Unscented Kalman Filter (UKF)

Sigma-point generation, the unscented transform, nonlinear state propagation, predicted mean and covariance, nonlinear measurement updates, tuning parameters, covariance weights, spacecraft nonlinear navigation, RPO applications, numerical stability, and comparison with the EKF.

4.2.10 Unscented Kalman Filter (UKF)

The Unscented Kalman Filter (UKF) is a nonlinear state-estimation method designed to handle systems in which the process dynamics, the measurement models, or both are strongly nonlinear.

Like the Extended Kalman Filter, it belongs to the broader family of Bayesian recursive estimators. However, the UKF takes a fundamentally different approach to nonlinear uncertainty propagation.

The EKF asks:

\[ \boxed{ \text{How can I linearize the nonlinear system locally?} } \]

The UKF instead asks:

\[ \boxed{ \text{How does a carefully selected set of representative state samples move through the actual nonlinear function?} } \]

These representative samples are called sigma points.

Instead of computing Jacobians such as

\[ \mathbf F = \frac{\partial f}{\partial \mathbf x} \]

and

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

the UKF constructs a deterministic collection of sigma points around the current estimated mean and covariance.

Each point is propagated through the actual nonlinear model:

\[ \boxed{ \boldsymbol\chi_i \rightarrow f(\boldsymbol\chi_i) } \]

and the propagated points are then recombined to reconstruct the predicted mean and covariance.

The central UKF philosophy is therefore

\[ \boxed{ \text{Represent the Distribution by Sigma Points} \rightarrow \text{Propagate Them Through the Nonlinear Model} \rightarrow \text{Reconstruct Mean and Covariance}. } \]

This makes the UKF particularly attractive for nonlinear spacecraft-navigation problems where Jacobian derivation is difficult, error-prone, or where first-order linearisation may not adequately represent uncertainty propagation.

4.2.10.1 Why the UKF Is Needed

The ordinary Kalman filter assumes linear dynamics:

\[ \boxed{ \mathbf x_k = \mathbf F_{k-1}\mathbf x_{k-1} + \mathbf B_{k-1}\mathbf u_{k-1} + \mathbf w_{k-1} } \]

and linear measurements:

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

For many spacecraft systems, however,

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

and

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

The functions

\[ f(\cdot) \]

and

\[ h(\cdot) \]

may contain

Therefore, nonlinear state estimation is unavoidable in practical spacecraft GNC.

4.2.10.2 Limitation of EKF Linearisation

The EKF handles nonlinear dynamics by linearizing around the current estimate.

For example,

\[ f(\mathbf x) \]

is approximated near

\[ \hat{\mathbf x} \]

as

\[ \boxed{ f(\mathbf x) \approx f(\hat{\mathbf x}) + \mathbf F ( \mathbf x-\hat{\mathbf x} ) } \]

where

\[ \mathbf F = \left. \frac{\partial f}{\partial\mathbf x} \right|_{\hat{\mathbf x}}. \]

Similarly,

\[ h(\mathbf x) \]

is approximated using

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

This method is powerful, but several problems can arise:

The UKF was developed to address many of these issues.

4.2.10.3 Core UKF Idea

Suppose the current state estimate is described by

\[ \boxed{ \mathbf x \sim \mathcal N( \hat{\mathbf x}, \mathbf P ) }. \]

Instead of linearizing a nonlinear function, the UKF constructs a set of points

\[ \boldsymbol\chi_i \]

whose weighted mean and covariance match

\[ \hat{\mathbf x} \]

and

\[ \mathbf P. \]

These points are passed through

\[ f(\cdot). \]

Thus,

\[ \boxed{ \boldsymbol\chi_i \rightarrow f(\boldsymbol\chi_i) } \]

for each sigma point.

From the transformed points, compute

\[ \boxed{ \text{Predicted Mean} } \]

and

\[ \boxed{ \text{Predicted Covariance}. } \]

This process is known as the unscented transform.

4.2.10.4 Why It Is Called “Unscented”

The name does not refer to sensors or measurement type.

The term unscented originated from the idea that nonlinear transformation of a probability distribution could be approximated without explicitly linearizing the nonlinear function.

The essential idea is:

\[ \boxed{ \text{Approximate the Distribution} \quad \text{Not the Function}. } \]

The EKF approximates the nonlinear function locally.

The UKF approximates the probability distribution with deterministic sigma points and propagates those points through the true nonlinear function.

4.2.10.5 The Unscented Transform

Consider a random variable

\[ \mathbf x \]

with mean

\[ \bar{\mathbf x} \]

and covariance

\[ \mathbf P_x. \]

Suppose

\[ \mathbf y = g(\mathbf x) \]

where

\[ g(\cdot) \]

is nonlinear.

We want

\[ E[\mathbf y] \]

and

\[ \operatorname{Cov}(\mathbf y). \]

The unscented transform performs the following steps:

\[ \boxed{ \text{Mean and Covariance} } \]

\[ \downarrow \]

\[ \boxed{ \text{Generate Sigma Points} } \]

\[ \downarrow \]

\[ \boxed{ \text{Propagate Every Sigma Point Through }g } \]

\[ \downarrow \]

\[ \boxed{ \text{Weighted Mean} } \]

\[ \downarrow \]

\[ \boxed{ \text{Weighted Covariance}. } \]

4.2.10.6 State Dimension

Let the state dimension be

\[ \boxed{ n }. \]

For example, if

\[ \mathbf x = \begin{bmatrix} x\\ y\\ z\\ v_x\\ v_y\\ v_z \end{bmatrix}, \]

then

\[ n=6. \]

A standard scaled UKF typically uses

\[ \boxed{ 2n+1 } \]

sigma points.

Thus, for a six-state system,

\[ 2(6)+1=13 \]

sigma points are generated.

4.2.10.7 Sigma-Point Set

The sigma-point set is usually written

\[ \boxed{ \boldsymbol\chi_0, \boldsymbol\chi_1, \dots, \boldsymbol\chi_{2n}. } \]

The center sigma point is

\[ \boxed{ \boldsymbol\chi_0 = \hat{\mathbf x}. } \]

The remaining points are distributed symmetrically around the mean according to the covariance.

Conceptually,

\[ \boxed{ \text{Mean} + \text{Covariance Geometry} \rightarrow \text{Sigma-Point Cloud}. } \]

4.2.10.8 Sigma-Point Scaling Parameter

Define

\[ \boxed{ \lambda = \alpha^2 ( n+\kappa ) - n. } \]

The parameters

\[ \alpha,\qquad \beta,\qquad \kappa \]

control the distribution and weighting of sigma points.

They are important UKF tuning quantities.

4.2.10.9 Sigma-Point Generation

Let

\[ \mathbf P \]

be the state covariance.

Compute a matrix square root of

\[ (n+\lambda)\mathbf P. \]

A common implementation uses Cholesky decomposition:

\[ \boxed{ \mathbf L = \operatorname{chol} \left( (n+\lambda)\mathbf P \right). } \]

Then

\[ \boxed{ \boldsymbol\chi_0 = \hat{\mathbf x} } \]

and for

\[ i=1,\dots,n, \]

\[ \boxed{ \boldsymbol\chi_i = \hat{\mathbf x} + \mathbf L_i } \]

and

\[ \boxed{ \boldsymbol\chi_{i+n} = \hat{\mathbf x} - \mathbf L_i. } \]

Here,

\[ \mathbf L_i \]

is the i-th column of the covariance square-root matrix.

4.2.10.10 Geometric Meaning of Sigma Points

The covariance defines the size and direction of the uncertainty ellipsoid.

The sigma points are placed along principal directions of this uncertainty.

Therefore,

\[ \boxed{ \text{Sigma Points} = \text{Deterministic Samples of the Current Uncertainty Geometry}. } \]

If uncertainty in one direction is large, the sigma points spread farther in that direction.

If uncertainty is small, they remain closer to the mean.

4.2.10.11 Why Sigma Points Are Not Random Monte Carlo Samples

Sigma points are deterministically chosen.

They are not generated randomly.

A Monte Carlo simulation may require

\[ 10^3, 10^4, 10^5 \]

or more random samples.

The UKF may use only

\[ \boxed{ 2n+1 } \]

carefully positioned samples.

Thus,

\[ \boxed{ \text{UKF Sigma Points} \neq \text{Monte Carlo Samples}. } \]

The sigma points are specifically constructed so that their weighted statistics reproduce the current mean and covariance.

4.2.10.12 Mean Weights

The standard scaled UKF uses mean weights

\[ \boxed{ W_0^{(m)} = \frac{\lambda} {n+\lambda} } \]

and for

\[ i=1,\dots,2n, \]

\[ \boxed{ W_i^{(m)} = \frac{1} {2(n+\lambda)}. } \]

The superscript

\[ (m) \]

indicates weights used for calculating the mean.

4.2.10.13 Covariance Weights

The first covariance weight is

\[ \boxed{ W_0^{(c)} = \frac{\lambda}{n+\lambda} + ( 1-\alpha^2+\beta ) } \]

and

\[ \boxed{ W_i^{(c)} = \frac{1}{2(n+\lambda)}, \qquad i=1,\dots,2n. } \]

The covariance weights are therefore slightly different from the mean weights.

4.2.10.14 The Parameter \alpha

The parameter

\[ \boxed{ \alpha } \]

controls how far the sigma points are spread from the mean.

Small

\[ \alpha \]

places sigma points close to the mean.

Larger

\[ \alpha \]

spreads them farther away.

Typical choices depend on implementation and problem scaling.

There is no universal value that is optimal for every spacecraft application.

4.2.10.15 The Parameter \beta

The parameter

\[ \boxed{ \beta } \]

incorporates information about the prior distribution.

For a Gaussian prior, a commonly used choice is

\[ \boxed{ \beta=2. } \]

This often improves covariance approximation for Gaussian state uncertainty.

4.2.10.16 The Parameter \kappa

The parameter

\[ \boxed{ \kappa } \]

provides another degree of freedom for sigma-point scaling.

Common choices include

\[ \kappa=0 \]

or

\[ \kappa=3-n. \]

Its effect must be considered together with

\[ \alpha \]

because both affect

\[ \lambda. \]

4.2.10.17 Why Tuning Parameters Matter

Poor choices of

\[ \alpha, \beta, \kappa \]

can produce

Therefore UKF tuning is not simply cosmetic.

It influences how the estimator represents nonlinear uncertainty.

4.2.10.18 State Prediction

Suppose the nonlinear dynamics are

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

Each sigma point is propagated independently:

\[ \boxed{ \boldsymbol\chi_{i,k}^{-} = f( \boldsymbol\chi_{i,k-1}^{+}, \mathbf u_{k-1} ). } \]

This is the central nonlinear prediction step.

No dynamic Jacobian is required.

4.2.10.19 Predicted State Mean

Once all sigma points have been propagated,

\[ \boldsymbol\chi_{i,k}^{-}, \]

the predicted state mean is

\[ \boxed{ \hat{\mathbf x}_k^{-} = \sum_{i=0}^{2n} W_i^{(m)} \boldsymbol\chi_{i,k}^{-}. } \]

Thus the predicted state is reconstructed from the transformed sigma-point cloud.

4.2.10.20 Predicted State Covariance

The predicted covariance is

\[ \boxed{ \mathbf P_k^{-} = \sum_{i=0}^{2n} W_i^{(c)} \left( \boldsymbol\chi_{i,k}^{-} - \hat{\mathbf x}_k^{-} \right) \left( \boldsymbol\chi_{i,k}^{-} - \hat{\mathbf x}_k^{-} \right)^T + \mathbf Q_k. } \]

This captures how nonlinear propagation stretches, rotates, or distorts the uncertainty distribution.

4.2.10.21 Physical Interpretation of Prediction

Consider an uncertainty ellipse around the current state.

If the dynamics are nonlinear, the ellipse may transform into a curved or distorted shape.

The EKF approximates this transformation using a tangent linear mapping.

The UKF instead propagates representative points on the uncertainty distribution.

Therefore,

\[ \boxed{ \text{UKF Prediction} = \text{Propagate the Distribution Geometry Through the Nonlinear Dynamics}. } \]

4.2.10.22 Process Noise Q

The covariance

\[ \boxed{ \mathbf Q } \]

represents uncertainty in the process model.

Examples include

As in other Kalman filters,

\[ Q \]

must have physically meaningful units and dimensions.

4.2.10.23 Additive Process Noise UKF

A common simplified formulation assumes process noise is additive:

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

Then one can propagate sigma points through

\[ f(\cdot) \]

and add

\[ Q \]

after reconstructing covariance.

This is computationally simpler.

4.2.10.24 Augmented-State UKF

For more general noise models, process and measurement noise may be included in an augmented state:

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

The corresponding covariance is

\[ \boxed{ \mathbf P_a = \begin{bmatrix} \mathbf P & 0 & 0\\ 0 & \mathbf Q & 0\\ 0 & 0 & \mathbf R \end{bmatrix}. } \]

Sigma points are then generated from the augmented distribution.

This allows nonlinear noise effects to be propagated directly.

4.2.10.25 Cost of State Augmentation

If the physical state dimension is

\[ n_x \]

and the augmented noise dimension is

\[ n_w+n_v, \]

then

\[ n_a = n_x+n_w+n_v. \]

The number of sigma points becomes

\[ \boxed{ 2n_a+1. } \]

Therefore augmentation increases computational cost.

4.2.10.26 Measurement Prediction

Suppose the nonlinear measurement model is

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

Pass every predicted sigma point through the measurement function:

\[ \boxed{ \boldsymbol\zeta_{i,k} = h( \boldsymbol\chi_{i,k}^{-} ). } \]

These transformed points exist in measurement space.

4.2.10.27 Predicted Measurement Mean

The predicted measurement is

\[ \boxed{ \hat{\mathbf z}_k = \sum_{i=0}^{2n} W_i^{(m)} \boldsymbol\zeta_{i,k}. } \]

Thus the UKF predicts not just the state but also the complete expected measurement through nonlinear transformation.

4.2.10.28 Innovation Covariance

The innovation covariance is

\[ \boxed{ \mathbf S_k = \sum_{i=0}^{2n} W_i^{(c)} \left( \boldsymbol\zeta_{i,k} - \hat{\mathbf z}_k \right) \left( \boldsymbol\zeta_{i,k} - \hat{\mathbf z}_k \right)^T + \mathbf R_k. } \]

This represents the predicted spread of measurement residuals.

4.2.10.29 State-Measurement Cross Covariance

The UKF also calculates

\[ \boxed{ \mathbf P_{xz,k} = \sum_{i=0}^{2n} W_i^{(c)} \left( \boldsymbol\chi_{i,k}^{-} - \hat{\mathbf x}_k^{-} \right) \left( \boldsymbol\zeta_{i,k} - \hat{\mathbf z}_k \right)^T. } \]

This cross covariance replaces the role played by

\[ P^-H^T \]

in an EKF.

4.2.10.30 Kalman Gain

The UKF Kalman gain is

\[ \boxed{ \mathbf K_k = \mathbf P_{xz,k} \mathbf S_k^{-1}. } \]

Thus the gain still answers the familiar question:

\[ \boxed{ \text{How strongly should the measurement change the predicted state?} } \]

4.2.10.31 Innovation

The innovation remains

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

The physical interpretation is unchanged from KF and EKF.

It is the difference between

\[ \boxed{ \text{What the Sensor Measured} } \]

and

\[ \boxed{ \text{What the Current State Estimate Predicted the Sensor Should Measure}. } \]

4.2.10.32 State Update

The posterior state estimate is

\[ \boxed{ \hat{\mathbf x}_k^{+} = \hat{\mathbf x}_k^{-} + \mathbf K_k \mathbf y_k. } \]

This resembles the classical Kalman update.

The major difference lies in how

\[ \hat x^-, P^-, \hat z, S, P_{xz} \]

were obtained.

4.2.10.33 Covariance Update

The posterior covariance is

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

This reduces state uncertainty according to the information supplied by the measurement.

4.2.10.34 Complete UKF Prediction Cycle

The prediction stage can be summarized as follows.

4.2.10.34.1 Start with Prior Mean and Covariance

\[ \boxed{ \hat x_{k-1}^{+}, \qquad P_{k-1}^{+}. } \]

4.2.10.34.2 Generate Sigma Points

\[ \boxed{ \chi_{i,k-1}^{+}. } \]

4.2.10.34.3 Propagate Through Nonlinear Dynamics

\[ \boxed{ \chi_{i,k}^{-} = f( \chi_{i,k-1}^{+} ). } \]

4.2.10.34.4 Reconstruct Predicted Mean

\[ \boxed{ \hat x_k^{-} = \sum W_i^{(m)} \chi_{i,k}^{-}. } \]

4.2.10.34.5 Reconstruct Predicted Covariance

\[ \boxed{ P_k^{-} = \sum W_i^{(c)} ( \chi_i^{-}-\hat x^{-} ) ( \chi_i^{-}-\hat x^{-} )^T + Q. } \]

4.2.10.35 Complete UKF Measurement Cycle

4.2.10.35.1 Transform Sigma Points into Measurement Space

\[ \boxed{ \zeta_i = h( \chi_i^{-} ). } \]

4.2.10.35.2 Predicted Measurement

\[ \boxed{ \hat z = \sum W_i^{(m)} \zeta_i. } \]

4.2.10.35.3 Innovation Covariance

\[ \boxed{ S = \sum W_i^{(c)} ( \zeta_i-\hat z ) ( \zeta_i-\hat z )^T + R. } \]

4.2.10.35.4 Cross Covariance

\[ \boxed{ P_{xz} = \sum W_i^{(c)} ( \chi_i^{-}-\hat x^{-} ) ( \zeta_i-\hat z )^T. } \]

4.2.10.35.5 Kalman Gain

\[ \boxed{ K = P_{xz}S^{-1}. } \]

4.2.10.35.6 State Correction

\[ \boxed{ \hat x^+ = \hat x^- + K( z-\hat z ). } \]

4.2.10.35.7 Covariance Correction

\[ \boxed{ P^+ = P^- - KSK^T. } \]

4.2.10.36 UKF vs EKF

The fundamental difference is:

EKF

\[ \boxed{ \text{Nonlinear Function} \rightarrow \text{Jacobian} \rightarrow \text{Linear Approximation}. } \]

UKF

\[ \boxed{ \text{Sigma Points} \rightarrow \text{Actual Nonlinear Function} \rightarrow \text{Reconstructed Distribution}. } \]

A comparison is:

Feature EKF UKF
Nonlinear model Yes Yes
Jacobians required Yes No
Sigma points required No Yes
Computational cost Lower Usually higher
Strong nonlinearities Can be sensitive Often better
Implementation complexity Jacobian-heavy Sigma-point-heavy
Covariance propagation Linearized Statistical transform
Model evaluation per step Usually once 2n+1 or more

4.2.10.37 Why the UKF Can Be More Accurate

Suppose

\[ y=f(x) \]

is nonlinear.

The EKF propagates uncertainty using a first-order local derivative.

The UKF propagates multiple representative points through the actual nonlinear function.

Therefore the UKF can capture effects that a tangent approximation may miss.

Conceptually,

\[ \boxed{ \text{EKF} \rightarrow \text{Approximate Function} } \]

while

\[ \boxed{ \text{UKF} \rightarrow \text{Approximate Distribution}. } \]

For many nonlinear transformations, the unscented transform reproduces mean and covariance to higher-order accuracy than first-order linearization under suitable assumptions.

4.2.10.38 But UKF Is Not Automatically Better

The UKF still relies on assumptions.

It commonly approximates uncertainty using only mean and covariance.

If the true posterior is

the UKF can still be inaccurate.

Therefore,

\[ \boxed{ \text{No Jacobian} \neq \text{No Approximation}. } \]

The UKF approximates the underlying probability distribution using a finite deterministic point set.

4.2.10.39 Computational Cost

For a state dimension

\[ n, \]

the standard UKF uses

\[ 2n+1 \]

sigma points.

If the nonlinear model is expensive, each propagation

\[ f(\chi_i) \]

may be computationally demanding.

Thus

\[ \boxed{ \text{UKF Accuracy Benefit} \leftrightarrow \text{Higher Computation}. } \]

This is important for onboard spacecraft processors.

4.2.10.40 Example: Nonlinear Range Measurement

Suppose

\[ \mathbf x = \begin{bmatrix} x\\ y\\ z \end{bmatrix} \]

and a range sensor measures

\[ \boxed{ \rho = \sqrt{x^2+y^2+z^2}. } \]

The measurement is nonlinear.

In an EKF, derive

\[ H_\rho = \begin{bmatrix} x/\rho & y/\rho & z/\rho \end{bmatrix}. \]

In a UKF, no analytical derivative is required.

Each sigma point

\[ \chi_i = \begin{bmatrix} x_i\\ y_i\\ z_i \end{bmatrix} \]

is transformed:

\[ \boxed{ \rho_i = \sqrt{ x_i^2+y_i^2+z_i^2 }. } \]

The predicted range mean and variance are reconstructed from

\[ \rho_i. \]

4.2.10.41 Bearing Measurement Example

A camera may measure azimuth

\[ \boxed{ \alpha = \tan^{-1} \left( \frac{y}{x} \right). } \]

Instead of deriving

\[ \frac{\partial \alpha}{\partial x}, \qquad \frac{\partial \alpha}{\partial y}, \]

the UKF calculates

\[ \boxed{ \alpha_i = \tan^{-1} \left( \frac{y_i}{x_i} \right) } \]

for every sigma point.

This is particularly attractive for complex optical measurement models.

4.2.10.42 Angle-Wrapping Problem

Angles create a special problem.

Suppose sigma-point angles are

\[ 179^\circ \]

and

\[ -179^\circ. \]

A naive arithmetic mean gives approximately

\[ 0^\circ, \]

which is physically wrong.

The correct mean is near

\[ 180^\circ. \]

Therefore angular means must be calculated using circular statistics.

For example,

\[ \boxed{ \bar\theta = \operatorname{atan2} \left( \sum_i W_i\sin\theta_i, \sum_i W_i\cos\theta_i \right). } \]

4.2.10.43 Quaternion States Require Special Treatment

Quaternions are even more challenging than ordinary angles.

A naive arithmetic mean of quaternion sigma points may not preserve

\[ \|q\|=1. \]

Also,

\[ q \equiv -q. \]

Therefore a UKF for attitude estimation should not simply treat quaternion components like ordinary Euclidean variables without care.

Possible approaches include

4.2.10.44 Error-State UKF for Attitude

One robust attitude approach is conceptually similar to the MEKF.

Maintain a nominal quaternion

\[ \hat q \]

and represent uncertainty using

\[ \delta\theta. \]

Sigma points are generated in the local rotational error state.

Each error sigma point is mapped to an error quaternion:

\[ \delta q_i. \]

Then

\[ \boxed{ q_i = \delta q_i \otimes \hat q. } \]

These quaternion sigma points can be propagated through nonlinear attitude dynamics.

This preserves the geometric structure more naturally.

4.2.10.45 UKF for Spacecraft Orbital Navigation

A spacecraft navigation state may be

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

Nonlinear dynamics may include

Each sigma point represents a slightly different possible orbital state.

Propagating all sigma points through the nonlinear orbital model captures how uncertainty evolves under the orbital dynamics.

4.2.10.46 UKF for RPO

A relative-navigation state might contain

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

or include attitude and biases:

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

Sensors may include

The UKF can naturally propagate sigma points through both nonlinear relative dynamics and nonlinear sensor models.

4.2.10.47 Camera-Based Relative Navigation

A camera measurement may depend on relative position through

\[ \boxed{ u = f_x \frac{x}{z} + c_x } \]

and

\[ \boxed{ v = f_y \frac{y}{z} + c_y. } \]

These are strongly nonlinear when range changes significantly.

The UKF simply transforms each sigma point:

\[ \boxed{ (x_i,y_i,z_i) \rightarrow (u_i,v_i). } \]

No analytical camera Jacobian is required.

4.2.10.48 LiDAR-Based RPO

LiDAR can provide

For pure range,

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

For relative pose, the measurement function may be much more complicated.

The UKF is useful because

\[ \boxed{ \text{Complex Nonlinear Measurement} \rightarrow \text{Direct Sigma-Point Transformation}. } \]

4.2.10.49 Multi-Sensor UKF

Different sensors may update the same UKF.

For example:

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

Each sensor has its own measurement function

\[ h_j(x) \]

and measurement covariance

\[ R_j. \]

When sensor j produces a measurement, sigma points are mapped through its corresponding

\[ h_j. \]

4.2.10.50 Multi-Rate Sensors

Different sensors may operate at different rates.

For example,

\[ f_{IMU} \gg f_{camera} \]

and

\[ f_{camera} \neq f_{GNSS}. \]

The UKF can

\[ \boxed{ \text{Predict at High Rate} } \]

and

\[ \boxed{ \text{Update Whenever a Measurement Arrives}. } \]

This is similar to EKF and ESKF operation.

4.2.10.51 Sensor Dropout

If a measurement sensor becomes unavailable,

\[ \boxed{ \text{Prediction Continues} } \]

but no correction occurs.

Covariance generally grows.

When measurements return,

\[ \boxed{ \text{Measurement Update} \rightarrow \text{Reduced State Uncertainty}. } \]

The UKF therefore naturally handles temporary sensor dropout, provided the model remains sufficiently valid.

4.2.10.52 Innovation Gating

The innovation is

\[ y=z-\hat z. \]

The innovation covariance is

\[ S. \]

A normalized statistic is

\[ \boxed{ NIS = y^TS^{-1}y. } \]

If the value is too large, the measurement may be inconsistent with the predicted state.

Possible causes include

4.2.10.53 Measurement Timing

As with all navigation filters, timestamp consistency is critical.

A sigma-point cloud propagated to

\[ t_k \]

must be compared with a measurement belonging to

\[ t_k. \]

Otherwise,

\[ \boxed{ \text{Correct Measurement} + \text{Wrong State Time} = \text{Incorrect Innovation}. } \]

This becomes particularly important during high-rate motion or spacecraft slews.

4.2.10.54 Numerical Stability

The UKF repeatedly performs matrix square-root operations.

Therefore

\[ P \]

must remain symmetric and positive semidefinite.

In practice, numerical errors may cause

\[ P\neq P^T. \]

A common numerical cleanup is

\[ \boxed{ P \leftarrow \frac12 ( P+P^T ). } \]

Small positive diagonal regularization may also be used when justified.

4.2.10.55 Cholesky Failure

Sigma-point generation often requires

\[ \operatorname{chol}(P). \]

Cholesky decomposition fails if

\[ P \]

is not positive definite.

This may indicate

A Cholesky failure should not simply be hidden.

It is often evidence of an estimator problem.

4.2.10.56 Square-Root UKF

A Square-Root UKF propagates a square-root covariance factor directly.

Instead of storing

\[ P, \]

maintain

\[ S \]

such that

\[ \boxed{ P = SS^T. } \]

Advantages can include

This can be attractive for flight-software implementations.

4.2.10.57 State Scaling

Suppose a state contains quantities with very different magnitudes:

\[ r\sim10^6\text{ m}, \]

\[ v\sim10^3\text{ m/s}, \]

\[ b_g\sim10^{-5}\text{ rad/s}. \]

Poor scaling can create numerical conditioning problems.

Therefore state normalization or careful covariance scaling may be important.

4.2.10.58 Process Noise Tuning

If

\[ Q \]

is too small,

the UKF trusts the model too strongly.

Consequences include

If

\[ Q \]

is too large,

the state prediction becomes overly uncertain.

Consequences include

4.2.10.59 Measurement Noise Tuning

If

\[ R \]

is too small,

the filter trusts measurements excessively.

If

\[ R \]

is too large,

the filter underuses useful measurement information.

Thus

\[ \boxed{ Q/R \text{ tuning still matters in the UKF exactly as it does in the EKF}. } \]

The absence of Jacobians does not remove the need for realistic stochastic modeling.

4.2.10.60 Initialization

The UKF requires

\[ \boxed{ \hat x_0 } \]

and

\[ \boxed{ P_0. } \]

These define the first sigma-point cloud.

If

\[ P_0 \]

is too small, sigma points are too tightly concentrated.

If

\[ P_0 \]

is unrealistically large, sigma points may enter physically invalid regions.

Therefore initialization directly affects the geometry of sigma-point propagation.

4.2.10.61 Physical Constraints

Some states have constraints.

Examples include

Sigma points can potentially move into physically impossible regions.

For constrained problems, special strategies may be required.

4.2.10.62 UKF and Non-Gaussian Uncertainty

Although the sigma points can capture nonlinear transformation better than first-order linearization, the standard UKF still ultimately represents the state primarily through

\[ \boxed{ \text{Mean} + \text{Covariance}. } \]

Therefore it does not fully represent arbitrary multimodal distributions.

For strongly multimodal problems, particle filters or Gaussian-mixture methods may be more appropriate.

4.2.10.63 UKF vs Particle Filter

A particle filter may use

\[ N_p \]

random weighted particles.

The UKF uses a deterministic set of

\[ 2n+1 \]

sigma points.

Therefore:

\[ \boxed{ \text{UKF} \rightarrow \text{Efficient Gaussian Approximation} } \]

while

\[ \boxed{ \text{Particle Filter} \rightarrow \text{General Distribution Representation}. } \]

Particle filters are generally more flexible but much more computationally expensive.

4.2.10.64 UKF vs ESKF

The ESKF and UKF address different aspects of nonlinear estimation.

The ESKF focuses on

\[ \boxed{ \text{Nominal State} + \text{Small Local Error State}. } \]

The UKF focuses on

\[ \boxed{ \text{Nonlinear Uncertainty Propagation Without Jacobians}. } \]

These concepts are not mutually exclusive.

One can construct an error-state UKF.

4.2.10.65 UKF vs MEKF

The MEKF specifically handles quaternion attitude geometry using multiplicative attitude errors.

A standard UKF does not automatically solve quaternion-manifold issues.

Therefore for attitude estimation one may use

\[ \boxed{ \text{Multiplicative UKF} } \]

or

\[ \boxed{ \text{Quaternion Error-State UKF}. } \]

The UKF handles nonlinear uncertainty propagation, while the multiplicative formulation handles attitude geometry.

4.2.10.66 When UKF Is Attractive

The UKF is particularly attractive when

4.2.10.67 When EKF May Still Be Better

The EKF may remain preferable when

Therefore,

\[ \boxed{ \text{UKF} \neq \text{Automatic Replacement for EKF}. } \]

Estimator selection should depend on mission requirements.

4.2.10.68 Computational Scaling

With

\[ n \]

states, the standard UKF evaluates the process model approximately

\[ 2n+1 \]

times per prediction.

For

\[ n=20, \]

this becomes

\[ 41 \]

model evaluations per update.

If each propagation includes expensive orbital dynamics, the computational load can become significant.

4.2.10.69 Parallelism

Sigma-point propagation is naturally parallelizable because each

\[ \chi_i \]

can be propagated independently.

Thus

\[ \boxed{ \chi_0,\chi_1,\dots,\chi_{2n} } \]

may be evaluated simultaneously on suitable hardware.

This can partly offset UKF computational cost.

4.2.10.70 Simulink / Software Architecture

A clean UKF software architecture can contain

\[ \boxed{\text{Current Mean + Covariance}} \]

\[ \downarrow \]

\[ \boxed{\text{Sigma-Point Generator}} \]

\[ \downarrow \]

\[ \boxed{\text{Nonlinear Dynamics for Each Sigma Point}} \]

\[ \downarrow \]

\[ \boxed{\text{Predicted Mean + Covariance}} \]

then

\[ \boxed{\text{Measurement Model for Each Sigma Point}} \]

\[ \downarrow \]

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

\[ \downarrow \]

\[ \boxed{\text{Innovation Covariance + Cross Covariance}} \]

\[ \downarrow \]

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

\[ \downarrow \]

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

4.2.10.71 Recommended Software Blocks

A modular implementation may separate

This makes verification much easier.

4.2.10.72 Verification Strategy

The UKF should be verified incrementally.

4.2.10.72.1 Identity Transform Test

Set

\[ f(x)=x. \]

Verify that reconstructed mean and covariance equal the original values.

4.2.10.72.2 Linear-System Test

Use a linear system.

The UKF should reproduce standard Kalman-filter behavior.

4.2.10.72.3 Scalar Nonlinearity Test

Use a simple nonlinear function such as

\[ y=x^2. \]

Compare unscented transformation with Monte Carlo simulation.

4.2.10.72.4 Zero-Noise Test

Remove process and measurement noise.

Verify convergence.

4.2.10.72.5 Initial Error Test

Introduce known state offsets.

4.2.10.72.6 Measurement Dropout Test

Disable measurements and observe covariance growth.

4.2.10.72.7 Outlier Test

Inject false measurements and verify gating.

4.2.10.72.8 Strong-Nonlinearity Test

Compare EKF and UKF under increasingly nonlinear conditions.

4.2.10.72.9 Parameter-Sensitivity Test

Vary

\[ \alpha,\beta,\kappa. \]

4.2.10.72.10 Numerical-Stability Test

Monitor positive definiteness of

\[ P. \]

4.2.10.73 Comparison with Monte Carlo Propagation

A useful verification approach is to generate many random states from

\[ \mathcal N( \hat x,P ) \]

and propagate them through

\[ f(x). \]

Then compare Monte Carlo statistics against the UKF predicted

\[ \hat x^- \]

and

\[ P^-. \]

Conceptually,

\[ \boxed{ \text{Monte Carlo Distribution} \leftrightarrow \text{Unscented Approximation}. } \]

This provides a direct test of the unscented transform.

4.2.10.74 NIS

The normalized innovation squared remains

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

It is useful for

UKF consistency should not be judged only by state RMSE.

4.2.10.75 NEES

When truth is available,

\[ e_k = x_{\text{true},k} - \hat x_k \]

or the appropriate manifold error can be formed.

Then

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

NEES compares actual state error against predicted estimator uncertainty.

4.2.10.76 Monte Carlo Validation

Randomize

Evaluate

\[ \boxed{RMSE} \]

\[ \boxed{NIS} \]

\[ \boxed{NEES} \]

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

\[ \boxed{\text{Maximum Error}} \]

\[ \boxed{\text{Divergence Rate}} \]

\[ \boxed{\text{Computation Time}}. \]

4.2.10.77 UKF for RPO Mission Phases

Far Range

Relative GNSS may dominate:

\[ \boxed{ \text{Relative GNSS} \rightarrow \text{UKF Update}. } \]

Mid Range

Fuse

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

Close Range

Fuse

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

Capture

Use high-precision

\[ \boxed{ \text{Relative Position} + \text{Velocity} + \text{Attitude} + \text{Angular Rate}. } \]

The UKF can maintain continuity as the measurement functions change across phases.

4.2.10.78 Sensor Handover

Suppose the navigation chain changes from

\[ \text{GNSS} \]

to

\[ \text{Camera} \]

to

\[ \text{LiDAR}. \]

The posterior state and covariance from the previous measurement regime become the prior for the next.

Thus,

\[ \boxed{ \text{Posterior} \rightarrow \text{Next Prior}. } \]

The filter should not normally be reset simply because the active sensor changes.

4.2.10.79 Capture Readiness

The UKF supplies both

\[ \hat x \]

and

\[ P. \]

Therefore capture criteria may consider

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

and

\[ \|\hat v_{rel}\| \]

together with uncertainty.

Conceptually,

\[ \boxed{ \text{Capture Readiness} = \text{Estimated Relative State} + \text{State Confidence}. } \]

4.2.10.80 Relationship to Guidance and Control

The UKF feeds estimated state to guidance and control:

\[ \boxed{ \text{Sensors} \rightarrow \text{UKF} \rightarrow \hat x \rightarrow \text{Guidance} \rightarrow \text{Control}. } \]

Therefore UKF characteristics influence

Estimator performance must ultimately be evaluated in the closed GNC loop.

4.2.10.81 Common UKF Implementation Errors

Typical mistakes include:

4.2.10.82 Key Equations at a Glance

Scaling

\[ \boxed{ \lambda = \alpha^2(n+\kappa)-n. } \]

Center Sigma Point

\[ \boxed{ \chi_0=\hat x. } \]

Sigma Points

\[ \boxed{ \chi_i = \hat x + \left[ \sqrt{(n+\lambda)P} \right]_i } \]

\[ \boxed{ \chi_{i+n} = \hat x - \left[ \sqrt{(n+\lambda)P} \right]_i. } \]

Mean Weights

\[ \boxed{ W_0^{(m)} = \frac{\lambda}{n+\lambda} } \]

\[ \boxed{ W_i^{(m)} = \frac{1}{2(n+\lambda)}. } \]

Covariance Weight

\[ \boxed{ W_0^{(c)} = \frac{\lambda}{n+\lambda} + 1-\alpha^2+\beta. } \]

Sigma-Point Propagation

\[ \boxed{ \chi_i^- = f(\chi_i^+). } \]

Predicted Mean

\[ \boxed{ \hat x^- = \sum_i W_i^{(m)} \chi_i^-. } \]

Predicted Covariance

\[ \boxed{ P^- = \sum_i W_i^{(c)} ( \chi_i^--\hat x^- ) ( \chi_i^--\hat x^- )^T + Q. } \]

Measurement Sigma Points

\[ \boxed{ \zeta_i = h(\chi_i^-). } \]

Predicted Measurement

\[ \boxed{ \hat z = \sum_i W_i^{(m)} \zeta_i. } \]

Innovation Covariance

\[ \boxed{ S = \sum_i W_i^{(c)} ( \zeta_i-\hat z ) ( \zeta_i-\hat z )^T + R. } \]

Cross Covariance

\[ \boxed{ P_{xz} = \sum_i W_i^{(c)} ( \chi_i^--\hat x^- ) ( \zeta_i-\hat z )^T. } \]

Kalman Gain

\[ \boxed{ K = P_{xz}S^{-1}. } \]

State Update

\[ \boxed{ \hat x^+ = \hat x^- + K( z-\hat z ). } \]

Covariance Update

\[ \boxed{ P^+ = P^- - KSK^T. } \]

4.2.10.83 UKF in One Diagram

The complete UKF concept can be summarized as

\[ \boxed{ \text{Prior Mean + Covariance} } \]

\[ \downarrow \]

\[ \boxed{ \text{Generate Sigma Points} } \]

\[ \downarrow \]

\[ \boxed{ \text{Propagate Through Nonlinear Dynamics} } \]

\[ \downarrow \]

\[ \boxed{ \text{Reconstruct Predicted Mean + Covariance} } \]

then

\[ \boxed{ \text{Transform Sigma Points Through Measurement Model} } \]

\[ \downarrow \]

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

\[ \downarrow \]

\[ \boxed{ \text{Innovation + Innovation Covariance} } \]

\[ \downarrow \]

\[ \boxed{ \text{Cross Covariance} } \]

\[ \downarrow \]

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

\[ \downarrow \]

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

The central philosophy is therefore

\[ \boxed{ \text{Do Not Linearize the Nonlinear Function; Propagate a Carefully Chosen Representation of the Uncertainty Through It}. } \]

4.2.10.84 Key Takeaways

The Unscented Kalman Filter should be understood through the following points:

\[ 2n+1 \]

sigma points.

\[ P_{xz}S^{-1}. \]

\[ \alpha,\beta,\kappa \]

control sigma-point scaling and weights.

The most important concept is

\[ \boxed{ \text{UKF} = \text{Sigma Points} + \text{Nonlinear Propagation} + \text{Statistical Reconstruction of Mean and Covariance}. } \]

Or physically:

\[ \boxed{ \text{Instead of asking how the nonlinear function behaves infinitesimally at one point, ask where representative possibilities around the current estimate actually move.} } \]

Next: 4.2.11 Information and Federated Filters

The Kalman-family filters developed so far have largely represented uncertainty using the covariance form

\[ \boxed{ \hat x,\qquad P. } \]

An equivalent representation can instead use the information matrix

\[ \boxed{ Y=P^{-1} } \]

and information vector

\[ \boxed{ y=Y\hat x. } \]

This representation can be particularly useful when independent information from several sensors or local estimators must be combined.

For independent information sources,

\[ \boxed{ Y_{\text{combined}} = Y_1+Y_2+\cdots } \]

under the appropriate assumptions.

This creates a natural bridge toward distributed and federated navigation architectures.

The next section should therefore develop 4.2.11 Information and Federated Filters, including:

4.2.11 Information and Federated Filters

Information-form Kalman filtering, information vectors and information matrices, decentralized estimation, local and master filters, federated architectures, information sharing, fault isolation, redundancy, covariance intersection, and distributed spacecraft navigation.

Continue to Information and Federated Filters →