Online Inertia Tensor Identification for Non-Cooperative Spacecraft via Augmented UKF

1. Problem Statement

Autonomous proximity operations (e.g., active debris removal, on-orbit servicing) require high-fidelity relative navigation. A critical challenge arises when dealing with non-cooperative targets (e.g., tumbling debris or defunct satellites) where:

• Mass properties are unknown: The inertia tensor is not known a priori, and pre-flight CAD models often fail to capture in-orbit mass redistribution (fuel usage, deployed configurations).
• Model Divergence: Standard model-based propagators assume known mass properties. When these are incorrect or missing, estimation errors accumulate rapidly, causing the guidance and control loops to destabilize.
• Observability Limitations: Identifying inertia requires specific trajectory excitation. Many existing methods rely on angular acceleration (often noisy) or require specific maneuver windows, making them difficult to apply in continuous, uncooperative scenarios.

The paper addresses the gap in simultaneously estimating the 6-DOF relative pose and the full inertia tensor of a non-cooperative target in real-time without ground-based pre-calibration.

2. Methodology

The authors propose an end-to-end pipeline integrating high-fidelity simulation, deep learning perception, and an augmented state estimation framework.

A. System Dynamics & Modeling

• Relative Motion: The system models the relative translation in the Local Vertical Local Horizontal (LVLH) frame using nonlinear equations of motion (including Coriolis and gravity gradient terms) and relative attitude kinematics using Modified Rodrigues Parameters (MRPs).
• Inertia Parameterization: The target's symmetric inertia tensor is parameterized by its six independent elements ($J_{xx}, J_{yy}, J_{zz}, J_{xy}, J_{xz}, J_{yz}$). These are treated as random-walk states (slowly varying) within the filter to allow online convergence without assuming a fixed prior.
• Torque-Free Dynamics: The rotational dynamics are modeled via the Euler equation, where the unknown inertia matrix is solved jointly with the kinematic states.

B. Perception & Sensor Simulation

• Simulation Environment: A high-fidelity Blender-based environment simulates the ENVISAT satellite. It generates synchronized RGB images and LiDAR depth data with realistic lighting, self-occlusion, and ray-cast visibility labeling.
• Visual Front-End: A lightweight Convolutional Neural Network (TinyCornerNET) detects 2D image-plane corners from monocular images. It uses a focal-style heatmap loss to handle class imbalance and visibility validation.
• Sensor Fusion: The system fuses 2D corner detections with LiDAR depth measurements to constrain the rigid-body dynamics.

C. Estimation Framework: Augmented Unscented Kalman Filter (UKF)

The core innovation is an Augmented UKF that jointly estimates:

1. Kinematic States: Relative position, velocity, attitude (MRPs), and angular velocity (12 states).
2. Inertial States: The 6 independent elements of the inertia tensor.
3. Bias State: A scalar depth-bias term to correct LiDAR systematic errors.

Key Algorithmic Features:

• Sigma-Point Propagation: Unlike the Extended Kalman Filter (EKF), the UKF uses deterministic sampling (sigma points) to propagate the mean and covariance through the highly nonlinear dynamics and camera projection models, preserving higher-order accuracy.
• Adaptive Process Noise: To prevent "covariance collapse" (where the filter becomes overconfident in constant parameters) and ensure physical consistency, the filter employs an adaptive process noise formulation.
  • Cross-Covariance-Guided Inflation: When measurements are lost (e.g., due to occlusion), the filter uses the cross-covariance between pre- and post-propagation sigma points to inject directional uncertainty. This aligns the covariance growth with the actual dynamics of the system, preventing drift without requiring a backward-pass smoother.
• Adaptive Measurement Noise: An innovation-based tuning mechanism inflates the measurement covariance when sensor data is unreliable (e.g., poor illumination or occlusion), down-weighting bad data automatically.

3. Key Contributions

1. Joint Estimation Architecture: A novel framework that simultaneously recovers the full 6-DOF pose and the complete 6-element inertia tensor of a non-cooperative target using only onboard sensors (monocular vision + LiDAR).
2. No Pre-Calibration Required: The system operates without prior knowledge of the target's mass distribution, starting from either an informed prior or a completely uninformative (isotropic) guess.
3. Robust Adaptive Filtering: The introduction of cross-covariance-guided process noise inflation ensures numerical stability and prevents state divergence during prolonged periods of measurement loss or high uncertainty, a common issue in online parameter identification.
4. High-Fidelity Validation: The pipeline is validated using a photorealistic Blender simulation of the ENVISAT satellite, bridging the gap between theoretical control and realistic sensor noise/occlusion.

4. Results

The authors conducted extensive Monte Carlo simulations ($N=100$) under two initialization scenarios:

• Scenario 1: Informed but Uncertain Prior:
  • Initial inertia estimates were centered near nominal values with Gaussian perturbations.
  • Result: Rapid convergence with steady-state standard deviations below 0.07% for diagonal inertia components. Mean bias was negligible ($10^{-5}$ to $10^{-7}$).
• Scenario 2: Uninformative (Isotropic) Initialization:
  • The filter started with a completely unknown inertia tensor (assumed isotropic, $J_{xx}=J_{yy}=J_{zz}=1/3$).
  • Result: All 100 runs converged to true values within 500 seconds.
  • Performance: Steady-state standard deviation for diagonal components was < 0.05%. The estimator proved asymptotically unbiased even from a worst-case prior.

The results demonstrate that the filter can simultaneously converge kinematic states and inertial parameters, enabling accurate long-term trajectory prediction.

5. Significance

This work provides a viable solution for long-horizon autonomous operations in deep space and debris removal missions. By accurately recovering a target's physical parameters online, the framework:

• Enables Robust Guidance: Allows guidance algorithms to function correctly even when the target's mass properties are unknown or changing.
• Reduces Ground Dependency: Eliminates the need for ground-based pre-calibration or specific "excitation maneuvers" to identify inertia, making the system more autonomous.
• Enhances Safety: Prevents the divergence of estimation loops that could lead to mission failure during close-proximity operations with tumbling objects.

Future work will extend this to include mass estimation under thruster excitation and validation with physical hardware-in-the-loop experiments.