Dual Star-Tracker Misalignment Estimation

• The paper introduces a novel approach for dual star-tracker misalignment estimation using a combined MEKF and Bayesian MMAE framework to achieve arcsecond-level precision.
• It details a methodology that integrates adaptive grid refinement and diversity metrics to balance accuracy and computational load on CubeSat-class platforms.
• Simulation results demonstrate accurate joint estimation of spacecraft attitude, gyroscopic biases, and star-tracker misalignments, confirming robust performance under realistic noise conditions.

Dual star-tracker misalignment estimation refers to the autonomous, in-flight determination and compensation of installation errors for two independent star trackers mounted on a spacecraft bus. This capability is vital for CubeSat-class vehicles in GPS-denied deep-space missions, where precise attitude knowledge and calibration are essential and external references are not available. Recent methodologies—most notably, adaptive multi-model estimation combining a Multiplicative Extended Kalman Filter (MEKF) with a Bayesian Multiple-Model Adaptive Estimation (MMAE) bank—enable joint estimation of spacecraft attitude, gyroscopic biases, and two sets of star-tracker misalignment parameters, achieving arcsecond-level accuracy under operational constraints (Ganganath et al., 3 Jan 2026).

1. Estimation Framework and State Representation

The state vector governing the MEKF is constructed as $x(t) = [q(t)^\top~\omega(t)^\top~b(t)^\top]^\top$, where $q(t)\in\mathbb{R}^4$ denotes the unit quaternion representing attitude (scalar-last), $\omega(t)\in\mathbb{R}^3$ is body-frame angular velocity, and $b(t)\in\mathbb{R}^3$ is the gyroscope bias. The true nonlinear dynamics are written as:

$$\dot q = \frac{1}{2}\,\Omega(\omega)\,q,\quad \dot\omega = J^{-1}\bigl(M_c - \omega\times J\,\omega\bigr),\quad \dot b = 0,$$

with $\Omega(\omega)$ denoting the rate-skew-symmetric matrix. The MEKF utilizes a multiplicative error-state representation: for small-angle attitude error $\delta\theta\in\mathbb{R}^3$, and additive errors $\delta\omega$, $\delta b$, the error state is

$$\delta x = [\delta\omega~\delta b~\delta\theta]^\top,$$

and the error quaternion is $$\delta q \approx [\tfrac{1}{2}\,\delta\theta~1]^\top$$, so $q = \delta q \otimes \hat{q}$.

Linearized error dynamics yield time propagation for both the filter state and covariance. Discrete-time propagation is implemented as

$$\Phi = \exp(F\,\Delta t), \quad P^- = \Phi\,P^+\,\Phi^\top + Q_d,$$

where $Q_d \approx Q\,\Delta t$ under small $\Delta t$.

2. Bayesian MMAE for Dual Star-Tracker Misalignments

Each of the two star trackers is assigned a constant small-angle misalignment parameter $\mu_k\in\mathbb{R}^3$, $k\in\{1,2\}$, which is mapped to a quaternion representation $\delta q(\mu_k)$ and corresponding direction-cosine matrix $C_{\mu_k}$. The MMAE formulation discretizes the misalignment space as a 6D grid with $N$ combinations of $(\mu_1^{(j)},\mu_2^{(j)})_{j=1}^N$, each representing a candidate sensor geometry.

The likelihood update for each hypothesis $j$ employs residuals $r_j$ formed from stacked line-of-sight (LOS) star-tracker measurements and assumed zero-mean Gaussian noise $R$:

$$\tilde w_j[k] = w_j[k-1]\,\exp\!\left(-\tfrac{1}{2}\,r_j^\top R^{-1}\,r_j\right),$$

$$w_j[k] = \frac{\tilde w_j[k]}{\sum_{i=1}^N\tilde w_i[k]}.$$

The MEKF runs in parallel for each misalignment hypothesis, leveraging appropriate $C_{\mu_k}$ values in the measurement model.

3. Diversity Metric Ψ and Grid Refinement Strategy

To prevent probability collapse onto poorly located hypotheses, a diversity metric $\Psi$ is introduced:

$$A_k = \left(\sum_{j=1}^N w_j[k]^2\right)^{-1},\quad \Psi[k]=100\,\frac{A_k}{N}\;\%\!.$$

Here, $\Psi=100\%$ indicates uniform model probabilities, while $\Psi \ll 100\%$ signals excessive concentration. When $\Psi[k] < \Psi_{\rm th}$ (e.g., $10\%$), the grid is adaptively refined: the weighted mean misalignment for each tracker

$$\hat{\mu}_1 = \sum_j w_j\,\mu_1^{(j)},\quad \hat{\mu}_2 = \sum_j w_j\,\mu_2^{(j)}$$

centers a new 6D grid with smaller angular spacing, all weights are reset evenly, and estimation continues, ensuring computational efficiency by focusing resources in high-probability regions.

4. Dual Star-Tracker Measurement Modelling and LOS Stacking

For camera $k$, inertial reference vectors $\{v^I_i\}$ are transformed under the true attitude and misalignment:

$v_{S,i,{\rm true}}^{(k)} = C_{\mu_k}\,C(q)\,v^I_i,$

with noisy sensor measurements modeled as

$$y_i^{(k)} = v_{S,i,{\rm true}}^{(k)} + \eta_i^{(k)},\quad \eta_i^{(k)}\sim\mathcal{N}(0,\sigma_\eta^2 I_3).$$

All six vectors from each star tracker are concatenated yielding an $18\times 1$ measurement vector $y^v$, which is stacked with gyro data $y_\omega = \omega + b + \eta_\omega$ to produce the full measurement vector $y$. The MEKF linearizes LOS residuals to form a $21\times 9$ measurement Jacobian $H$ and block-diagonal measurement covariance $$R = \mathrm{blockdiag}(\sigma_\eta^2 I_{18}, R_\omega)$$. This enables simultaneous attitude and sensor geometry uncertainty propagation.

5. Simulation Protocols and Estimation Performance

Monte Carlo experiments were conducted with $N_{\rm MC}=100$ runs per configuration. Dual-misalignment tests used $T=250{,}000$ s trajectories, $\Delta t=0.5$ s time steps, and inertia diag(100, 60, 50) kg·m². Noise parameters included star-tracker noise $\sigma_\eta=8.73\times10^{-4}$ rad, and gyro noise $\sigma_\omega=5\times10^{-4}$ rad/s. Initial grids used $5\times5\times5\times5\times5\times5$ resolutions.

Attitude RMSE stabilized below $0.05^\circ$ in the dual case and below $1^\circ$ in the single case. Misalignment RMSE converged to $O(10^{-4})$ deg (approximately $0.3$–$0.5$ arcseconds) for both trackers. Gyro bias and angular velocity errors settled at $O(10^{-5})$–$O(10^{-6})$ rad/s.

Key summary statistics for the dual-misalignment scenario, averaged over 100 runs, are:

Quantity	Final Mean Error
Attitude	$[-0.021,\,0.041,\,−0.055]^\circ$
Misalignment 1	$[-0.00052,\,0.000058,\,−0.000012]^\circ$
Misalignment 2	$[-0.00051, −0.00061, 0.00044]^\circ$
$\omega$-error	$O(10^{-5})$ rad/s
Bias-error	$O(10^{-6})$ rad/s

Plots produced for verification included RMSE time series for all parameters, temporal evolution of $\Psi$, model probability trends before/after refinement, and $\pm3\sigma$ consistency envelopes.

6. Practical Implementation Considerations and Computational Load

The architecture is designed for real-time operation on CubeSat-class CPUs (update rate: 0.5 s). Despite the initial dimensionality of the grid, adaptive refinement and grid pruning sustain a manageable number of active models—typically a few dozen—making parallel MEKF evaluations feasible even with limited resources. The approach maintains a fixed continuous-state dimension, with misalignment hypotheses handled discretely via the MMAE bank. All likelihood computations and MEKF updates are parallelizable and do not require augmentation of the Kalman filter state with misalignment parameters.

A plausible implication is that the use of diversity-triggered grid refinement selectively deploys computational resources, allowing robust and accurate sensor calibration in resource-constrained deep-space platforms, and establishes dual star-tracker misalignment estimation as a practical methodology for next-generation CubeSat missions (Ganganath et al., 3 Jan 2026).