ADCS.estimators.attitude_estimators.attitude_AugmentedUKF module

Unscented attitude estimator with augmented parameter blocks.

1. Construct the augmented tangent-state estimate

Validate the physical, bias, and disturbance-parameter blocks and retain the right-error tangent covariance.

\[\hat{\mathbf{x}}_a = [\hat{\boldsymbol{\omega}},\hat{\mathbf{q}},\hat{\mathbf{h}}, \hat{\mathbf{b}}_a,\hat{\mathbf{b}}_s,\hat{\mathbf{d}}]^T, \qquad \delta\mathbf{x}_a = [\delta\boldsymbol{\omega},\delta\boldsymbol{\theta},\delta\mathbf{h}, \delta\mathbf{b}_a,\delta\mathbf{b}_s,\delta\mathbf{d}]^T \in\mathbb{R}^{n_a}, \qquad n_a=6+n_h+n_{b_a}+n_{b_s}+n_d.\]

The covariance is in tangent coordinates; the nominal quaternion stays on \(\mathbb{S}^{3}\).

2. Generate and propagate augmented sigma points

Each sigma point carries its own parameter realization; those values are synchronized into EstimatedSatellite before nonlinear propagation.

\[ \begin{align}\begin{aligned}\lambda=\alpha^2(n_a+\kappa)-n_a,\qquad \gamma=\sqrt{n_a+\lambda},\qquad \mathbf{P}_a=\mathbf{S}^{T}\mathbf{S},\\\boldsymbol{\xi}_0=\mathbf{0},\qquad \boldsymbol{\xi}_i=\pm\gamma\mathbf{S}_{i,:}^{T},\qquad \mathbf{X}_{i,k}^{+}=\hat{\mathbf{x}}_{a,k}^{+} \boxplus\boldsymbol{\xi}_i.\end{aligned}\end{align} \]

The standard unscented mean and covariance weights \(W_i^{(m)}\) and \(W_i^{(c)}\) are computed with this augmented tangent dimension.

3. Form the unscented process statistics

Compute the weighted manifold mean and sigma deviations, then discretize and add the augmented process noise.

\[ \begin{align}\begin{aligned}\mathbf{X}_{i,k+1}^{-}= f_a(\mathbf{X}_{i,k}^{+},\mathbf{u}_k,\mathbf{o}_k,\mathbf{o}_{k+1},\Delta t), \qquad \sum_iW_i^{(m)} (\mathbf{X}_{i,k+1}^{-}\boxminus\hat{\mathbf{x}}_{a,k+1}^{-})=\mathbf{0}.\\\mathbf{d}_{i,k}^{x}=\mathbf{X}_{i,k+1}^{-} \boxminus\hat{\mathbf{x}}_{a,k+1}^{-},\qquad \mathbf{P}_{a,k+1}^{-}= \sum_iW_i^{(c)}\mathbf{d}_{i,k}^{x}(\mathbf{d}_{i,k}^{x})^{T} +\mathbf{Q}_{d,a,k}^{model}+\mathbf{Q}_{d,a,k}^{state}.\end{aligned}\end{align} \]

The augmented parameter values are held by each deterministic propagation; their uncertainty enters through the state and model process covariances.

4. Predict measurements for every sigma point

Transform sigma points through the shared measurement stack and form measurement and state cross-covariances.

\[ \begin{align}\begin{aligned}\mathbf{Z}_{i,k}=h(\mathbf{X}_{i,k}^{-},\mathbf{o}_k),\qquad \mathbf{d}_{i,k}^{z}=\mathbf{Z}_{i,k}\boxminus\hat{\mathbf{z}}_k,\\\mathbf{P}_{zz}=\sum_iW_i^{(c)}\mathbf{d}_{i,k}^{z} (\mathbf{d}_{i,k}^{z})^{T}+\mathbf{R}_k,\qquad \mathbf{P}_{xz}=\sum_iW_i^{(c)}\mathbf{d}_{i,k}^{x} (\mathbf{d}_{i,k}^{z})^{T}.\end{aligned}\end{align} \]

5. Apply the unscented correction

Compute the gain, update the physical state and augmented parameters, and transport the posterior covariance.

\[ \begin{align}\begin{aligned}\mathbf{K}_k=\mathbf{P}_{xz}\mathbf{P}_{zz}^{-1},\qquad \mathbf{r}_k=\mathbf{z}_k\boxminus\hat{\mathbf{z}}_k,\qquad \delta\mathbf{x}_{a,k}=\mathbf{K}_k\mathbf{r}_k,\\\mathbf{P}_{a,k,raw}^{+}=\mathbf{P}_{a,k}^{-}- \mathbf{K}_k\mathbf{P}_{zz}\mathbf{K}_k^{T}.\end{aligned}\end{align} \]

6. Retract the attitude and retain the augmented estimate

Keep the right-error tangent chart and preserve all bias and disturbance parameter blocks for the next sigma-point generation.

\[\mathbf{x}_{a,k}^{+}=\mathbf{x}_{a,k}^{-}\boxplus\delta\mathbf{x}_{a,k}, \qquad \mathbf{q}_k^{+}=\hat{\mathbf{q}}_k^{-} \otimes\phi^{-1}(\delta\boldsymbol{\theta}_k), \qquad \mathbf{P}_{a,k}^{+}=\mathbf{J}_{reset}\mathbf{P}_{a,k,raw}^{+} \mathbf{J}_{reset}^{T}.\]

class ADCS.estimators.attitude_estimators.attitude_AugmentedUKF.AugmentedUKF(satellite, state, *, dt, unmodeled_dynamics_psd=0.0, quaternion_mode='quaternion_vector', alpha=1.0, beta=2.0, kappa=0.0)

Bases: UKF

UKF supporting sensor-bias and disturbance-parameter state blocks.

Parameters:

• satellite (Any)

• state (EstimatorState)

• dt (float)

• unmodeled_dynamics_psd (Any)

• quaternion_mode (str)

• alpha (float)

• beta (float)

• kappa (float)

supports_augmented_parameters = True