. } \]

The classical observability question is:

\[ \boxed{ \text{Can }\mathbf x(0)\text{ be reconstructed from }\mathbf y(t)? } \]

For linear time-invariant systems, this can be tested using the observability matrix.

4.2.17.9 Observability Matrix

For an n-state linear system, define

\[ \boxed{ \mathcal O = \begin{bmatrix} C\\ CA\\ CA^2\\ \vdots\\ CA^{n-1} \end{bmatrix}. } \]

The system is completely observable if

\[ \boxed{ \operatorname{rank}(\mathcal O)=n. } \]

This is the classical Kalman observability rank condition.

4.2.17.10 Why the Matrix Contains C,CA,CA^2,\dots

The first measurement is

\[ y=Cx. \]

Differentiate:

\[ \dot y = C\dot x = CAx. \]

Differentiate again:

\[ \ddot y = CA^2x. \]

Therefore successive measurement derivatives reveal additional combinations of the state.

The observability matrix stacks this information.

4.2.17.11 Rank Interpretation

If

\[ \operatorname{rank}(\mathcal O)=n, \]

all state directions affect the measurement history independently.

If

\[ \operatorname{rank}(\mathcal O)

some state directions remain indistinguishable.

The dimension of the unobservable subspace is

\[ \boxed{ n-\operatorname{rank}(\mathcal O). } \]

4.2.17.12 Simple Example

Consider

\[ A= \begin{bmatrix} 0&1\\ 0&0 \end{bmatrix}, \qquad C= \begin{bmatrix} 1&0 \end{bmatrix}. \]

Then

\[ CA = \begin{bmatrix} 0&1 \end{bmatrix}. \]

Thus

\[ \mathcal O = \begin{bmatrix} 1&0\\ 0&1 \end{bmatrix}. \]

Therefore

\[ \boxed{ \operatorname{rank}(\mathcal O)=2. } \]

Both position-like state x_1 and velocity-like state x_2 are observable even though only x_1 is directly measured.

4.2.17.13 Unobservable Linear Example

Now suppose

\[ A= \begin{bmatrix} 0&0\\ 0&0 \end{bmatrix}, \qquad C= \begin{bmatrix} 1&0 \end{bmatrix}. \]

Then

\[ CA= \begin{bmatrix} 0&0 \end{bmatrix}. \]

Hence

\[ \mathcal O = \begin{bmatrix} 1&0\\ 0&0 \end{bmatrix}. \]

Therefore

\[ \boxed{ \operatorname{rank}(\mathcal O)=1<2. } \]

The second state is unobservable.

4.2.17.14 Observability Is a Property of the System Pair

For the linear system, observability depends on

\[ \boxed{ (A,C) } \]

together.

The dynamics matrix A determines how states influence each other.

The measurement matrix C determines what combinations of states are sensed.

Thus the same sensor model can produce different observability when combined with different dynamics.

4.2.17.15 Continuous-Time Observability Gramian

Another useful quantity is the observability Gramian:

\[ \boxed{ W_O = \int_{t_0}^{t_1} \Phi(t,t_0)^T C(t)^T C(t) \Phi(t,t_0) \,dt. } \]

For an LTI system,

\[ \Phi(t,t_0) = e^{A(t-t_0)}. \]

If W_O is positive definite over the interval, the system is observable over that interval.

4.2.17.16 Physical Meaning of the Observability Gramian

The Gramian measures how strongly changes in different state directions affect the accumulated measurements.

If a state direction produces almost no measurement change, the corresponding Gramian eigenvalue becomes small.

Thus Gramian eigenvalues provide information about observability strength, not only rank.

4.2.17.17 Smallest Observability Eigenvalue

Let the eigenvalues of W_O be

\[ \lambda_1,\lambda_2,\ldots,\lambda_n. \]

If

\[ \lambda_{\min} \]

is very small, at least one state direction is weakly observable.

Therefore,

\[ \boxed{ \lambda_{\min}(W_O) } \]

can serve as a practical observability metric.

4.2.17.18 Observability Condition Number

A useful measure is

\[ \boxed{ \kappa_O = \frac{ \lambda_{\max}(W_O) }{ \lambda_{\min}(W_O) }. } \]

If \kappa_O is very large, observability is highly uneven.

Some states are much better constrained than others.

4.2.17.19 Singular Values of the Observability Matrix

Similarly, perform singular value decomposition:

\[ \boxed{ \mathcal O = U\Sigma V^T. } \]

The singular values

\[ \sigma_i \]

indicate observability strength.

A very small singular value implies a weakly observable state combination.

4.2.17.20 Why Rank Alone Is Not Enough Numerically

A matrix may technically have full rank but still contain one singular value such as

\[ 10^{-10}. \]

Mathematically, the system is observable.

Practically, estimating the associated state may require extremely accurate measurements.

Therefore engineering analysis should examine both:

\[ \boxed{ \operatorname{rank}(\mathcal O) } \]

and

\[ \boxed{ \sigma_{\min}(\mathcal O). } \]

4.2.17.21 Discrete-Time Observability

For

\[ \boxed{ x_{k+1} = F x_k } \]

and

\[ \boxed{ z_k = H x_k, } \]

the discrete-time observability matrix is

\[ \boxed{ \mathcal O_d = \begin{bmatrix} H\\ HF\\ HF^2\\ \vdots\\ HF^{n-1} \end{bmatrix}. } \]

Complete observability requires

\[ \boxed{ \operatorname{rank}(\mathcal O_d)=n. } \]

4.2.17.22 Time-Varying Systems

Spacecraft navigation systems are often time varying:

\[ \boxed{ x_{k+1} = F_kx_k. } \]

Measurements may also vary:

\[ \boxed{ z_k = H_kx_k. } \]

Then observability depends on the sequence

\[ F_k, H_k \]

over an interval.

This is especially relevant when sensor geometry changes with orbital motion.

4.2.17.23 Time-Varying Observability Matrix

Over several measurement times, a local observability matrix can be constructed:

\[ \boxed{ \mathcal O = \begin{bmatrix} H_0\\ H_1F_0\\ H_2F_1F_0\\ \vdots \end{bmatrix}. } \]

This reveals whether measurements collected over time constrain all state directions.

4.2.17.24 Nonlinear Spacecraft Systems

Spacecraft dynamics and measurement models are usually nonlinear:

\[ \boxed{ \dot x = f(x,u) } \]

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

Examples include:

• quaternion attitude dynamics,
• orbital motion,
• camera projection,
• LiDAR range,
• relative navigation.

Classical linear observability analysis must therefore be extended.

4.2.17.25 Local Linearized Observability

A practical approach is to linearize around a nominal trajectory:

\[ \boxed{ \delta\dot x = F(t)\delta x } \]

\[ \boxed{ \delta z = H(t)\delta x. } \]

Then time-varying linear observability analysis can be applied locally.

This is particularly relevant to EKF and ESKF design.

4.2.17.26 Local Observability

A nonlinear system may be observable only in a neighborhood of a particular trajectory.

Thus:

\[ \boxed{ \text{Observability Can Depend on the Current State and Trajectory}. } \]

The same sensor configuration may be well observable in one geometry and weakly observable in another.

4.2.17.27 Nonlinear Observability Using Lie Derivatives

For

\[ \dot x=f(x) \]

and

\[ y=h(x), \]

define the first Lie derivative:

\[ \boxed{ L_f h = \frac{\partial h}{\partial x}f(x). } \]

Higher derivatives are

\[ \boxed{ L_f^2h = \frac{\partial L_fh}{\partial x}f(x) } \]

and so on.

4.2.17.28 Nonlinear Observability Matrix

A local nonlinear observability matrix may be formed from gradients:

\[ \boxed{ \mathcal O_N = \begin{bmatrix} \nabla h\\ \nabla L_fh\\ \nabla L_f^2h\\ \vdots \end{bmatrix}. } \]

If this matrix has full rank locally, the state is locally observable.

4.2.17.29 Why Nonlinear Observability Matters for GNC

Camera and LiDAR measurements are nonlinear functions of spacecraft state.

For example:

\[ \boxed{ z_{\text{camera}} = \pi( C(q) (r_T-r_C) ) } \]

where \pi is the perspective projection.

The observability of range, attitude, and bias therefore depends on geometry and motion.

4.2.17.30 Sensor Geometry

Sensor geometry strongly affects observability.

Suppose two spacecraft lie along the same line of sight.

A camera may determine direction accurately but be insensitive to motion along that line.

Thus the uncertainty ellipsoid may be long in the radial direction.

This is a geometric observability limitation.

4.2.17.31 Line-of-Sight Measurement

For relative position

\[ r_{rel}, \]

the unit line-of-sight vector is

\[ \boxed{ \ell = \frac{ r_{rel} }{ \|r_{rel}\| }. } \]

A bearing sensor measures approximately the direction

\[ \ell. \]

But the magnitude

\[ \|r_{rel}\| \]

does not directly appear.

Therefore instantaneous range is not directly observable from bearing alone.

4.2.17.32 Bearing-Only Navigation

Suppose the measurement is azimuth and elevation:

\[ \boxed{ z = \begin{bmatrix} \alpha\\ \beta \end{bmatrix}. } \]

The measurement determines direction to the target.

However,

\[ \boxed{ r_{rel} = \rho\,\ell } \]

contains unknown range

\[ \rho. \]

With one static bearing observation, infinitely many points along the ray produce the same measurement.

4.2.17.33 Bearing-Only Range Observability

Range can become observable through relative motion.

If the observer maneuvers, line-of-sight direction changes differently for different possible ranges.

Thus:

\[ \boxed{ \text{Observer Motion} \rightarrow \text{Parallax} \rightarrow \text{Range Information}. } \]

This is why maneuvering can improve bearing-only navigation.

4.2.17.34 No-Parallax Geometry

If observer and target move such that the line-of-sight direction remains nearly constant, range may remain poorly observable.

This occurs when motion is mostly along the line of sight.

Thus:

\[ \boxed{ \text{Pure Radial Relative Motion} \rightarrow \text{Weak Bearing-Only Range Observability}. } \]

4.2.17.35 Lateral Motion Improves Range Observability

Suppose the chaser moves sideways relative to the line of sight.

Then the apparent bearing changes.

The resulting parallax depends strongly on range.

Therefore:

\[ \boxed{ \text{Lateral Baseline} \rightarrow \text{Improved Range Observability}. } \]

This principle appears in optical navigation, stereo vision, and angles-only navigation.

4.2.17.36 Range-Only Navigation

A range sensor measures:

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

One scalar measurement constrains the target to a sphere.

Instantaneously, many different directions produce the same range.

Therefore range alone does not fully determine 3-D relative position.

4.2.17.37 Geometric Interpretation of Range-Only Measurement

For measured range

\[ \rho_m, \]

the target lies somewhere on:

\[ \boxed{ \|r_{rel}\| = \rho_m. } \]

This is a spherical surface.

Thus one range measurement eliminates radial uncertainty but leaves angular ambiguity.

4.2.17.38 Range-Only Observability Through Motion

As spacecraft dynamics evolve, successive spheres intersect.

Known vehicle motion can then recover position direction.

Therefore:

\[ \boxed{ \text{Range History} + \text{Known Motion} \rightarrow \text{More Complete Relative-State Information}. } \]

4.2.17.39 Range-Rate Measurement

Range rate is

\[ \boxed{ \dot\rho = \ell^T v_{rel}. } \]

It measures the component of relative velocity along the line of sight.

Transverse velocity remains less directly constrained.

4.2.17.40 Range and Range-Rate Together

Using

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

provides direct information about radial position and radial velocity.

However, lateral position and velocity still depend strongly on geometry and dynamics.

4.2.17.41 Full Relative Position Measurement

If a sensor directly provides

\[ \boxed{ r_{rel} = \begin{bmatrix} x\\ y\\ z \end{bmatrix}, } \]

relative position is instantaneously observed.

Velocity may still need to be inferred through dynamics or measured separately.

4.2.17.42 Position Measurements Make Velocity Observable Through Dynamics

With translational dynamics

\[ \dot r=v, \]

a sequence of position measurements contains velocity information.

Thus relative velocity can be observable even when the sensor does not directly measure velocity.

This is exactly analogous to the simple linear example.

4.2.17.43 GNSS Position and Velocity

GNSS often provides both:

\[ \boxed{ r_{GNSS} } \]

and

\[ \boxed{ v_{GNSS}. } \]

This directly constrains translational states strongly.

When GNSS is removed, those states may become significantly less observable depending on remaining sensors.

4.2.17.44 Relative GNSS

Relative or differential GNSS can directly constrain relative position:

\[ \boxed{ r_{rel} = r_C-r_T. } \]

Depending on implementation, relative velocity may also be available.

This can strongly improve far- and mid-range RPO observability.

4.2.17.45 Camera Plus GNSS

Consider

\[ \boxed{ \text{GNSS} + \text{Camera}. } \]

GNSS may provide strong absolute or relative position information.

The camera provides target bearing and attitude-related information.

Together, ambiguities present in either sensor alone can be reduced.

This demonstrates the fundamental advantage of complementary sensor geometry.

4.2.17.46 Camera Plus LiDAR

Near the target:

\[ \boxed{ \text{Camera} + \text{LiDAR} } \]

is a powerful combination.

The camera may provide precise angular information.

LiDAR may provide precise range or relative 3-D geometry.

Therefore:

\[ \boxed{ \text{Angular Information} + \text{Range Information} \rightarrow \text{Improved Relative Pose Observability}. } \]

4.2.17.47 Sensor Fusion Improves Observability, Not Just Noise

Sensor fusion is often described as improving accuracy by averaging noise.

But an equally important role is:

\[ \boxed{ \text{Combining Complementary Measurements to Eliminate Unobservable Directions}. } \]

For example, bearing plus range can provide information that neither alone gives completely.

4.2.17.48 Attitude Observability

Spacecraft attitude has three rotational degrees of freedom.

A single vector observation does not generally determine all three.

Suppose a known inertial vector is

\[ v^I \]

and its measured body-frame direction is

\[ v^B. \]

This constrains spacecraft orientation.

But rotation about that vector remains ambiguous.

4.2.17.49 Single-Vector Attitude Ambiguity

If

\[ v^B = C_{BI}v^I, \]

many attitude matrices can map the inertial vector onto the same body vector while differing by rotation about v^B.

Therefore:

\[ \boxed{ \text{One Vector Observation} \rightarrow \text{One Unobservable Rotation}. } \]

4.2.17.50 Two Non-Collinear Vector Observations

Suppose the spacecraft observes two known non-collinear inertial vectors:

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

Their body-frame measurements are

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

Together they can determine full attitude.

Thus:

\[ \boxed{ \text{Two Non-Collinear Reference Vectors} \rightarrow \text{Full Attitude Observability}. } \]

4.2.17.51 Sun Sensor and Magnetometer

A classical spacecraft example is:

\[ \boxed{ \text{Sun Vector} + \text{Magnetic Field Vector}. } \]

Provided the vectors are not nearly parallel, they provide strong attitude information.

If they become nearly aligned, attitude observability weakens.

4.2.17.52 Star Tracker Observability

A star tracker typically determines full attitude directly because the observed star pattern provides multiple inertial references.

Thus it offers strong attitude observability.

However, gyroscope bias still requires dynamic information over time.

4.2.17.53 Gyroscope Measurement

A gyroscope measures

\[ \boxed{ \omega_m = \omega+b_g+n_g. } \]

The measurement contains both true angular rate and gyro bias.

Without an independent attitude reference, it is difficult to distinguish persistent bias from true rotation.

4.2.17.54 Gyro Bias Observability

Gyro bias becomes observable when an external attitude reference is available.

For example:

\[ \boxed{ \text{Gyro Propagation} + \text{Star Tracker Attitude} } \]

allows the estimator to see systematic drift between propagated and measured attitude.

That drift contains information about b_g.

4.2.17.55 Bias Observability Requires Time

A constant gyro bias may not be determined instantaneously.

Its effect accumulates:

\[ \boxed{ \delta\theta \approx b_g\Delta t. } \]

Therefore a sequence of attitude measurements is required to estimate it reliably.

4.2.17.56 Accelerometer Bias

Accelerometer measurement is approximately

\[ \boxed{ a_m = a+b_a+n_a. } \]

Bias

\[ b_a \]

causes velocity error:

\[ \boxed{ \delta v \approx b_a t } \]

and position error:

\[ \boxed{ \delta r \approx \frac12b_at^2. } \]

External position or velocity measurements can therefore make accelerometer bias observable.

4.2.17.57 Accelerometer Bias with GNSS

The combination

\[ \boxed{ \text{Accelerometer} + \text{GNSS Velocity} } \]

is particularly useful.

If integrated accelerometer velocity consistently diverges from GNSS velocity, the filter can estimate accelerometer bias.

4.2.17.58 Bias Observability and Maneuvering

Bias observability often improves during maneuvers.

A static spacecraft may not sufficiently separate:

• sensor bias,
• attitude error,
• scale factor.

Controlled excitation introduces measurement signatures that distinguish these states.

4.2.17.59 Persistent Excitation

A system may be theoretically observable but require sufficient motion to reveal the states.

This is often called persistent excitation.

Conceptually:

\[ \boxed{ \text{Rich Motion} \rightarrow \text{Distinct Measurement Signatures} \rightarrow \text{Improved Parameter Observability}. } \]

4.2.17.60 Why Stationary Calibration Can Be Insufficient

Suppose a spacecraft remains nearly motionless.

A scale-factor error may produce almost no observable effect.

Likewise, certain misalignments may remain hidden.

Therefore some calibration states require deliberate motion.

4.2.17.61 Maneuver Design for Observability

Maneuvers can be selected specifically to improve observability.

Examples include:

• attitude slews,
• roll/pitch/yaw excitation,
• lateral translation,
• orbital baseline changes.

The objective is not merely motion.

It is motion that excites weak state directions.

4.2.17.62 Observability-Aware Trajectory Design

For autonomous navigation, guidance can sometimes choose trajectories that improve state estimation.

Thus:

\[ \boxed{ \text{Guidance} \rightarrow \text{Better Sensor Geometry} \rightarrow \text{Better Observability} \rightarrow \text{Better Navigation}. } \]

This creates an important coupling between guidance and navigation.

4.2.17.63 Example: Bearing-Only RPO Maneuver

Suppose the chaser observes only target bearing.

A purely radial approach may provide poor range observability.

A small lateral maneuver produces parallax.

Thus:

\[ \boxed{ \text{Lateral Excitation} \rightarrow \text{Range Becomes More Observable}. } \]

4.2.17.64 Example: Attitude Bias Calibration Maneuver

Suppose gyro scale factor and bias are uncertain.

A sequence of rotations about different body axes can separate:

• true angular motion,
• bias,
• scale factor,
• axis misalignment.

Therefore multi-axis motion is more informative than a single-axis rotation.

4.2.17.65 Sensor Boresight Observability

Suppose camera boresight error is represented by

\[ \delta\theta_c. \]

If the spacecraft always observes the target with nearly identical geometry, boresight error may be strongly correlated with attitude error.

Changing viewing geometry helps separate the two.

4.2.17.66 Lever-Arm Observability

Suppose a sensor is displaced from the spacecraft center of mass by

\[ r_{S/CG}. \]

Sensor position is

\[ \boxed{ r_S = r_{CG} + C_{IB}r_{S/CG}. } \]

Without rotational excitation, the lever-arm contribution may appear nearly constant.

Rotational motion changes its inertial projection and improves observability.

4.2.17.67 Clock-Bias Observability

Suppose a sensor time offset is

\[ \delta t. \]

Measurement may be

\[ \boxed{ z_k = h( x(t_k-\delta t) ). } \]

For small \delta t,

\[ z_k \approx h(x_k) - H\dot x_k\delta t. \]

Thus time offset becomes observable when

\[ \dot x_k \]

produces measurable change.

4.2.17.68 Static Motion and Clock Bias

If

\[ \dot x\approx0, \]

then

\[ \frac{\partial h}{\partial\delta t} \approx0. \]

Therefore timing offset becomes poorly observable.

This is why dynamic motion helps time-calibration estimation.

4.2.17.69 Scale-Factor Observability

Consider

\[ \boxed{ z = (1+s)x. } \]

If x varies very little, scale factor s may be difficult to separate from bias or state error.

Exciting the measured quantity over a broad range improves scale-factor observability.

4.2.17.70 Sensor Misalignment Observability

Suppose a sensor frame differs from its nominal alignment by

\[ \delta\theta_s. \]

During limited motion, this error may appear similar to spacecraft attitude error.

Multi-axis excitation helps separate:

\[ \boxed{ \text{Sensor Misalignment} } \]

from

\[ \boxed{ \text{Vehicle Attitude Error}. } \]

4.2.17.71 Relative Navigation State

For a generic RPO application, a typical relative-navigation state may include

\[ \boxed{ x = \begin{bmatrix} r_{rel}\\ v_{rel}\\ q_{rel}\\ \omega_{rel}\\ b_g\\ b_a \end{bmatrix}. } \]

Not every sensor measures every part of this state.

Observability analysis determines whether the complete state can be reconstructed from the sensor suite.

4.2.17.72 Relative Position Observability

Relative position may be constrained by:

• relative GNSS,
• LiDAR range/pose,
• camera bearing,
• stereo vision.

The quality and dimensionality of information differ substantially between these sensors.

4.2.17.73 Relative Velocity Observability

Velocity may be measured directly through:

• GNSS velocity,
• Doppler,
• range rate.

Or it may be inferred dynamically from a sequence of relative-position measurements.

If position measurements are slow or noisy, relative velocity can become weakly observable.

4.2.17.74 Relative Attitude Observability

Relative attitude may be estimated from:

• visual target pose,
• fiducial markers,
• target feature geometry,
• star tracker plus known target attitude.

A simple range sensor does not provide relative attitude.

Therefore close-range RPO often requires pose-sensitive sensing.

4.2.17.75 Relative Angular Rate Observability

Relative angular rate may be:

• measured from gyros on both vehicles,
• estimated from successive relative-attitude measurements.

If attitude updates are sparse, angular-rate observability may degrade.

4.2.17.76 RPO Far-Range Observability

Far from the target, useful sensors may include:

\[ \boxed{ \text{GNSS / Relative GNSS} } \]

with perhaps coarse optical bearing.

Relative position and velocity may be well observed through GNSS.

Relative attitude may remain less important or be determined independently.

4.2.17.77 RPO Mid-Range Observability

At mid range:

\[ \boxed{ \text{GNSS} + \text{Camera} } \]

may overlap.

This is advantageous because GNSS constrains translation while the camera improves relative direction and target geometry.

The overlap also helps detect cross-sensor inconsistencies.

4.2.17.78 RPO Close-Range Observability

Near the target:

\[ \boxed{ \text{Camera} + \text{LiDAR} + \text{IMU} } \]

may dominate.

Camera provides angular or pose information.

LiDAR provides range and geometry.

IMU propagates high-rate motion.

Together they can provide strong relative-state observability.

4.2.17.79 Precision Relative-Navigation Observability

During precision RPO, the navigation system may require accurate estimates of

\\[ \boxed{ r_{rel} } \\]

\\[ \boxed{ v_{rel} } \\]

\\[ \boxed{ \theta_{rel} } \\]

\\[ \boxed{ \omega_{rel}. } \\]

Observability should therefore be considered for translation, attitude, and angular-rate states together rather than relative position alone.

4.2.17.80 Observability After Sensor Failure

Suppose LiDAR fails.

Camera plus IMU may still provide adequate relative navigation.

But range uncertainty may grow.

Whether the mission can continue depends on whether the remaining sensor suite keeps all required states sufficiently observable.

Thus:

\[ \boxed{ \text{Sensor Failure Analysis} \rightarrow \text{Observability Analysis}. } \]

4.2.17.81 GNSS Failure

If GNSS is removed, absolute position observability may degrade.

For RPO, relative optical measurements may still maintain relative position.

Therefore:

\[ \boxed{ \text{Absolute Navigation Observability} } \]

and

\[ \boxed{ \text{Relative Navigation Observability} } \]

should be analyzed separately.

4.2.17.82 Camera Failure

If camera fails during close approach, LiDAR may still provide range or relative position.

But if LiDAR does not provide full pose, relative attitude may become weakly observable.

Thus the effect depends on LiDAR measurement type.

4.2.17.83 LiDAR Failure

If LiDAR fails and only monocular camera remains, bearing may remain excellent while range becomes less directly constrained.

Relative motion can still provide range information, but uncertainty may increase.

4.2.17.84 IMU Failure

Without IMU, high-rate state propagation weakens.

Slow camera and LiDAR updates may still determine state periodically, but high-frequency motion between updates becomes less observable.

Thus the system may have sufficient low-rate observability but inadequate control-bandwidth navigation.

4.2.17.85 Observability vs Update Rate

A state can be theoretically observable but poorly estimated if measurements arrive too slowly.

For example, relative angular rate may be observable from camera pose changes.

But if camera update rate is only

\[ 1~\mathrm{Hz}, \]

it may not support a control loop requiring

\[ 50~\mathrm{Hz} \]

state information.

This is a practical observability issue.

4.2.17.86 Observability vs Measurement Noise

High noise does not necessarily make a state mathematically unobservable.

However, it can make the information so weak that the state is practically unusable.

Therefore:

\[ \boxed{ \text{Observability} + \text{Measurement Precision} \rightarrow \text{Achievable Estimation Accuracy}. } \]

4.2.17.87 Observability and Fisher Information

Measurement information can be approximated by the Fisher information matrix:

\[ \boxed{ \mathcal I = H^TR^{-1}H. } \]

Large information in a state direction means the measurement constrains that direction strongly.

This links observability to uncertainty reduction.

4.2.17.88 Information Accumulation Over Time

Over several independent measurements,

\[ \boxed{ \mathcal I_{\text{total}} = \sum_k H_k^TR_k^{-1}H_k. } \]

Changing geometry can cause successive measurements to constrain different state directions.

Therefore information can accumulate into a fully observable solution.

4.2.17.89 Geometry and Information Ellipsoid

The inverse information matrix is related to uncertainty.

Poor geometry creates an elongated uncertainty ellipsoid.

Good geometry creates a more compact ellipsoid.

This is similar to dilution-of-precision concepts in GNSS.

4.2.17.90 GNSS Geometry

GNSS position observability depends on satellite geometry.

If satellites are clustered in similar directions, position uncertainty becomes large in some directions.

If they are well distributed, geometry improves.

Thus even a nominally available GNSS solution may have weak directional observability.

4.2.17.91 Geometric Dilution of Precision

GNSS geometry is often summarized using dilution-of-precision metrics.

Conceptually:

\[ \boxed{ \text{Poor Geometry} \rightarrow \text{Large DOP} \rightarrow \text{Large Position Uncertainty}. } \]

This is an operational form of observability-strength analysis.

4.2.17.92 Camera Feature Geometry

Camera pose estimation also depends on feature geometry.

Features that are:

• widely distributed,
• non-collinear,
• at varying depth,

provide stronger pose constraints.

Poorly distributed features can create weak directions.

4.2.17.93 Collinear Camera Features

If several tracked features lie nearly on one image line, pose estimation may become poorly conditioned.

Certain rotational or translational directions may become difficult to distinguish.

Thus feature count alone does not determine observability.

Geometry matters.

4.2.17.94 Planar Target Geometry

A planar relative-navigation target can create specific pose ambiguities depending on camera viewpoint.

Depth estimation may be weaker when the camera views the plane almost frontally and feature baseline is small.

Changing viewpoint can improve conditioning.

4.2.17.95 Stereo Vision Observability

With two cameras separated by baseline B, depth can be inferred from disparity.

Approximately:

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

where

• f is focal length,
• d is disparity.

Thus stereo geometry directly improves range observability.

4.2.17.96 Stereo Depth Sensitivity

Differentiating approximately:

\[ \boxed{ \delta Z \approx -\frac{fB}{d^2}\delta d. } \]

At long range, disparity becomes small.

Therefore depth becomes increasingly sensitive to pixel error.

Thus stereo range observability weakens with distance.

4.2.17.97 LiDAR Geometry

A LiDAR point cloud contains geometry distributed across the target surface.

Pose observability depends on target shape.

A richly three-dimensional target may constrain full pose well.

A highly symmetric object may create ambiguities.

4.2.17.98 Symmetry-Induced Ambiguity

Suppose a target is rotationally symmetric.

A rotation about the symmetry axis may produce almost identical LiDAR measurements.

Therefore that attitude direction may be weakly observable.

This is an example of object geometry limiting observability.

4.2.17.99 Target Model Helps Observability

If the navigation system knows the target geometry accurately, measurements can be compared against a detailed model.

This provides richer pose information.

Thus:

\[ \boxed{ \text{Known Target Shape} + \text{Sensor Geometry} \rightarrow \text{Improved Relative Pose Observability}. } \]

4.2.17.100 Relative Orbit Dynamics

For one common linearized RPO example, translational dynamics may be approximated using Clohessy-Wiltshire equations:

\[ \boxed{ \ddot x - 3n^2x - 2n\dot y = 0 } \]

\[ \boxed{ \ddot y + 2n\dot x = 0 } \]

\[ \boxed{ \ddot z + n^2z = 0. } \]

These couplings help propagate information between position and velocity components.

4.2.17.101 CW Dynamics and Observability

Because radial and along-track motion are dynamically coupled, measuring one component over time can provide information about another.

Thus the orbital dynamics themselves contribute to observability.

This is why relative navigation should not be analyzed using measurement geometry alone.

4.2.17.102 Unforced Motion Can Improve Observability

Even without active maneuvers, orbital dynamics evolve the geometry.

A state direction poorly observable at one instant may become better observed later.

Therefore observability should often be evaluated over a time window rather than at one instant.

4.2.17.103 Maneuver-Induced Observability

Active thrust changes the relative trajectory.

This may improve geometry.

However, maneuvers also introduce process uncertainty and actuator errors.

Therefore observability-enhancing maneuvers should balance:

\[ \boxed{ \text{Information Gain} } \]

against

\[ \boxed{ \text{Control Cost and Risk}. } \]

4.2.17.104 Observability and Process Noise

Process noise does not directly determine mathematical observability.

However, large Q can overwhelm weak measurement information.

Thus a weakly observable state may experience covariance growth despite technically being observable.

4.2.17.105 Observability and Covariance

In a Kalman filter,

\[ \boxed{ P^+ = (I-KH)P^-. } \]

Measurements reduce covariance primarily in state directions to which they are sensitive.

Unobservable directions receive little or no covariance reduction.

4.2.17.106 Covariance Signature of Unobservability

Suppose a state direction is unobservable.

Prediction repeatedly adds process uncertainty.

But measurements do not constrain that direction.

Therefore its covariance tends to grow:

\[ \boxed{ P_{\text{unobs}} \uparrow. } \]

This is a useful practical indicator.

4.2.17.107 Covariance Cross-Coupling

Even if a sensor does not directly measure state x_i, covariance cross-correlation with measured states may allow the update to reduce its uncertainty.

This is another manifestation of dynamic observability.

4.2.17.108 Kalman Gain and Observability

Kalman gain is

\[ \boxed{ K = PH^T ( HPH^T+R )^{-1}. } \]

If a state has no covariance coupling to measured quantities, the corresponding rows of K may be small.

Thus the measurement has little effect on that state.

4.2.17.109 Filter Convergence Requires Observability

A well-tuned Kalman filter cannot estimate an unobservable state accurately simply by waiting longer.

Without information,

\[ \boxed{ \text{No Estimator Algorithm Can Recover What the Measurements Do Not Contain}. } \]

This is a fundamental limitation.

4.2.17.110 Tuning Cannot Create Observability

Increasing

\[ Q \]

or decreasing

\[ R \]

cannot make an unobservable state physically observable.

It may change filter behavior, but not the information content.

Therefore:

\[ \boxed{ \text{Filter Tuning} \neq \text{Observability Creation}. } \]

4.2.17.111 More Complex Filters Cannot Create Observability

Similarly, replacing an EKF with UKF or Particle Filter does not automatically solve an observability problem.

A more sophisticated estimator may better handle nonlinearities.

But:

\[ \boxed{ \text{No Filter Can Estimate Information That the Sensors and Dynamics Do Not Provide}. } \]

4.2.17.112 Adding a Sensor Can Create Observability

Suppose bearing-only navigation leaves range weakly observable.

Adding LiDAR range measurement introduces:

\[ \boxed{ \rho } \]

directly.

The previously weak radial direction may become strongly observable.

Thus sensor selection is fundamentally an observability design problem.

4.2.17.113 Sensor Placement and Observability

Sensor location on the spacecraft matters.

For example, two cameras separated by a baseline provide stereo information.

Two nearly co-located sensors may provide redundant noise reduction but little new geometric information.

Thus:

\[ \boxed{ \text{Sensor Placement} \rightarrow \text{Measurement Geometry} \rightarrow \text{Observability}. } \]

4.2.17.114 Baseline Length

For stereo or multi-antenna systems, larger baseline often improves angular or depth sensitivity.

However, baseline is constrained by:

• spacecraft dimensions,
• structural alignment,
• thermal distortion,
• calibration.

Therefore observability is part of mechanical design.

4.2.17.115 Multi-Antenna GNSS Attitude

Multiple GNSS antennas separated by known baselines can provide attitude information.

The carrier-phase difference depends on baseline orientation relative to satellite directions.

Thus antenna geometry determines attitude observability.

4.2.17.116 Differential GNSS Relative Navigation

If target and chaser both have GNSS, differencing can reduce common errors.

Relative position can become highly observable.

However, performance still depends on:

• common satellite visibility,
• baseline,
• signal environment,
• synchronization.

4.2.17.117 Observability and Sensor Synchronization

Two individually informative sensors may produce poor combined estimation if their measurements are not synchronized.

Timing errors can introduce apparent states or biases.

Thus:

\[ \boxed{ \text{Observability Analysis Assumes Correct Measurement Timing}. } \]

This connects observability to Section 4.2.14.

4.2.17.118 Observability and Frame Calibration

Incorrect frame transformations can appear as state errors.

For example, camera boresight error may be confused with spacecraft attitude error.

If both are estimated simultaneously, their separability depends on motion and geometry.

Therefore calibration states must be included in observability analysis.

4.2.17.119 State Augmentation Changes Observability

Suppose the original state is

\[ x. \]

Now augment with:

\[ \boxed{ x_a = \begin{bmatrix} x\\ b_g\\ b_a\\ \delta t\\ \delta\theta_c \end{bmatrix}. } \]

The original system may have been observable.

The augmented system may not be.

Every new estimated parameter requires measurement information.

4.2.17.120 Too Many Calibration States

Adding many calibration parameters can create strong correlations.

For example:

• camera boresight,
• spacecraft attitude bias,
• target pose bias,

may generate similar measurement effects.

If the data cannot distinguish them, the augmented filter becomes weakly observable.

4.2.17.121 State Correlation as an Observability Indicator

Very high covariance correlation between two states may indicate they are difficult to distinguish.

For states x_i,x_j, define

\[ \boxed{ \rho_{ij} = \frac{ P_{ij} }{ \sqrt{P_{ii}P_{jj}} }. } \]

If

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

the states may be strongly coupled.

4.2.17.122 Correlation Does Not Always Mean Unobservable

High correlation can occur even in observable systems.

However, persistent near-perfect correlation is a useful warning that two states may be weakly distinguishable.

Thus covariance diagnostics complement formal observability analysis.

4.2.17.123 Observability and Estimator Initialization

An unobservable state cannot be corrected later by measurements.

Therefore its initial value may remain strongly influential.

This means initialization becomes especially important for weakly observable states.

4.2.17.124 Prior Information

A Bayesian estimator can use prior information:

\[ \boxed{ p(x_0). } \]

A prior can reduce uncertainty in an unobservable direction.

But this does not mean the state becomes observable from measurements.

The information originates from the prior, not the sensor data.

4.2.17.125 Prior vs Measurement Observability

Suppose a calibration parameter is fixed using ground calibration.

The filter may know it accurately.

But if in-flight measurements contain no sensitivity to that parameter, it is still unobservable in flight.

Thus:

\[ \boxed{ \text{Known from Prior} \neq \text{Observable from Measurements}. } \]

4.2.17.126 Fixing Unobservable States

If a state is known to be unobservable and sufficiently calibrated, one option is not to estimate it online.

Instead:

\[ \boxed{ x_i=x_{i,\text{calibrated}}. } \]

This reduces filter dimension and avoids artificial covariance interactions.

4.2.17.127 When to Estimate a Calibration State

Estimate a parameter online only if:

• it may change significantly,
• measurements contain information about it,
• the required motion exists,
• its observability has been verified.

Otherwise keeping it fixed may be better.

4.2.17.128 Observability and Sensor Dropout

Suppose the nominal suite is:

\[ \boxed{ \text{GNSS} + \text{Camera} + \text{LiDAR} + \text{IMU}. } \]

If one sensor drops out, the measurement matrix changes.

Therefore observability changes immediately.

This should be analyzed for each failure combination.

4.2.17.129 Fault-Case Observability Matrix

For failure case j, construct

\[ \boxed{ \mathcal O^{(j)}. } \]

Then evaluate:

\[ \boxed{ \operatorname{rank} \left( \mathcal O^{(j)} \right). } \]

This determines whether all required states remain observable after the fault.

4.2.17.130 N-Minus-One Observability

A useful fault-tolerance requirement is:

\[ \boxed{ \text{All Required States Remain Observable After Loss of Any One Non-Critical Sensor}. } \]

This can be called an N-1 observability requirement.

It is especially valuable for critical RPO missions.

4.2.17.131 Sensor Criticality from Observability

A sensor is observability-critical if removing it causes:

\[ \boxed{ \operatorname{rank} ( \mathcal O )

This provides a rigorous way to classify sensor criticality.

4.2.17.132 Weak Observability After Sensor Loss

Rank may remain full after sensor loss, but the smallest singular value may collapse.

Thus:

\[ \boxed{ \sigma_{\min} \downarrow } \]

can reveal severe performance degradation even when rank remains unchanged.

4.2.17.133 Observability Margin

A practical metric may be based on:

\[ \boxed{ M_O = \sigma_{\min}(\mathcal O) } \]

or the smallest Gramian eigenvalue.

Larger M_O indicates stronger worst-direction observability.

4.2.17.134 Observability Margin and Mission Phase

The margin may vary with range:

\[ \boxed{ M_O=M_O(\rho). } \]

A sensor suite may be adequate at 100 m but poorly conditioned at 5 m, or vice versa.

Therefore observability should be evaluated throughout the RPO trajectory.

4.2.17.135 Observability Map

For RPO, one can compute an observability metric over relative geometry.

For example:

\[ \boxed{ M_O(x,y,z). } \]

This creates an observability map showing favorable and unfavorable approach regions.

Such maps can support trajectory design.

4.2.17.136 Observability Along a Planned Trajectory

Evaluate:

\[ \boxed{ \sigma_{\min} ( \mathcal O(t) ) } \]

or

\[ \boxed{ \lambda_{\min} ( W_O(t) ) } \]

along the mission trajectory.

This shows where observability becomes weak.

4.2.17.137 Guidance Constraint from Observability

Guidance may impose:

\[ \boxed{ M_O > M_{O,\min}. } \]

If observability falls below the threshold, guidance may:

• alter trajectory,
• introduce lateral motion,
• wait for better geometry,
• activate another sensor.

This is observability-aware guidance.

4.2.17.138 Observability and Hold Points

RPO hold points are often used for:

• navigation verification,
• sensor handover,
• state convergence.

A hold location should ideally also provide favorable sensor geometry.

Thus hold-point selection can include observability criteria.

4.2.17.139 Static Hold Limitation

However, remaining completely stationary at a hold point may reduce observability of certain parameters.

For example:

• range from bearing-only sensing,
• timing offset,
• scale factor.

Therefore small calibration maneuvers may sometimes be useful.

4.2.17.140 Observability and Sensor Handover

During handover:

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

The overlap period provides observability redundancy.

Before removing sensor A, verify that sensor B plus remaining sensors maintain required observability.

4.2.17.141 Handover Criterion

A robust handover condition may require:

\[ \boxed{ \text{New Sensor Valid} } \]

\[ \land \]

\[ \boxed{ \text{New Sensor Consistent} } \]

\[ \land \]

\[ \boxed{ \text{Required States Observable}. } \]

This is stronger than simply checking that the new sensor produces data.

4.2.17.142 Observability During Sensor Reacquisition

After a sensor dropout, the estimator covariance may have grown significantly.

When the sensor returns, some state directions may reconverge quickly while others remain weak.

Therefore reacquisition time depends on observability strength.

4.2.17.143 Convergence Rate and Observability

Strongly observable states tend to converge quickly.

Weakly observable states may require many measurements.

Thus:

\[ \boxed{ \text{Observability Strength} \rightarrow \text{Estimator Convergence Rate}. } \]

This provides the connection to Section 4.2.18.

4.2.17.144 Observability vs Controllability

Observability should not be confused with controllability.

Observability asks:

\[ \boxed{ \text{Can the state be inferred?} } \]

Controllability asks:

\[ \boxed{ \text{Can the state be driven by control inputs?} } \]

They are dual concepts for linear systems but represent different physical questions.

4.2.17.145 Why GNC Needs Both

A spacecraft may be able to control a state that it cannot accurately observe.

That is dangerous because the controller has poor knowledge of what it is controlling.

Similarly, a state can be well observed but not effectively actuated.

Successful GNC requires both.

4.2.17.146 Detectability

A weaker concept than observability is detectability.

A system is detectable if all unobservable modes are stable.

Thus unobservable states may exist, but they do not grow dangerously.

For estimator stability, detectability can sometimes be sufficient.

4.2.17.147 Why Detectability Matters

If an unobservable mode is unstable, estimation error can grow without bound.

If the unobservable mode is stable, its error may naturally decay or remain bounded.

Therefore detectability is relevant to observer design.

4.2.17.148 Unobservable Bias Modes

Bias states often behave as:

\[ \boxed{ \dot b = w_b. } \]

They are approximately random walks.

An unobservable bias therefore does not naturally decay.

Its uncertainty tends to grow.

This makes bias observability especially important.

4.2.17.149 Gauge Freedoms

Some estimation problems contain inherent coordinate freedoms.

For example, if only relative positions are measured in a multi-spacecraft system, global translation may be unobservable.

The complete formation can shift by the same vector without changing relative measurements.

This is called a gauge freedom.

4.2.17.150 Relative Navigation Gauge

If measurements depend only on

\[ \boxed{ r_i-r_j, } \]

then adding the same translation

\[ c \]

to all spacecraft positions leaves measurements unchanged.

Therefore absolute position is unobservable unless an external reference is added.

4.2.17.151 Absolute vs Relative Observability

A system may have excellent relative-state observability while lacking absolute-state observability.

This is often acceptable for docking.

The mission requirement determines which states actually need to be observable.

4.2.17.152 Mission-Required State Set

Define the required navigation state:

\[ \boxed{ x_{\text{req}}. } \]

Not every internal estimator state must necessarily be observable to the same level.

For RPO, the most critical states may be:

\[ \boxed{ r_{rel}, v_{rel}, q_{rel}, \omega_{rel}. } \]

Observability requirements should be tied to mission needs.

4.2.17.153 Reduced Observability Requirement

A mission may tolerate unobservable absolute yaw while requiring full relative pose.

Therefore the appropriate question is not always:

\[ \boxed{ \operatorname{rank}(\mathcal O)=n? } \]

but:

\[ \boxed{ \text{Are All Mission-Critical State Combinations Observable?} } \]

4.2.17.154 Observability of State Combinations

Sometimes individual states cannot be separated, but a combination can be observed.

For example:

\[ \boxed{ x_1+x_2 } \]

may be observable while x_1 and x_2 separately are not.

This is important when interpreting covariance and filter states.

4.2.17.155 Unobservable Subspace

If the observability matrix has null space

\[ \boxed{ \mathcal N(\mathcal O), } \]

then any perturbation

\[ \delta x \in \mathcal N(\mathcal O) \]

produces no observable measurement change.

This subspace identifies exactly which state combinations are invisible.

4.2.17.156 Computing the Unobservable Directions

Using SVD:

\[ \mathcal O=U\Sigma V^T. \]

Columns of V associated with zero or near-zero singular values represent unobservable or weakly observable state combinations.

This is often more informative than rank alone.

4.2.17.157 State-Combination Interpretation

Suppose an unobservable singular vector is approximately

\[ \boxed{ v_u = \begin{bmatrix} 0.7\\ -0.7\\ 0 \end{bmatrix}. } \]

This means a change that increases one state while decreasing another in that combination produces little measurement change.

Such interpretation can guide sensor or maneuver design.

4.2.17.158 Observability and Covariance Eigenvectors

Covariance can also be decomposed:

\[ \boxed{ P = V_P\Lambda_PV_P^T. } \]

The eigenvector corresponding to the largest covariance often points toward a poorly observed state direction.

This provides a practical diagnostic.

4.2.17.159 Navigation Uncertainty Ellipsoid

For position covariance

\[ P_r, \]

the uncertainty ellipsoid satisfies approximately

\[ \boxed{ \delta r^TP_r^{-1}\delta r = c. } \]

The long axis indicates the weakest position direction.

Sensor geometry should ideally reduce this axis.

4.2.17.160 Observability and Precision Relative-Navigation Corridor

Suppose a precision relative-navigation corridor is narrow laterally.

If navigation uncertainty is largest laterally, the spacecraft may be unable to verify that it lies safely inside the corridor.

Thus observability must be aligned with mission-critical directions.

4.2.17.161 Directional Accuracy Requirement

Instead of one scalar position uncertainty requirement, define

\[ \boxed{ \sigma_x, \sigma_y, \sigma_z. } \]

This allows sensor geometry to be evaluated against directional mission requirements.

4.2.17.162 Along-Track, Cross-Track and Radial Observability

For orbital relative motion, useful axes include:

• radial,
• along-track,
• cross-track.

Sensor sensitivity may differ strongly across these axes.

Therefore covariance should often be transformed into the Hill frame for interpretation.

4.2.17.163 Hill-Frame Covariance

Let

\[ C_{HI} \]

transform inertial position error into Hill frame.

Then

\[ \boxed{ P_H = C_{HI} P_I C_{HI}^T. } \]

This reveals radial, along-track, and cross-track uncertainty.

4.2.17.164 Observability in the Hill Frame

A sensor may constrain radial motion strongly but cross-track motion weakly.

Plotting:

\[ \boxed{ \sigma_R, \sigma_T, \sigma_N } \]

can make this immediately visible.

This is highly useful in RPO analysis.

4.2.17.165 Observability and Relative Orbital Elements

RPO can also be represented with relative orbital elements.

Certain sensor combinations may observe some ROE components better than others.

Observability analysis can therefore be performed in either Cartesian or orbital-element state coordinates.

4.2.17.166 State Coordinate Transformation

Observability itself is invariant under nonsingular coordinate transformations.

However, interpretation can become much easier in a physically meaningful coordinate system.

Thus:

\[ \boxed{ \text{Same Information} } \]

may be viewed as:

\[ \boxed{ \text{Cartesian State} } \]

or

\[ \boxed{ \text{Relative Orbital Elements}. } \]

4.2.17.167 Numerical Observability Analysis

A practical analysis workflow is:

1. define state vector,
2. define dynamic model,
3. define sensor models,
4. linearize along trajectory,
5. construct observability matrix or Gramian,
6. compute rank,
7. compute singular values,
8. identify weak state directions,
9. repeat for different mission phases and sensor failures.

4.2.17.168 Observability Simulation

One practical test is to perturb one state:

\[ \boxed{ x_0 \rightarrow x_0+\delta x_i. } \]

Then propagate both trajectories and compare predicted measurements.

If the measurement histories remain nearly identical, that state direction is weakly observable.

4.2.17.169 Sensitivity Matrix

Define measurement sensitivity to initial state:

\[ \boxed{ S_z = \frac{\partial z}{\partial x_0}. } \]

Stack sensitivities over time:

\[ \boxed{ \mathcal S = \begin{bmatrix} S_{z,1}\\ S_{z,2}\\ \vdots \end{bmatrix}. } \]

This matrix provides a numerical observability measure for nonlinear trajectories.

4.2.17.170 Finite-Difference Sensitivity

If analytical Jacobians are difficult, use:

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

This can estimate local sensitivity numerically.

Care is required when selecting \epsilon.

4.2.17.171 Automatic Differentiation

For complex navigation models, automatic differentiation can generate accurate Jacobians for:

\[ F \]

and

\[ H. \]

This reduces finite-difference errors and can improve observability analysis.

4.2.17.172 Empirical Observability Gramian

For nonlinear systems, an empirical observability Gramian can be constructed by perturbing initial states and observing output differences.

This is useful when analytical nonlinear observability calculations become impractical.

4.2.17.173 Monte Carlo Observability Interpretation

Monte Carlo runs can reveal weakly observable states indirectly.

Symptoms include:

• large dispersion,
• slow convergence,
• sensitivity to initialization,
• strong state correlations.

However, Monte Carlo does not replace formal observability analysis.

4.2.17.174 Observability and NEES

If a state direction is weakly observable, actual errors may grow.

A correctly tuned covariance should grow accordingly.

NEES can help check whether the filter's uncertainty reflects this weakness.

4.2.17.175 Observability and NIS

NIS mainly tests measurement-space consistency.

A filter may have acceptable NIS while some internal states remain weakly observable.

Therefore:

\[ \boxed{ \text{Good NIS} \not\Rightarrow \text{All States Well Observable}. } \]

4.2.17.176 Innovation Cannot Reveal Every Hidden State

If a state lies in an unobservable direction, changing it may produce almost no innovation.

Thus innovation monitoring alone cannot diagnose all observability problems.

Formal state-information analysis is still necessary.

4.2.17.177 Observability and Filter Divergence

Weak observability can lead to:

• slow convergence,
• large covariance,
• sensitivity to model errors.

But divergence may also occur because of:

• incorrect tuning,
• bad linearization,
• numerical problems.

Therefore observability should be checked before attributing convergence issues solely to the filter implementation.

4.2.17.178 Observability and Covariance Collapse

A dangerous failure occurs when the filter reports small covariance in a weakly observable state.

Then:

\[ \boxed{ \text{Reported Confidence} \gg \text{Actual Information}. } \]

This can occur through incorrect tuning or inconsistent constraints.

4.2.17.179 Artificial Constraints

Sometimes engineers artificially constrain states using tiny process noise or very small prior covariance.

This can make the filter numerically stable.

But it does not create real observability.

The filter may simply become overconfident.

4.2.17.180 Pseudo-Measurements

A pseudo-measurement introduces known physical information.

For example:

\[ \boxed{ v_z\approx0 } \]

during a known hold condition.

Pseudo-measurements can improve estimation if the physical constraint is valid.

They should not be used merely to hide unobservable dynamics.

4.2.17.181 Physical Constraints Can Add Information

If a docking mechanism physically constrains certain motion, that constraint is real information.

Then the constraint can legitimately improve observability.

The key requirement is that the assumption reflects actual physics.

4.2.17.182 Observability and Target Cooperation

A cooperative target may provide:

• GNSS position,
• attitude telemetry,
• inter-satellite ranging,
• beacon signals.

This greatly improves observability compared with a fully non-cooperative target.

4.2.17.183 Non-Cooperative Target

For a non-cooperative target, navigation may rely on:

• passive camera,
• LiDAR,
• radar.

Target pose and motion may then be harder to estimate.

Sensor geometry and target features become especially important.

4.2.17.184 Unknown Target Motion

If target dynamics are also estimated,

\[ x \]

may include target angular rate or translational acceleration.

Adding these states can reduce observability unless measurements provide sufficient information.

4.2.17.185 Target Dynamic Model Helps

A known or constrained target motion model adds dynamic relationships.

For example, assuming near-constant angular velocity couples successive pose measurements.

This can make angular-rate states observable.

4.2.17.186 But Wrong Dynamic Assumptions Are Dangerous

An assumed constant target motion may artificially constrain the estimator.

If the target actually maneuvers, residuals may grow.

Therefore strong model constraints should be used only when justified.

4.2.17.187 Observability and Multi-Rate Sensors

High-rate IMU plus low-rate camera is a common architecture.

IMU provides temporal propagation.

Camera provides absolute corrections.

The combination can make states observable even though neither is sufficient alone for long-duration accuracy.

4.2.17.188 Low-Rate Absolute Sensor

Suppose camera updates every

\[ 1~\mathrm{s}. \]

Bias information may accumulate during the interval.

The next camera update corrects the resulting drift.

Thus bias observability exists, but convergence speed depends on update rate.

4.2.17.189 Observability During Long Sensor Gaps

During long gaps, the system temporarily operates with reduced measurement information.

Some states may become effectively unobservable over that interval.

Covariance should grow until measurements return.

4.2.17.190 Instantaneous vs Interval Observability

A state may be unobservable at one instant but observable over a time interval.

Therefore:

\[ \boxed{ \text{Instantaneous Measurement Rank} } \]

is not the same as

\[ \boxed{ \text{Dynamic Observability Over Time}. } \]

This distinction is crucial in spacecraft navigation.

4.2.17.191 Sensor Geometry Can Change Continuously

As spacecraft move, line-of-sight vectors, ranges, and target orientation change.

Thus:

\[ \boxed{ H=H(t). } \]

Observability can improve or deteriorate continuously throughout the mission.

4.2.17.192 Periods of Poor Geometry

A temporary loss of strong observability may be acceptable if:

• duration is short,
• process model is accurate,
• uncertainty remains within limits.

The mission should therefore consider both observability strength and duration.

4.2.17.193 Observability Recovery

After geometry improves, measurements may again constrain the previously weak states.

The covariance should then decrease.

Thus:

\[ \boxed{ \text{Poor Geometry} \rightarrow P\uparrow } \]

followed by

\[ \boxed{ \text{Improved Geometry} \rightarrow P\downarrow. } \]

4.2.17.194 Observability and Filter Initialization

At startup, some states may not yet be observable because sufficient motion has not occurred.

For example, gyro bias may require time and attitude-reference updates.

Therefore initialization strategy should distinguish:

\[ \boxed{ \text{States Known Initially} } \]

from

\[ \boxed{ \text{States That Must Converge Dynamically}. } \]

4.2.17.195 Delayed Observability

Some states become observable only after enough measurements are accumulated.

Thus initialization may involve:

\[ \boxed{ \text{Initial Guess} \rightarrow \text{Data Accumulation} \rightarrow \text{Convergence}. } \]

This leads directly to the next section on initialization and convergence.

4.2.17.196 Observability Test for Filter Design

Before implementing a large EKF or ESKF:

1. list every state,
2. identify which sensor constrains it,
3. identify whether that constraint is direct or dynamic,
4. identify required maneuvers,
5. test each sensor-loss case.

This prevents building filters with unnecessary or unobservable states.

4.2.17.197 State-to-Sensor Mapping Table

A useful engineering table may be:

Such a table provides an intuitive observability overview before formal analysis.

4.2.17.198 Sensor-Loss Matrix

Similarly, define scenarios:

This helps connect observability with FDIR.

4.2.17.199 Observability Should Be Verified, Not Assumed

A sensor suite that looks intuitively rich may still contain hidden unobservable directions.

Therefore:

\[ \boxed{ \text{Sensor Count} \neq \text{Observability Proof}. } \]

Formal and numerical analysis should be performed.

4.2.17.200 Observability Verification with Simulation

A verification campaign can:

• perturb each state,
• simulate measurements,
• run the estimator,
• examine convergence,
• inspect covariance,
• compute singular values.

This combines theoretical and practical verification.

4.2.17.201 State Perturbation Test

For each state x_i:

\[ \boxed{ x_i(0) \rightarrow x_i(0)+\Delta x_i. } \]

Run the estimator.

If the error consistently converges, this supports practical observability.

If it persists, investigate weak observability.

4.2.17.202 Multiple Initial Conditions

Repeat with different perturbation directions.

Nonlinear observability can depend on initial geometry.

Therefore one initial condition is not sufficient.

4.2.17.203 Sensor-Dropout Observability Test

Disable each sensor individually.

For each case, monitor:

\[ \boxed{ P(t), } \]

\[ \boxed{ \text{state error}, } \]

\[ \boxed{ \sigma_{\min}(\mathcal O). } \]

This connects sensor failure directly to navigation capability.

4.2.17.204 Geometry Sweep

For RPO, vary:

• range,
• lateral offset,
• approach direction,
• target orientation.

Compute observability metrics.

This identifies weak geometric configurations.

4.2.17.205 Maneuver Sweep

Test different excitation maneuvers.

Compare:

\[ \boxed{ \lambda_{\min}(W_O). } \]

The maneuver giving larger minimum eigenvalue generally provides better worst-direction observability.

4.2.17.206 Range Sweep

For camera navigation, evaluate observability at:

\[ 100~\mathrm{m}, \quad 50~\mathrm{m}, \quad 15~\mathrm{m}, \quad 5~\mathrm{m}, \quad 1~\mathrm{m}. \]

Sensor sensitivity can vary greatly with range.

4.2.17.207 Monte Carlo Observability Verification

Randomize:

• state errors,
• sensor noise,
• target orientation,
• geometry,
• biases.

Evaluate whether weakly observable states converge consistently.

This provides statistical confidence.

4.2.17.208 HIL Observability Verification

Hardware-in-the-loop tests can verify whether the actual sensors provide the information predicted by simulation.

This catches:

• calibration errors,
• real noise levels,
• timing limitations,
• sensor geometry imperfections.

4.2.17.209 Observability Diagnostic Plots

Useful plots include:

\[ \boxed{ \sigma_{\min}(\mathcal O) \text{ vs time} } \]

\[ \boxed{ \lambda_{\min}(W_O) \text{ vs range} } \]

\[ \boxed{ P_{ii} \text{ vs time} } \]

\[ \boxed{ \rho_{ij} \text{ vs time}. } \]

These show when observability weakens.

4.2.17.210 Covariance Eigenvalue Plot

Plot eigenvalues of

\[ P. \]

A rapidly growing eigenvalue indicates an increasingly uncertain state combination.

The associated eigenvector identifies the weak direction.

4.2.17.211 Observability Metric Telemetry

For advanced autonomous navigation, the system may monitor an online metric such as:

\[ \boxed{ M_O. } \]

Guidance could then respond if observability becomes poor.

This is especially useful for long autonomous RPO sequences.

4.2.17.212 Online Observability Monitoring

A sliding-window information matrix can be constructed:

\[ \boxed{ \mathcal I_W = \sum_{k=t-W}^{t} \Phi_k^TH_k^TR_k^{-1}H_k\Phi_k. } \]

Its minimum eigenvalue indicates recent observability.

4.2.17.213 Online Observability Warning

If:

\[ \boxed{ \lambda_{\min} ( \mathcal I_W ) < \lambda_{\text{warn}}, } \]

navigation may raise:

\[ \boxed{ \text{OBSERVABILITY\_DEGRADED}. } \]

This can support autonomous mission management.

4.2.17.214 Observability and Navigation Health

A more complete navigation-health model is:

\[ \boxed{ H_{\text{nav}} = f( P, \text{sensor health}, \text{state age}, M_O ). } \]

This improves on simply checking whether sensors are online.

4.2.17.215 Observability and Precision-Navigation Readiness

Precision-navigation readiness logic can include:

\[ \boxed{ M_O > M_{O,\min}. } \]

This ensures that the relative state is not only within limits but also sufficiently observable.

4.2.17.216 Why Low Covariance Alone Is Not Enough

A filter can occasionally report low covariance because of incorrect assumptions.

Therefore:

\[ \boxed{ P\text{ Small} } \]

does not by itself prove observability.

Formal or empirical observability analysis should support the covariance interpretation.

4.2.17.217 Observability and Model Fidelity

Incorrect dynamics may artificially create or destroy apparent observability.

For example, assuming a fixed target when the target actually rotates can cause incorrect state coupling.

Thus observability analysis should use realistic dynamics.

4.2.17.218 Observability and Process Constraints

Strong deterministic dynamics can make states appear observable.

But if real process uncertainty is large, those states may be only weakly constrained in practice.

Therefore theoretical observability and stochastic estimation performance should both be considered.

4.2.17.219 Observability and Sensor Selection

When comparing candidate sensors, ask not only:

\[ \boxed{ \text{Which Sensor Has Lower Noise?} } \]

but also:

\[ \boxed{ \text{Which Sensor Adds New State Information?} } \]

A moderately accurate sensor with complementary geometry may be more valuable than another extremely accurate measurement of an already well-observed direction.

4.2.17.220 Example: Camera vs Second Camera

A second camera mounted almost at the same location and orientation may mostly reduce noise.

A second camera with useful baseline can add stereo depth information.

Thus the second configuration improves both accuracy and observability.

4.2.17.221 Example: LiDAR Addition

Adding LiDAR to monocular optical navigation provides direct range.

This specifically strengthens the radial state direction.

Therefore its value is not merely lower noise but removal of a geometric ambiguity.

4.2.17.222 Observability in Sensor Procurement

Sensor specifications should therefore include not only:

• accuracy,
• bias,
• noise,
• update rate,

but also how the sensor contributes to state observability in the mission geometry.

This connects sensor selection directly to navigation architecture.

4.2.17.223 Observability and Field of View

A camera may technically make states observable only while the target remains inside the field of view.

If target geometry leaves the FOV, the measurement disappears and observability changes.

Thus sensor coverage is part of observability analysis.

4.2.17.224 Occlusion

Target features may become occluded.

This reduces the number and geometry of available features.

Therefore camera pose observability can change abruptly even if the camera hardware remains healthy.

4.2.17.225 Illumination

Poor illumination may remove usable camera features.

Therefore environmental conditions can indirectly affect observability by reducing the available measurement set.

4.2.17.226 LiDAR Visibility

LiDAR requires sufficient target surface returns.

At shallow incidence angles or poor geometry, the usable point-cloud structure may weaken.

Thus target orientation affects LiDAR pose observability.

4.2.17.227 Sensor Saturation and Observability

If a sensor saturates,

\[ z \]

no longer represents true state sensitivity.

During saturation, the effective information content may collapse.

The sensor should then be considered unavailable for observability purposes.

4.2.17.228 Observability and Measurement Gating

A sensor may be nominally available but repeatedly rejected by innovation gates.

Then its effective information contribution is approximately zero.

Thus practical observability depends on accepted measurements, not merely generated measurements.

4.2.17.229 Effective Sensor Set

Define:

\[ \boxed{ \mathcal S_{\text{effective}} = \{ \text{Healthy, Valid, Accepted Sensors} \}. } \]

Observability should be evaluated using this effective sensor set.

4.2.17.230 Fault Isolation Can Change Observability Immediately

When FDIR excludes a sensor,

\[ H \]

changes immediately.

Therefore the navigation system should reassess whether the remaining state estimate still satisfies mission requirements.

This directly connects Sections 4.2.16 and 4.2.17.

4.2.17.231 Observability-Aware FDIR

A sensor should not necessarily be isolated solely because it is suspicious if removing it would make navigation unsafe.

Instead the system may choose:

• reduced weight,
• monitor-only operation,
• mission hold,

while resolving the fault.

This is an architecture-level trade.

4.2.17.232 Degraded Navigation Decision

A degraded navigation mode should consider:

\[ \boxed{ \text{Remaining Sensors} } \]

\[ + \]

\[ \boxed{ P } \]

\[ + \]

\[ \boxed{ M_O } \]

\[ + \]

\[ \boxed{ \text{Mission Phase}. } \]

This determines whether continued operation is safe.

4.2.17.233 Observability and Safe Hold

A hold maneuver may be selected when observability deteriorates.

At hold, the spacecraft can:

• stop approach,
• wait for sensor recovery,
• perform calibration motion,
• reacquire target features.

Thus hold points serve both guidance and navigation functions.

4.2.17.234 Observability and Retreat

If critical state directions become unobservable during close-proximity RPO, retreat may improve:

• range,
• field of view,
• target visibility,
• sensor geometry.

Therefore retreat can restore observability as well as increase physical separation.

4.2.17.235 Observability and Abort

If required observability cannot be restored within safe limits, the mission may need to abort.

The trigger should be based on navigation performance and safety requirements, not merely one sensor-health flag.

4.2.17.236 Observability Requirements

A navigation requirement may state:

\[ \boxed{ \operatorname{rank}(\mathcal O) = n_{\text{req}} } \]

for required states.

A stronger practical requirement may also impose:

\[ \boxed{ \sigma_{\min}(\mathcal O) > \sigma_{\min,\text{req}}. } \]

4.2.17.237 Covariance-Based Requirement

Mission requirements may instead specify:

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

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

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

These are operational consequences of sufficient observability.

4.2.17.238 Why Both Formal and Covariance Requirements Are Useful

Rank confirms that information theoretically exists.

Covariance confirms whether the information is sufficient under realistic noise.

Therefore a robust design should consider both.

4.2.17.239 Observability Design Flow

A practical design flow is:

\[ \boxed{ \text{Mission State Requirements} } \]

\[ \downarrow \]

\[ \boxed{ \text{Required Observable States} } \]

\[ \downarrow \]

\[ \boxed{ \text{Candidate Sensors} } \]

\[ \downarrow \]

\[ \boxed{ \text{Geometry and Dynamics Analysis} } \]

\[ \downarrow \]

\[ \boxed{ \text{Observability Metrics} } \]

\[ \downarrow \]

\[ \boxed{ \text{Covariance / Monte Carlo Verification}. } \]

4.2.17.240 Observability Checklist for Each State

For each state, ask:

1. Which sensor directly measures it?
2. Which dynamics couple it into measured states?
3. Which maneuvers make it visible?
4. What happens if one sensor fails?
5. How quickly does its covariance grow?
6. Is its accuracy sufficient for the mission?

This is a practical engineering method.

4.2.17.241 Common Observability Mistakes

Typical mistakes include:

• assuming a state is observable because it is included in the filter,
• assuming more sensors automatically mean full observability,
• checking only instantaneous H,
• ignoring dynamic coupling,
• checking only matrix rank and not conditioning,
• estimating too many calibration states,
• ignoring sensor geometry,
• ignoring motion dependence,
• ignoring sensor-dropout cases,
• assuming small covariance proves observability,
• trying to fix unobservability by tuning Q or R,
• failing to analyze mission-phase transitions.

4.2.17.242 Observability in One Diagram

The complete idea can be summarized as:

\[ \boxed{ \text{State} } \]

\[ \downarrow \]

\[ \boxed{ \text{Dynamics} } \]

\[ \downarrow \]

\[ \boxed{ \text{Sensor Geometry and Motion} } \]

\[ \downarrow \]

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

\[ \downarrow \]

\[ \boxed{ \text{Observable State Information}. } \]

If a state change produces no distinguishable measurement change:

\[ \boxed{ \text{State Direction} \rightarrow \text{Unobservable}. } \]

4.2.17.243 Key Equations at a Glance

Linear System

\[ \boxed{ \dot x=Ax } \]

\[ \boxed{ y=Cx. } \]

Observability Matrix

\[ \boxed{ \mathcal O = \begin{bmatrix} C\\ CA\\ CA^2\\ \vdots\\ CA^{n-1} \end{bmatrix}. } \]

Rank Condition

\[ \boxed{ \operatorname{rank}(\mathcal O)=n. } \]

Discrete-Time Observability Matrix

\[ \boxed{ \mathcal O_d = \begin{bmatrix} H\\ HF\\ HF^2\\ \vdots\\ HF^{n-1} \end{bmatrix}. } \]

Observability Gramian

\[ \boxed{ W_O = \int \Phi^TC^TC\Phi \,dt. } \]

Nonlinear Lie Derivative

\[ \boxed{ L_fh = \frac{\partial h}{\partial x}f(x). } \]

Local Nonlinear Observability Matrix

\[ \boxed{ \mathcal O_N = \begin{bmatrix} \nabla h\\ \nabla L_fh\\ \nabla L_f^2h\\ \vdots \end{bmatrix}. } \]

Range

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

Line of Sight

\[ \boxed{ \ell = \frac{r_{rel}}{\|r_{rel}\|}. } \]

Range Rate

\[ \boxed{ \dot\rho = \ell^Tv_{rel}. } \]

Information Matrix

\[ \boxed{ \mathcal I = H^TR^{-1}H. } \]

Accumulated Information

\[ \boxed{ \mathcal I_{\text{total}} = \sum_k H_k^TR_k^{-1}H_k. } \]

State Correlation

\[ \boxed{ \rho_{ij} = \frac{ P_{ij} }{ \sqrt{P_{ii}P_{jj}} }. } \]

4.2.17.244 Key Takeaways

The most important concepts of observability in sensor fusion are:

• Observability determines whether internal states can be reconstructed from measurements and dynamics.
• A state does not need to be directly measured to be observable.
• Dynamic coupling can reveal unmeasured states.
• If different states produce identical measurement histories, their difference is unobservable.
• For linear systems, observability is tested with the observability matrix.
• Full rank means complete mathematical observability.
• Rank alone does not describe observability strength.
• Singular values and Gramian eigenvalues reveal weak directions.
• A state may be mathematically observable but practically difficult to estimate.
• Nonlinear spacecraft systems often require local trajectory-dependent observability analysis.
• Sensor geometry strongly affects observability.
• Bearing-only measurements generally provide weak instantaneous range information.
• Relative motion and parallax can make range observable.
• Range-only measurements do not instantly determine 3-D direction.
• Position histories can make velocity observable.
• One inertial vector does not generally determine complete attitude.
• Two non-collinear vector observations can determine full attitude.
• Gyro bias becomes observable through comparison with external attitude references.
• Accelerometer bias becomes observable through position or velocity references.
• Calibration states require sufficient excitation.
• Clock offset becomes observable when system motion creates measurable timing sensitivity.
• Sensor boresight and lever-arm calibration depend on changing geometry.
• Adding states to a filter can reduce overall observability.
• A more advanced filter cannot create missing information.
• Tuning Q and R cannot create observability.
• Adding complementary sensors can remove unobservable directions.
• Camera and LiDAR are complementary because they provide different geometric information.
• Sensor placement affects observability.
• Redundancy should be evaluated in terms of independent information, not merely hardware count.
• Observability may change with mission phase.
• Observability may change after sensor failure.
• Each degraded sensor configuration should be analyzed separately.
• Full rank after a fault does not guarantee good conditioning.
• The smallest singular value can provide a practical observability margin.
• Guidance can deliberately improve observability through maneuver design.
• Lateral motion can improve bearing-only range observability.
• Hold points can be chosen partly for favorable sensor geometry.
• Sensor handover should verify that the new sensor suite preserves required observability.
• Covariance growth often reveals weak or missing observability.
• Unobservable states should not become artificially overconfident.
• Prior information can constrain a state but does not make it measurement-observable.
• Not every estimator state necessarily needs full observability; mission-critical state combinations do.
• Relative navigation may remain observable even when absolute position is not.
• Online observability metrics can support autonomous navigation health monitoring.
• RPO precision-navigation readiness should consider observability together with state error, covariance, sensor health, and measurement freshness.

The central observability principle is:

\[ \boxed{ \text{The Estimator Can Only Recover State Information That Is Actually Encoded in the Measurements and Dynamics}. } \]

The most important engineering warning is:

\[ \boxed{ \text{A State Existing in the Filter Does Not Mean That the State Is Observable}. } \]

And the most important sensor-fusion principle is:

\[ \boxed{ \text{Complementary Sensors Are Valuable Not Only Because They Reduce Noise, but Because They Observe Different State Directions}. } \]

Next: 4.2.18 Filter Initialization and Convergence

Once the sensor suite provides sufficient observability, another fundamental question remains:

\[ \boxed{ \text{How Should the Estimator Be Started?} } \]

A Kalman filter, EKF, ESKF, MEKF, or UKF requires an initial state:

\[ \boxed{ \hat x_0 } \]

and an initial covariance:

\[ \boxed{ P_0. } \]

If the initial state is poor, the estimator may require significant time to converge.

If the initial covariance is unrealistically small, valid measurements may be rejected and the filter may become overconfident.

If it is excessively large, the filter may experience large transients.

Attitude initialization is particularly important because quaternion estimation and small-angle error filters assume specific error representations.

Bias states may also need special treatment because they are often only gradually observable.

Therefore the next section, 4.2.18 Filter Initialization and Convergence, should develop:

• initial-state selection,
• initial covariance,
• prior uncertainty,
• position and velocity initialization,
• quaternion and attitude initialization,
• coarse alignment,
• gyro-bias initialization,
• accelerometer-bias initialization,
• initializing calibration states,
• initialization from multiple sensors,
• covariance consistency,
• filter transients,
• convergence time,
• convergence criteria,
• covariance contraction,
• covariance collapse,
• filter overconfidence,
• poor initialization and innovation gating,
• delayed bias convergence,
• initialization after sensor reset,
• estimator reinitialization,
• sensor reacquisition,
• warm start vs cold start,
• Monte Carlo convergence testing,
• RPO phase initialization,
• and practical initialization logic for flight software.