⚡ electrical engineering

MPC for underactuated spacecraft control with a Lyapunov supervised physics-informed neural network correction layer

This paper proposes a hierarchical control architecture for underactuated spacecraft that combines a nonlinear model predictive controller with a physics-informed neural network for disturbance estimation and a Lyapunov-based supervisory layer to ensure stability, thereby achieving robust attitude maneuvering and reduced steady-state errors despite inertia uncertainty and environmental disturbances.

Original authors: Amirhossein Ayanmanesh Motlaghmofrad, Carlo Cena, Mauro Martini, Marcello Chiaberge

Published 2026-06-12

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Amirhossein Ayanmanesh Motlaghmofrad, Carlo Cena, Mauro Martini, Marcello Chiaberge

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 steer a spaceship, but one of its three main steering wheels has broken. This is what engineers call an "underactuated spacecraft." It's like driving a car where the left wheel is stuck; you can still move forward, but turning left is incredibly difficult, and a tiny bump in the road (like space dust or gravity) could send you spinning out of control.

The paper presents a new way to steer this "broken" spaceship using a three-part team: a Smart Planner, a Learning Assistant, and a Strict Safety Supervisor.

Here is how they work together, explained simply:

1. The Smart Planner (The NMPC)

Think of this as the experienced captain of the ship.

What it does: It looks ahead, calculates the best path to point the ship in the right direction, and makes sure the ship doesn't spin too fast or run out of fuel.
The Problem: The captain is working with a map that isn't 100% perfect. The ship's weight might be slightly different than the map says, or there are invisible winds (disturbances) pushing the ship. Because the map is slightly wrong, the captain's plan might miss the target by a little bit.

2. The Learning Assistant (The PINN)

This is a super-smart student who has studied thousands of practice runs.

What it does: It watches the ship and the "invisible winds." It learns to predict exactly how much the ship is being pushed off course by things the captain's map missed. It then whispers a tiny correction to the captain: "Hey, push a little harder to the right to counter that wind."
The Catch: Because this student is learning from data, it might sometimes get overconfident or make a wild guess. If the student is wrong, it could accidentally make the ship spin out of control.

3. The Strict Safety Supervisor (The Lyapunov Layer)

This is the strict safety inspector standing between the student and the captain.

What it does: Before the student's correction is actually used, the inspector checks it against a strict rulebook (math called "Lyapunov stability"). The rule is simple: "Will this correction make the ship safer or more stable?"
The Result:
- If the student's guess looks safe and helpful, the inspector says, "Go ahead, use it!"
- If the student's guess looks risky or unreliable, the inspector says, "No way!" and blocks the correction. The ship then reverts to just following the Captain's original plan.

The Big Experiment

The authors tested this team in a high-tech computer simulation that mimics real space conditions (gravity, air resistance, and broken wheels). They ran the test 100 times with different starting positions.

What they found:

The Team vs. The Captain Alone: When the Captain worked with the Student (and the Inspector), the ship pointed much more accurately than when the Captain worked alone.
The Numbers:
- The average error in pointing the ship dropped by about 3.8%.
- In the worst-case scenarios (where the ship was most likely to fail), the error dropped by a huge 24%.
- The ship was much more consistent; it didn't just get lucky on one run, it performed well every time.

The "Safety vs. Speed" Trade-off

The paper notes an interesting detail: The team performed best when the Student was allowed to speak freely (without the Inspector). However, that is risky because the Student might make a mistake.

When the Inspector was active, the performance was slightly lower than the "free" Student, but it was still much better than the Captain alone, and most importantly, it was guaranteed to be safe. The Inspector ensures that if the Student gets confused, the ship doesn't crash; it just goes back to the Captain's safe, standard plan.

In Summary

This paper proposes a control system for broken spacecraft that combines:

A reliable planner (the baseline).
A learning AI that fixes small errors the planner misses.
A safety guard that stops the AI if it gets too risky.

The result is a spaceship that can steer itself more accurately than before, even with broken parts and imperfect maps, without ever losing its safety guarantees.

Technical Summary: MPC for Underactuated Spacecraft Control with a Lyapunov-Supervised Physics-Informed Neural Network Correction Layer

Problem Statement
Underactuated spacecraft, which lack full control authority along all rotational axes due to design constraints or actuator failures, face significant challenges in attitude maneuvering and stabilization. Conventional controllers struggle with these systems because they cannot directly stabilize all attitude components without complex reference planning strategies. Furthermore, Model Predictive Control (MPC) approaches, while effective for constraint handling, remain sensitive to inertia uncertainty and unmodeled dynamic couplings. These mismatches between the physical plant and the predictive model often lead to degraded tracking performance and potential instability in the presence of environmental disturbances.

Methodology
The authors propose a hierarchical control architecture integrating three distinct layers to address underactuation and model uncertainty while maintaining safety guarantees:

Nonlinear Model Predictive Controller (NMPC) Baseline: This layer serves as the primary controller, responsible for constraint-aware maneuver planning and ensuring nominal closed-loop stability under actuator limits (e.g., reaction wheel speed and torque saturation). It operates on a discrete prediction model using a fixed-step Runge-Kutta (RK4) discretization with a 0.5s update period and a 12s horizon.
Physics-Informed Neural Network (PINN) Correction: To mitigate model-plant mismatch (specifically inertia uncertainty and unmodeled couplings), a PINN is trained offline on simulation data to estimate residual disturbance torques ( $\tau_d$ ). The network utilizes a NARX structure with three hidden layers (128 neurons each) and takes 16 features as input, including angular velocities and reaction wheel torques. The training objective minimizes a multi-step rollout loss, enforcing consistency with rigid-body rotational dynamics to improve short-horizon prediction accuracy.
Lyapunov-Based Supervisory Safety Mechanism: This deterministic layer acts as a safety filter between the NMPC and the PINN. It evaluates the learned correction in real-time against a conservative, near worst-case assumption based on the nominal inertia model. The supervisor scales or suppresses the PINN's output to ensure the non-increase of an energy-like Lyapunov measure. This guarantees that if the learned correction is deemed unreliable or unsafe, the system gracefully degrades to the purely model-based NMPC policy without switching modes.

Key Contributions
The paper outlines three primary contributions:

Hybrid Control Framework: A novel architecture combining a model-based baseline with a PINN residual compensator to address underactuation and structured modeling uncertainties while preserving theoretical safety guarantees.
Deterministic Supervision: The development of a Lyapunov-based supervisory mechanism that constrains neural network contributions in real-time, adhering to formal verification principles to ensure closed-loop stability.
Performance Validation: Demonstration of improved steady-state tracking performance and reduced error variance compared to a standalone NMPC, validated through high-fidelity simulations.

Experimental Results
The proposed architecture was evaluated using a high-fidelity simulation environment modeling reaction wheel dynamics, actuator saturation, and environmental disturbances (gravity-gradient and aerodynamic drag). A Monte Carlo campaign consisting of 100 simulations with diverse initial attitudes (75° and 125° slew maneuvers) and structured inertia uncertainty was conducted.

Key statistical findings include:

Steady-State Error: The supervised PINN configuration achieved a 3.8% reduction in average steady-state Root Mean Square Error (RMSE) compared to the baseline NMPC. The worst-case steady-state error was reduced by 24.16%.
Final Attitude Error: The system demonstrated a 12.7% reduction in average final error, a 52.9% reduction in maximum final error, and a 29.42% reduction in final error variance.
Statistical Significance: Paired comparisons using the Wilcoxon signed-rank test confirmed that these improvements are statistically significant.
Safety vs. Performance Trade-off: While the unsupervised NMPC+PINN variant showed slightly higher performance gains, the supervised version (NMPC+PINN+Supervisor) maintained strict stability guarantees. The supervisor occasionally rejected beneficial corrections that appeared unsafe under nominal assumptions, prioritizing robustness over peak performance.

Significance and Claims
The paper claims that this hierarchical approach successfully bridges the gap between the robustness of model-based control and the adaptability of data-driven methods. By integrating a PINN as a residual estimator and enforcing stability through a Lyapunov-based supervisor, the system achieves improved tracking accuracy for underactuated spacecraft without compromising the safety properties required for critical aerospace operations. The architecture ensures "graceful degradation," allowing the system to revert to a certifiable, purely model-based controller if the learning-based augmentation becomes unreliable, thereby addressing the safety-critical nature of space missions.