ADCS.estimators.attitude_estimators.attitude_AugmentedMEKF module

Multiplicative EKF with joint bias and disturbance-parameter estimation.

1. Construct the augmented tangent-state estimate

Validate the physical state and append bias and disturbance-parameter blocks while retaining the three-coordinate right-attitude error.

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

The nominal quaternion has four coefficients, while \(\delta\boldsymbol{\theta}\) is its three-coordinate right error.

2. Propagate the augmented nominal state

Synchronize the nominal augmented parameters into EstimatedSatellite and propagate the nonlinear spacecraft model.

\[\begin{split}\hat{\mathbf{x}}_{p,k+1}^{-} = f(\hat{\mathbf{x}}_{p,k}^{+},\hat{\mathbf{b}}_{a,k}^{+}, \hat{\mathbf{d}}_k^{+},\mathbf{u}_k,\mathbf{o}_k,\mathbf{o}_{k+1},\Delta t), \qquad \begin{bmatrix}\hat{\mathbf{b}}_a\\\hat{\mathbf{b}}_s\\\hat{\mathbf{d}}\end{bmatrix}_{k+1}^{-} = \begin{bmatrix}\hat{\mathbf{b}}_a\\\hat{\mathbf{b}}_s\\\hat{\mathbf{d}}\end{bmatrix}_{k}^{+}.\end{split}\]

3. Linearize the augmented process model

Construct the tangent error dynamics and discretize physical, bias, and disturbance-parameter process noise.

\[\delta\dot{\mathbf{x}}_a=\mathbf{F}_{a,k}\delta\mathbf{x}_a+ \mathbf{w}_{a,k},\qquad \mathbf{\Phi}_{a,k}\approx e^{\mathbf{F}_{a,k}\Delta t}, \qquad \mathbf{Q}_{d,a,k}^{VL}= \int_0^{\Delta t}e^{\mathbf{F}_{a,k}\tau}\mathbf{Q}_{c,a,k} e^{\mathbf{F}_{a,k}^{T}\tau}\,d\tau.\]

The parameter-error rows of \(\mathbf{F}_{a,k}\) are zero for the deterministic random-walk model; their configured noise remains in \(\mathbf{Q}_{c,a,k}\).

4. Predict the tangent covariance

Apply the linear covariance recursion in the right-error coordinates.

\[\mathbf{P}_{a,k+1}^{-}= \mathbf{\Phi}_{a,k}\mathbf{P}_{a,k}^{+}\mathbf{\Phi}_{a,k}^{T} +\mathbf{Q}_{d,a,k}^{model}+\mathbf{Q}_{d,a,k}^{state}.\]

5. Predict measurements and form the innovation

Use the shared measurement stack and its bias-aware residual definitions.

\[\hat{\mathbf{z}}_k=h(\hat{\mathbf{x}}_{p,k}^{-},\mathbf{o}_k) +\mathbf{D}_{s,k}\hat{\mathbf{b}}_{s,k}^{-}, \qquad \mathbf{r}_k=\mathbf{z}_k\boxminus\hat{\mathbf{z}}_k.\]

6. Form the augmented measurement Jacobian

Include physical-state, sensor-bias, and disturbance-parameter sensitivities.

\[\mathbf{H}_{a,k}= \begin{bmatrix} \dfrac{\partial h}{\partial\delta\mathbf{x}_p} & \dfrac{\partial h}{\partial\mathbf{b}_a} & \mathbf{D}_{s,k} & \dfrac{\partial h}{\partial\mathbf{d}} \end{bmatrix}_{\hat{\mathbf{x}}_{a,k}^{-}}, \qquad \mathbf{R}_k=\mathbb{E}[\mathbf{v}_k\mathbf{v}_k^T].\]

7. Apply the Kalman correction

Correct the physical state and every enabled augmented parameter block.

\[ \begin{align}\begin{aligned}\mathbf{S}_k=\mathbf{H}_{a,k}\mathbf{P}_{a,k}^{-}\mathbf{H}_{a,k}^{T}+ \mathbf{R}_k,\qquad \mathbf{K}_k=\mathbf{P}_{a,k}^{-}\mathbf{H}_{a,k}^{T}\mathbf{S}_k^{-1}, \qquad \delta\mathbf{x}_{a,k}=\mathbf{K}_k\mathbf{r}_k.\\\mathbf{P}_{a,k,raw}^{+}=(\mathbf{I}-\mathbf{K}_k\mathbf{H}_{a,k}) \mathbf{P}_{a,k}^{-}(\mathbf{I}-\mathbf{K}_k\mathbf{H}_{a,k})^T+ \mathbf{K}_k\mathbf{R}_k\mathbf{K}_k^T.\end{aligned}\end{align} \]

8. Retract the attitude and reset the tangent covariance

Apply the multiplicative quaternion retraction and transport the covariance about the new tangent origin.

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

class ADCS.estimators.attitude_estimators.attitude_AugmentedMEKF.AugmentedMEKF(satellite, state, *, dt, unmodeled_dynamics_psd=0.0, quaternion_mode='quaternion_vector')

Bases: MEKF

MEKF for the full EstimatorState layout.

The state may contain actuator biases, attitude-sensor biases, and estimated disturbance parameters in addition to angular velocity, attitude, and reaction-wheel momentum. Their dynamics and measurement couplings are supplied by EstimatedSatellite and the shared process model/measurement stack; the covariance remains in right tangent coordinates, just as for MEKF.

Parameters:

• satellite (Any)

• state (EstimatorState)

• dt (float)

• unmodeled_dynamics_psd (Any)

• quaternion_mode (str)

supports_augmented_parameters = True