⚡ electrical engineering

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

This paper proposes a robust low-thrust rendezvous guidance architecture that tightly couples a Kalman filter with adaptive homotopy-based optimal control, dynamically relaxing the control strategy to a conservative regime during sensor degradation to achieve sub-meter terminal accuracy where traditional fixed-epsilon methods fail.

Original authors: Batu Candan, Simone Servadio

Published 2026-05-07

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Batu Candan, Simone Servadio

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to park a very slow, fuel-efficient spaceship next to a giant, tumbling piece of space junk (like a broken satellite) that you can't talk to. You have to do this blindly, relying only on a camera that sometimes gets blinded by the sun or confused by shadows.

This paper presents a new "brain" for the spaceship that solves a major problem: How do you drive aggressively to save fuel without crashing when your eyes are playing tricks on you?

Here is the breakdown of their solution using simple analogies:

The Problem: The "Aggressive Driver" vs. The "Bad Camera"

The Goal: The spaceship wants to use the absolute minimum amount of fuel to get close to the target. In math terms, this creates a "bang-bang" driving style. Think of this like a race car driver who only does two things: floor the gas or slam the brakes. There is no coasting.
The Risk: This aggressive style is incredibly efficient, but it is also "brittle." If the driver's camera (the sensor) sees a shadow and thinks the target is in a different spot, the driver might slam the brakes or gas at the wrong time, causing a crash or a huge miss.
The Old Way: Usually, the "navigator" (who looks at the camera) and the "driver" (who controls the thrusters) work separately. Even if the navigator screams, "Hey, I'm not sure what I'm seeing!" the driver keeps driving aggressively because that's the plan to save fuel.

The Solution: A Tightly-Coupled Team

The authors created a system where the navigator and the driver are best friends who talk constantly. They don't just share where the target is; they share how confident they are about that location.

1. The "Suspicious Shadow" Detector (MTF)

First, the system has a special filter (called MTF) that acts like a skeptical detective.

How it works: If the camera sees a sudden, weird jump in the target's position (like a glitch or a glare), the detective says, "That doesn't look right. I'm going to lower my trust in this specific piece of data."
The Analogy: Imagine you are walking in the fog. You see a shape that looks like a person, but it flickers. Instead of assuming it's a person and running toward it, you say, "That's probably just a trick of the light," and you slow down your mental model of where that person is.

2. The "Confidence Dial" (Adaptive Homotopy)

This is the magic part. The system takes the detective's confidence level and turns a dial on the driver's controls.

High Confidence (Clear Vision): When the camera is clear and the detective is sure, the dial is set to 0. The driver goes back to being the aggressive "race car" (floor gas, slam brakes) to save fuel.
Low Confidence (Foggy/Glaring): When the camera is glitching, the detective turns the dial up. This tells the driver: "Stop being a race car. Drive like a cautious grandparent."
The Result: The driver switches from "bang-bang" (jerky, fuel-efficient) to "smooth and steady" (slightly less fuel-efficient, but much safer).

What Happened in the Test?

The researchers simulated a scenario where the camera was deliberately messed up with bad data.

The Old Drivers (Fixed Aggressive): Even with the "detective" helping them see better, the drivers kept trying to drive like race cars. Because they were too aggressive, they missed the target by about 160 meters (roughly the length of a football field).
The New Team (Adaptive): When the camera got glitchy, the system automatically told the driver to slow down and smooth out the approach.
- The Result: They missed the target by less than 1 meter.
- The Cost: They used a bit more fuel (about 27% more than the perfect theoretical plan), but they actually got the job done.

The Key Takeaway

The paper proves that flexibility is better than stubborn efficiency.

By letting the driver's "aggressiveness" change based on how much the navigator trusts the camera, the spaceship can handle bad sensors without crashing.

The Detective (MTF) makes the trust signal sharper.
The Dial (Homotopy) makes the driving style safer when trust is low.

Together, they turned a system that would have crashed (missing by 160m) into one that parked perfectly (missing by less than 1m), all while keeping the fuel usage reasonable. The paper concludes that this "tight coupling" is the secret to making autonomous space rendezvous robust enough for real-world missions.

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

Problem Statement
Autonomous rendezvous with uncooperative objects (e.g., space debris or non-cooperative spacecraft) requires robust estimation, planning, and control under degraded sensing conditions. While low-thrust propulsion offers high fuel efficiency, minimum-fuel guidance typically yields "bang-bang" control structures. These structures are numerically sensitive to initial co-states, regularization parameters, and estimation errors. In uncooperative scenarios, non-Gaussian sensor anomalies (such as optical shadowing or glare) can abruptly degrade state estimates. Existing approaches often decouple the guidance law's numerical aggressiveness from the estimator's instantaneous confidence, leading to brittle closed-loop performance where the controller continues to pursue an aggressive fuel-optimal solution despite unreliable state information.

Methodology
The paper proposes a tightly-coupled estimation and guidance architecture that dynamically modulates the numerical regularization of an indirect optimal control solver based on navigation confidence. The system operates within the Clohessy–Wiltshire (CW) frame for relative motion dynamics.

Estimation Framework: The translational state is estimated using a linear Kalman Filter (KF). To handle non-Gaussian anomalies, the filter employs a Measurement-Side Multiple Tuning Factors (MTF) mechanism. This mechanism identifies measurement directions where the innovation energy exceeds nominal expectations and inflates the covariance matrix only along those specific axes. This suppresses the filter's trust in suspicious data while maintaining sensitivity in well-behaved directions.
Guidance Framework: The guidance law utilizes a receding-horizon indirect solver based on Pontryagin's Minimum Principle. To ensure numerical tractability, a homotopy parameter ( $\epsilon$ ) is introduced to smooth the discontinuous bang-bang control structure.
Adaptive Coupling: The core innovation is the online mapping of a composite confidence score to the homotopy parameter $\epsilon$ . The confidence score is derived from:
- The normalized innovation squared ( $s_{NIS}$ ).
- The relative magnitude of the MTF correction ( $\rho_{MTF}$ ).
- A low-pass filtered composite score ( $\bar{s}_k$ ).
When the confidence score is low (indicating high innovation or MTF activity), the homotopy parameter $\epsilon$ is increased. This relaxes the control problem toward a smoother, more conservative, energy-optimal regime. As confidence recovers, $\epsilon$ decreases, allowing the controller to revert toward the fuel-optimal bang-bang structure.

Key Contributions
The authors identify three primary contributions:

Tightly-Coupled Architecture: A receding-horizon framework where estimator confidence directly modulates the continuation process of an indirect low-thrust solver, rather than treating the homotopy parameter as a static offline variable.
MTF Adaptation for Relative Motion: The adaptation of MTF-based covariance tuning concepts specifically for translational relative motion guidance in uncooperative rendezvous scenarios.
Interpretable Trade-off: A demonstration of how innovation-adaptive homotopy enables a continuous, physically interpretable trade-off between fuel optimality and robustness during close-proximity operations.

Numerical Results
The proposed method was evaluated in a simulation derived from a prior vision-based relative navigation environment involving a tumbling target (ENVISAT benchmark). The scenario involved severe measurement degradation over an 800-second horizon.

Performance of Fixed Controllers: Both a plain Kalman Filter with fixed bang-bang control and an MTF-KF with fixed bang-bang control failed to meet the terminal objective under degraded sensing, resulting in terminal miss distances of approximately 160 meters ( $1.60 \times 10^{-1}$ km).
Performance of Adaptive Controller: The proposed MTF-adaptive homotopy controller reduced the terminal miss by roughly two orders of magnitude to the sub-meter level ( $9.28 \times 10^{-4}$ km, or ~0.93 m). This improvement came with a moderate fuel penalty of 27.4% relative to the open-loop fuel-optimal benchmark.
Role of Components:
- Adaptive Homotopy: Identified as the dominant robustness mechanism. An adaptive controller without MTF still succeeded qualitatively (miss distance ~14.2 m) but incurred a significantly higher fuel penalty (82%) and exhibited lower computational reliability (67.95% solve success rate).
- MTF: Acted as a secondary enhancement. When added to the adaptive controller, MTF reduced the terminal miss from ~14.2 m to ~0.93 m and reduced the fuel penalty from 82% to 27.4%. Crucially, the MTF-enhanced architecture achieved a 100% solve success rate with significantly faster mean re-solve times (54.89 ms vs. 324.18 ms), indicating improved numerical stability.

Significance and Claims
The paper claims that the primary benefit of the proposed method is not merely improved filtering, but the closed-loop coupling that allows estimator confidence to directly regularize the guidance law online. The results suggest that robust low-thrust close-proximity operations require breaking the traditional separation between estimation confidence and guidance aggressiveness.

The authors assert that their framework enables a continuous trade-off between fuel optimality and robustness, preventing the controller from blindly pursuing aggressive solutions when sensing is degraded. The computational results support the practical viability of the method, demonstrating consistently fast and reliable solution times suitable for online execution in the tested linearized scenario. Future work is noted to extend these methods to nonlinear dynamics, richer measurement models, and the inclusion of explicit keep-out regions for tumbling spacecraft appendages.