Neural Network Boost Cuts Energy Cost of Steering Deformable Atmospheric Sails

A thin film hurtling through the upper reaches of the atmosphere, buffeted by sparse but relentless streams of molecules, has long been one of the most delicate control problems in modern aerospace engineering. Atmospheric sails, large membrane structures designed to exploit aerodynamic drag in very low Earth orbit, promise a new way to deorbit satellites, adjust formations, and even breathe propulsion from the residual air at the edge of space. Yet the same gossamer structure that makes them attractive also makes them maddeningly difficult to steer. Now, a team of researchers at Beihang University, working with a collaborator at Nanyang Technological University, has unveiled a hybrid control strategy that slashes the energy needed to keep such a sail pointed in the right direction, with torque reductions of up to 55.2 percent compared with conventional methods.

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