The study, published in the International Journal of Aeronautical and Space Sciences, tackles a problem that has shadowed every membrane-based spacecraft concept since the earliest solar sail designs: the film itself deforms. When a sail twists under aerodynamic loading, its shape changes, and that change feeds back into the very torques that caused it. This coupling between torsional deformation and aerodynamic forces means that treating the sail as a rigid plate is not merely inaccurate, it can be fundamentally misleading. The research team, led by Shuiyuan Wu and Ming Xu with corresponding author Lincheng Li, built their analysis from the ground up, beginning with the physics of the film itself.

Their first step was to apply classical elasticity theory to calculate the torsional deformation of the thin membrane. Rather than stopping at an idealized analytical solution, the team cross-checked their results against a finite element simulation built on solid-shell elements, a numerical approach capable of capturing the behavior of structures that are thin in one dimension but extended in the other two. This dual verification matters because the deformation field directly determines how air molecules strike the surface, and therefore how much torque the sail experiences. An error of even a few degrees in the local surface inclination can shift the aerodynamic torque distribution across the entire membrane.

To translate the deformed geometry into forces, the researchers turned to the Schaaf–Chambre model, a foundational framework from the study of rarefied gas flows. At the altitudes where atmospheric sails operate, the air is so thin that molecules almost never collide with one another before hitting the spacecraft; the flow is instead governed by individual molecule-surface interactions. In this free molecular regime, the momentum transferred to the sail depends on how molecules reflect off the film, characterized by normal and tangential accommodation coefficients, and on the local angle of incidence. By integrating these molecular momentum exchanges over the deformed sail surface, the team derived the torques acting about all three axes and assembled a complete three-axis attitude dynamics model of the flexible sail.

One of the study’s quieter but more consequential findings is that the torques involved are remarkably small, all on the order of newton-meters. That may sound like good news, and in terms of structural loading it is, but it presents a severe challenge for control engineers. Actuators must respond to disturbances that are tiny, nonlinear, and constantly shifting as the membrane flexes. A controller that reacts too aggressively wastes precious energy and can excite vibrations in the film; one that reacts too sluggishly lets the sail drift off its intended attitude. The team’s answer to this dilemma is a composite scheme that pairs a radial basis function neural network, or RBFNN, with sliding mode control, a robust technique long favored for spacecraft attitude regulation.

Sliding mode control works by driving the system’s state onto a predefined sliding surface and then holding it there, offering strong guarantees against bounded disturbances. Its weakness is that it must treat unknown nonlinearities as disturbances to be overpowered, which demands larger control effort and can induce chattering. The RBFNN changes that equation. Because a radial basis function network can approximate arbitrary nonlinear functions given enough nodes, it can learn the unknown terms in the sail’s dynamics, specifically the nonlinear functions the researchers label f1(x), f3(x), and f5(x), and compensate for them directly. The simulations show that the network tracks these nonlinear terms rapidly and efficiently, effectively handing the sliding mode controller a much cleaner problem to solve.

The payoff is striking. When the RBFNN–SMC composite controller was applied, the torque required about the z-axis, Mz, dropped by as much as 55.2 percent compared with sliding mode control alone. Just as importantly, the physical motion of the sail films themselves became gentler. The rotation angles of the film segments, denoted β2, β3, and β4 in the study, were notably reduced under the hybrid scheme, with the motor rotation amplitude for β4 shrinking by 35.6 percent. For a membrane structure whose service life is limited by flexing, folding, and fatigue, smaller rotation angles translate directly into longer operational lifetimes. The researchers summarize this with an accumulated energy index, which came out lower under the composite controller, confirming that the neural network approach reduces total energy consumption rather than merely shifting it elsewhere in the system.

The implications extend well beyond a single simulation study. Atmospheric sails sit at the intersection of several pressing trends in space technology. Satellites destined for very low Earth orbit, where they can image the planet at higher resolution or serve as drag-based formation flyers, must constantly fight atmospheric drag, and sails offer a way to harness rather than resist it. Distributed drag sails have already been proposed for transverse satellite formation flying, and atmosphere-breathing electric propulsion concepts rely on the same rarefied flow physics that the Schaaf–Chambre model describes. Every one of these applications needs attitude control that is both precise and frugal, because small satellites cannot carry generous power budgets or oversized reaction hardware. A control scheme that cuts torque demand by more than half could be the difference between a feasible mission concept and an impractical one.

The study also contributes to a broader effort to model sail deformation rigorously. Previous work has examined the structural response of large solar sails during attitude maneuvers, analyzed wrinkling of solar-photon sails, and used point cloud methods to characterize sail deformation. What distinguishes the new research is its end-to-end treatment: the deformation is computed analytically, verified numerically, fed into a free molecular flow model to produce torques, and then closed into a full attitude dynamics loop where the controller must cope with the coupled torsional-aerodynamic effects. This chain matters because the coupling is precisely what makes deformable sails hard to control. A film that twists changes its own aerodynamic loading, which changes the torque, which changes the twist again. A controller blind to this loop would either fight it wastefully or be destabilized by it.

For now, the results come from simulation, and the transition to flight hardware will demand further validation, including how the neural network performs under model uncertainties, thermal-induced membrane vibrations, and the messy realities of actuator saturation. But the direction is clear. As membranes grow larger and spacecraft grow smaller, the era of treating flexible structures as rigid bodies is ending. Machine learning is increasingly finding its way into spacecraft control, from fault-tolerant attitude tracking with neural estimators to adaptive neural sliding controllers for rotorcraft and hypersonic vehicles. This study adds a compelling data point to that trend, showing that a well-trained network can shoulder the nonlinear burden of a flexing sail and leave the robust core of the controller free to do its job with a fraction of the effort. If atmospheric sails are to sweep through the thin air of low orbit in the years ahead, they may well do so with a neural network quietly trimming their course, saving energy with every gentle turn of the film.

Subject of Research: Energy-efficient neural network attitude control of deformable atmospheric sails under coupled torsional and aerodynamic effects

Article Title: Energy-Efficient RBFNN-SMC Attitude Control of Deformable Atmospheric Sail Under Coupled Torsional-Aerodynamic Effects

Article References: Wu, S., Xu, M., Li, L., Wang, H., & Meng, D. (2026). Energy-Efficient RBFNN-SMC Attitude Control of Deformable Atmospheric Sail Under Coupled Torsional-Aerodynamic Effects. International Journal of Aeronautical and Space Sciences. https://doi.org/10.1007/s42405-026-01241-4

Image Credits: AI Generated

DOI: 10.1007/s42405-026-01241-4

Keywords: atmospheric sail, attitude control, RBF neural network, sliding mode control, torsional deformation, free molecular flow, very low Earth orbit, spacecraft dynamics, membrane structures, energy efficiency, aerodynamic torque, flexible spacecraft

The research, led by Anuj Mangal of GLA University in Mathura, India, together with colleagues from several Indian engineering institutions, tackles a persistent gap in aviation diagnostics. Deep learning has transformed fields such as object identification and computer vision over the past decade, and convolutional neural networks have recently surged in popularity for diagnosing rotating machinery of all kinds. Yet gas turbine diagnosis remains an area where these powerful techniques have seen comparatively limited adoption. The authors argue that the complexity of engine thermodynamics, the sheer variety of operating conditions, and the difficulty of obtaining labeled fault data have all slowed progress, and their framework is an attempt to close that gap with tools that are both data-driven and grounded in engine physics.

At the heart of the approach is a two-stage architecture. The first stage is a physics-driven efficiency trend-monitoring system that tracks how engine performance degrades over time. Rather than waiting for a hard failure, the system continuously compares observed sensor readings against expected values, records degradation-related performance changes, and creates a new performance baseline whenever the engine’s condition shifts in a lasting way. This baseline management is critical, because an engine that has slowly lost a fraction of its efficiency through normal wear is not malfunctioning, while a sudden deviation from the current baseline may signal a genuine fault. The trend monitor also produces fault signatures, the characteristic patterns that specific gas path problems leave in the sensor data.

Those physics-based fault signatures then serve a second purpose: they train the fault detection and isolation system. Because real engine failures are rare and dangerous, researchers cannot simply wait for engines to break in order to gather training data. By using a physics-based model to generate realistic fault signatures for a range of gas path problems, the team could teach their diagnostic networks to recognize and categorize faults at the component level, distinguishing, for example, between degradation in a compressor and trouble in a turbine. This marriage of simulation and machine learning is a growing trend in aerospace engineering, where digital models supply the labeled examples that real-world operations cannot provide safely or economically.

The second stage of the framework is built around a radial basis function, or RBF, neural network, a type of network whose hidden neurons respond to inputs based on their distance from learned center points. RBF networks are prized for fast training and strong local approximation abilities, making them well suited to mapping the nonlinear relationships between engine sensor readings and engine health. The proposed pipeline integrates sensor-based degradation indicators with normalization, sliding window feature extraction, and outlier detection, a sequence designed to improve prediction reliability when raw data are noisy or incomplete. Sliding windows allow the network to see not just instantaneous readings but the shape of their evolution over time, which is often where degradation reveals itself first.

To evaluate the system rigorously, the researchers turned to the NASA C-MAPSS turbofan engine simulation dataset, a widely used benchmark that models the behavior of a three-shaft turbofan under realistic operating profiles. Crucially, the team deliberately tested their methods under conditions of substantial measurement noise to guarantee model robustness, since real aircraft sensors rarely deliver clean signals. Multiple failure scenarios were simulated, and the fault detection and isolation performance was assessed against a convolutional neural network-based classification approach and a deep long short-term memory, or LSTM, method, allowing a direct comparison of how different neural architectures handle the same diagnostic task.

The results were measured using root mean square error, or RMSE, along with score and average score metrics, the standard yardsticks in the remaining useful life prediction community. The proposed RBF neural network model achieved improved prediction performance compared with conventional approaches such as multilayer perceptrons and support vector machines. In practical terms, that means the network could estimate how much useful life remained in an engine more accurately, giving maintenance planners a tighter, more trustworthy window in which to schedule inspections and part replacements. Accurate remaining useful life estimation is one of the central goals of prognostics, because it converts maintenance from a calendar-driven ritual into a condition-driven decision.

The study also introduces a dedicated health assessment model built around exhaust gas temperature, one of the most telling indicators of gas turbine condition. An artificial neural network was used to identify the EGT parameter, while multiple regression analysis quantified how flight factors influence it. By the end of the research, the team had constructed a network capable of forecasting the EGT variable with minimal error. The engine’s state can be monitored instantly through an interface created in MATLAB Simulink, where performance degradation appears as a numerical value or a graphical representation that is easy to interpret. The authors even suggest the model may serve as a new kind of indicator that alerts pilots when the EGT sensor itself develops a fault in flight, a sensor-of-sensors safeguard that could add another layer of redundancy to cockpit instrumentation.

The motivation behind all of this is threefold: reduce fuel consumption, improve aircraft safety, and lower maintenance costs. Engines that run efficiently burn less fuel, and fuel is one of the largest operating expenses for any airline. Early fault detection prevents small defects from cascading into in-flight shutdowns or unscheduled landings. And condition-based maintenance, in which components are serviced when their actual condition warrants it rather than on fixed schedules, can trim billions from global maintenance budgets while keeping aircraft in revenue service longer. Performance degradation in gas turbine engines, as the literature defines it, is a function of vibration, oil pressure, motor fan velocity, oil temperatures, exhaust gas temperature, and fuel flow, and the framework monitors precisely these channels.

The broader significance of the work lies in its demonstration that hybrid approaches, pairing physics-based models with neural networks, can outperform purely data-driven methods in domains where training data are scarce and safety margins are unforgiving. As aviation moves toward more autonomous health management systems, and as digital twins of individual engines become standard practice, frameworks like the one described here may become as routine a part of an airliner’s ground infrastructure as the weather radar. The next time a flight lands smoothly and the passengers disembark unaware, there is a reasonable chance that a network of artificial neurons has already been quietly reviewing the engine’s every breath, deciding whether it is fit to fly again tomorrow.

Subject of Research: Machine learning-based performance monitoring and fault diagnosis of aircraft gas turbine engines

Article Title: Performance Monitoring and Fault Diagnosis of Aircraft Engines via RBF Neural Network Models

Article References: Mangal, A., Sanmugam, S., Alagarsamy, M., Murugan, J. S., Kuppusamy, S., & Rajaram, A. (2026). Performance Monitoring and Fault Diagnosis of Aircraft Engines via RBF Neural Network Models. International Journal of Aeronautical and Space Sciences. https://doi.org/10.1007/s42405-026-01225-4

Image Credits: AI Generated

DOI: 10.1007/s42405-026-01225-4

Keywords: aircraft engines, gas turbines, RBF neural network, fault diagnosis, engine health monitoring, remaining useful life, NASA C-MAPSS, exhaust gas temperature, machine learning, turbofan, predictive maintenance, convolutional neural networks

The research, led by Yiran Wei and Ming He of the Army Engineering University of the PLA in Nanjing, together with colleagues at the Beijing Institute of Technology and the National Key Laboratory on Near-Surface Detection, tackles two goals that usually pull against each other: rapid response and stability maintenance. A formation that reacts quickly to command changes tends to oscillate, while a formation tuned for smoothness tends to lag. The team’s answer is a decentralized adaptive tracking control method built on a leader-follower coordination strategy, in which each aircraft in the cluster is responsible for its own tracking behavior while the group as a whole follows a virtual leader.

The virtual leader idea is central to how the scheme scales. Rather than designating one physical drone as the boss whose failure would doom the mission, the researchers define a mathematical reference trajectory that every aircraft in the cluster tries to track, offset by its assigned position in the formation. Each CRA then adjusts its relative position to its neighbors to hold the overall shape. Because the coordination is decentralized, no single aircraft has to broadcast commands to the whole swarm, and the control burden does not grow explosively as the cluster expands. This design philosophy echoes a broad body of work on multi-agent formation control, from graph-based model predictive approaches to flocking-inspired swarm navigation, but the new study adapts it specifically to the dynamics of coaxial rotors.

To get there, the authors first built a mathematical model of a single CRA and then extended it to the cluster system. The model separates the aircraft’s motion into two subsystems with distinct jobs. The position subsystem maintains the formation by adjusting where each aircraft sits relative to the others, while the attitude subsystem is responsible for driving each aircraft’s orientation to a desired stable state. This split matters because coaxial aircraft are underactuated: they have fewer independent control inputs than degrees of freedom, so position is controlled indirectly through attitude. Getting the attitude loop right is therefore a precondition for getting the position loop right.

For the position subsystem, the team turned to back-stepping control, a well-established nonlinear design method that works backward through the layers of a system’s dynamics. In a back-stepping design, the controller treats some of the internal states of the aircraft, such as velocity components, as virtual controls for the outer states, such as position. It designs a stabilizing law for the outer loop, computes what the virtual controls would need to be to achieve it, and then steps inward, designing control laws for each successive layer until it reaches the true physical inputs. The result is a controller that respects the nonlinear coupling in the aircraft’s equations of motion rather than approximating them with a linearized model, which is why back-stepping has become a favorite tool for quadrotors and, increasingly, for coaxial platforms.

The attitude tracking controller is where the machine learning enters. Coaxial rotors generate complicated aerodynamic interactions, including blade-vortex interactions between the upper and lower rotors, that are difficult to model precisely. Rather than trying to capture every disturbance term analytically, the researchers designed an adaptive radial basis function neural network, or RBFNN, that learns the unknown nonlinearities online. An RBFNN is a particular class of neural network in which each neuron responds to inputs within a localized region of the input space, using a radial basis function such as a Gaussian centered at a particular point. This locality gives RBFNNs a powerful theoretical property: under mild assumptions, they can approximate any continuous nonlinear function over a compact domain. In the control context, that means the network can represent the lumped uncertainty in the attitude dynamics, whatever its exact form, and the adaptive tuning laws adjust the network weights in real time as the aircraft flies.

Critically, the team did not simply trust the neural network to behave. The stability of both the position and attitude controllers is mathematically proven using Lyapunov theory, the standard framework for demonstrating that a controlled system’s energy-like function decreases over time, driving the tracking errors to zero or to a bounded neighborhood of zero. The Lyapunov analysis ties the neural network’s weight adaptation to the error dynamics, ensuring that the learning process itself cannot destabilize the aircraft. This combination of learning-based approximation with rigorous stability proof is what distinguishes the approach from purely empirical machine learning controllers, and it addresses one of the biggest obstacles to deploying neural networks on real flight hardware.

Verification proceeded in two stages. First, the team ran numerical simulations of intelligent tracking scenarios, and then they moved to software-in-the-loop, or SITL, simulations, in which the flight control code runs on a simulated autopilot environment closer to real deployment conditions. SITL testing is a standard bridge between pure simulation and flight tests because it exposes the controller to computational delays, discretization effects, and software architecture issues that idealized numerical models ignore. Across both verification stages, the reported trajectories of the CRAs were smooth and continuous, with excellent position response and minimal amplitude vibration, the two qualities the study set out to reconcile.

The choice of coaxial platforms for cluster flight is not arbitrary. Compared with conventional multirotors, coaxial aircraft offer higher thrust-to-footprint ratios and better tolerance to rotor damage, and recent work has explored their use in everything from micro-scale twin-rotor systems with internet-of-things control to coaxial tilt-rotor designs with finite-time and adaptive attitude controllers. Earlier studies have applied back-stepping sliding mode control to coaxial trajectory tracking, fuzzy logic methods with time-delay estimation, dynamic inversion for compound coaxial helicopters, and adaptive neural network approaches for coaxial micro aerial vehicles. The new study’s contribution is to bring these threads together at the cluster level, pairing the back-stepping structure that has proven effective for single-aircraft tracking with neural network adaptation and a decentralized formation architecture.

The implications reach beyond any single aircraft type. Formation tracking of aerial swarms underpins applications ranging from perimeter surveillance and cooperative sensing to search-and-rescue and coordinated defense missions, and the field has been moving steadily toward controllers that can handle unmodeled dynamics, switching communication topologies, and uncertain environments. By demonstrating that an RBFNN-augmented back-stepping controller can deliver both fast formation response and stable flight for a coaxial cluster, validated through Lyapunov-guaranteed stability analysis and software-in-the-loop testing, the researchers offer a template that other swarm platforms could adapt. The work was supported in part by the National Natural Science Foundation of China, the National Key Research and Development Program of China, and the National Key Laboratory on Near-Surface Detection, and the authors report no conflicts of interest. The next step for this line of research, as with all simulation-validated control schemes, will be demonstrating the same smooth, vibration-free tracking on physical aircraft flying in the open air.

Subject of Research: Decentralized adaptive formation tracking control of coaxial-rotor aircraft clusters using back-stepping and RBF neural networks

Article Title: Intelligent tracking control of CRA cluster via back-stepping and RBFNN strategy

Article References: Wei, Y., He, M., Pan, Z., Wu, H., Liu, C., & Jing, P. (2026). Intelligent tracking control of CRA cluster via back-stepping and RBFNN strategy. International Journal of Machine Learning and Cybernetics, 17(9), Article 445. https://doi.org/10.1007/s13042-026-03215-0

Image Credits: AI Generated

DOI: 10.1007/s13042-026-03215-0

Keywords: coaxial rotor aircraft, formation control, back-stepping control, RBF neural network, adaptive control, leader-follower strategy, Lyapunov stability, UAV swarm, software-in-the-loop simulation, flight control, intelligent tracking, multi-agent systems