Tightly-Coupled Estimation and Guidance for Robust Low-Thrust Uncooperative Rendezvous via Adaptive Homotopy

Technical Summary: Tightly-Coupled Estimation and Guidance for Robust Low-Thrust Uncooperative Rendezvous via Adaptive Homotopy

Problem Statement
Autonomous rendezvous with uncooperative objects (e.g., active debris removal) requires robust guidance under degraded sensing conditions, such as non-Gaussian sensor anomalies, optical glare, or imperfect target knowledge. While low-thrust propulsion offers high fuel efficiency, minimum-fuel guidance typically yields "bang-bang" control structures. These structures are numerically brittle, possessing a narrow convergence basin for indirect optimal control solvers and high sensitivity to estimation errors. In standard architectures, the guidance law operates aggressively to minimize fuel regardless of the instantaneous confidence in the state estimate. If the estimator is corrupted by anomalies, the guidance law continues to pursue a fragile, fuel-optimal trajectory, often leading to large terminal miss distances or solver failure.

Methodology
The paper proposes a tightly-coupled estimation and guidance architecture where navigation confidence directly modulates the numerical regularization of an indirect optimal control solver. The system operates within a receding-horizon framework using the Clohessy–Wiltshire (CW) relative motion model.

1. Estimation Side (MTF Kalman Filter):

   • A linear Kalman filter estimates the translational relative state.
   • To handle non-Gaussian anomalies, a Multiple Tuning Factors (MTF) mechanism is employed. This strategy constructs a composite innovation score based on the normalized innovation and the activity of MTF covariance inflation.
   • When suspicious measurement directions are detected (where innovation energy exceeds nominal expectations), the MTF inflates the effective measurement covariance ($R_{k,eff}$) only along those specific axes. This reduces the filter's trust in corrupted data while preserving sensitivity in well-behaved directions.

2. Guidance Side (Adaptive Homotopy):

   • The guidance problem is formulated as a minimum-fuel optimal control problem solved via Pontryagin's Minimum Principle.
   • To address the numerical difficulty of the discontinuous bang-bang solution, a scalar homotopy parameter ($\epsilon$) is introduced. This parameter regularizes the control law, creating a family of problems ranging from energy-optimal (smooth, $\epsilon=1$) to fuel-optimal (bang-bang, $\epsilon \to 0$).
   • Innovation-Adaptive Scheduling: Unlike traditional offline continuation where $\epsilon$ is reduced on a fixed schedule, the proposed method maps the estimator's confidence score (derived from the MTF-augmented innovation) online to $\epsilon$.
     • High Confidence: $\epsilon$ is kept low, allowing the solver to recover a fuel-efficient, aggressive bang-bang solution.
     • Low Confidence (Anomalies): $\epsilon$ is increased automatically, relaxing the problem toward a smoother, more robust, and conservative energy-optimal regime.

3. Tight Coupling:

   • The architecture creates a unidirectional feedback loop where the estimator provides not just the state estimate ($\hat{x}_k$) but also a confidence signal ($\epsilon_k$) that reshapes the guidance optimization problem itself. The guidance solver solves a single boundary-value problem per update at the requested $\epsilon_k$, avoiding the sequential continuation steps typical of offline homotopy.

Key Contributions

• Tightly-Coupled Architecture: The formulation of a receding-horizon system where estimator confidence directly modulates the continuation process of an indirect low-thrust solver, rather than treating the homotopy parameter as a purely offline or fixed variable.
• MTF Adaptation for Relative Motion: The adaptation of Multiple Tuning Factors covariance inflation specifically for translational relative motion guidance in uncooperative rendezvous scenarios.
• Innovation-Adaptive Homotopy: The demonstration of a continuous, physically interpretable trade-off between fuel optimality and robustness. The system dynamically selects a guidance path that accounts for both mission performance and the quality of available information.

Numerical Results
The method was tested in a simulated terminal rendezvous scenario (800s horizon) involving a tumbling target, with measurement degradation introduced between $t=250$ and $t=400$ seconds. Degradation included a 300x inflation of measurement noise and a sustained additive bias.

• Performance Comparison:
  • Fixed Bang-Bang ($\epsilon=0$): Both plain and MTF-enhanced Kalman filters with fixed bang-bang guidance failed to meet terminal tolerances, resulting in miss distances of 4.40 m and 7.61 m, respectively.
  • Adaptive Homotopy (Plain Filter): Using the adaptive schedule with a plain filter (no MTF) improved the miss distance to 2.33 m but incurred a massive 212% fuel penalty, still failing to meet the strict tolerance.
  • Proposed MTF-Adaptive Homotopy: The full coupled system reduced the terminal miss to 0.335 m (sub-meter level) with a fuel penalty of only 30.04% relative to the open-loop fuel-optimal benchmark.
• Computational Efficiency: The proposed method achieved 100% solver reliability with a median solve time of 65 ms. In contrast, a "failure-triggered" homotopy approach (which retries with increasing $\epsilon$ upon failure) achieved 100% reliability but suffered from high latency (median 2542 ms) due to serial retries, making it unsuitable for real-time implementation.
• Ablation Study: The results confirmed that neither mechanism is sufficient alone. The MTF filter alone could not solve a numerically intractable guidance problem, and the adaptive schedule alone failed when driven by a filter that had not rejected corrupted measurements. The robustness arises from the interaction of both.

Significance and Claims
The paper claims that effective guidance for uncooperative rendezvous requires a continuous interaction between estimation and control. The primary contribution is not merely a better filter or a better solver, but the interaction between them. By allowing the guidance law to "retreat" to a more robust, conservative regime precisely when the navigation solution degrades, the system avoids the brittleness of aggressive fuel-optimal control under uncertainty.

The authors emphasize that the method is practically viable for onboard execution, as it requires only a single solver call per update (avoiding sequential continuation) and adds negligible computational overhead via the scalar confidence mapping. The work demonstrates that the desired trajectory should not be determined solely by the control objective but must be adapted according to the informativeness of measurements and the confidence of the state estimate. Future work is noted to include extensions to nonlinear dynamics, richer measurement models, and the inclusion of explicit keep-out regions for tumbling appendages.