One of the most significant constraints on systems designed for space is size. Everything going into space must fit inside a rocket fairing, which is intended to protect the rocket’s payload. Inside the fairing, systems like satellites and probes are secured to withstand the forces of liftoff. Traditionally, the fairing itself is considered expendable, and it can be ejected after launch to get its payload into the correct orbit.

Afterward, the fairing deorbits itself, without sacrificing the entire rocket. However, since the fairing must fit within the rocket housing, its size is constrained — and so is the size of the satellites and probes that need to hitch a ride inside it.

One trending technique among satellite designers is to engineer systems and components that fold away, making them easier to pack into the rocket fairing. These "flexible" systems must deploy correctly once released by the fairing, which can be a challenge in a microgravity environment. So how do they manage it?

What are flexible satellites?

There are three main types of flexible appendages on modern-day satellites:

• Antennas
• Solar panels
• Robotic arms

Artist's depiction of an unfolding, flexible satellite. Source: Exemplar SolutionsEach of these systems benefits greatly from the increased area they can access once they extend beyond the confines of a rocket fairing after reaching orbit. Antennas can reach different frequencies at higher signal strengths, solar panels can collect more light and robotic arms can perform additional tasks with the extra space afforded by the extension of flexible appendages.

Typically, these flexible appendages are made from specialized materials, like Kapton, which has very high thermal stability. Carbon-fiber composites are another popular choice due to their combination of stiffness and strength, despite being lightweight - a key consideration in space systems.

No matter the material, its physical properties become an integral part of the overall satellite system, especially when it comes to modeling the satellite’s operation and controlling its orbit. Every satellite has a system called the attitude and orbit control system (AOCS), which is responsible for ensuring the satellite stays in the right place and doesn’t spin out of control during its mission. However, most modern-day AOCS require two things to accurately control the direction and movement of a satellite: a reliable system model and an understanding of the original state of that system.

Unfortunately, providing these is easier said than done, especially for flexible satellites. Minor imperfections in the Kapton or carbon fiber manufacturing process could cause unexpected variances in how the satellite reacts to motion. For example, the carbon fiber of a boom might be slightly thinner in one section due to a manufacturing flaw. Most likely, the model developed by system engineers to feed into the orbital control system will not account for that slightly thinner section. Over time, this discrepancy between the physical reality and the model can lead to large overcorrections in the control system, potentially destabilizing the entire satellite.

Additionally, known changes in the satellite, such as fuel consumption during an orbital maneuver, can alter the satellite’s physical model in difficult-to-understand ways. These challenges can disrupt the control system and, in turn, threaten the satellite’s stability.

Controlling the system

A recent paper published in Acta Astronautica by Dr. Alex Elliott and his colleagues at Cranfield University’s School of Aerospace, Transport, and Manufacturing tackles this problem. The paper reviews the current method of modeling how an AOCS controls a traditional "rigid-body" satellite - essentially using a finite-element analysis (FEA) of the entire system and hoping it is accurate.

This approach doesn't work well for more complex systems, so other researchers have begun using more advanced control techniques that specialize in systems that aren’t fully understood when the original model was developed. One of the most common of these is the Kalman filter.

Widely used throughout control engineering, Kalman filters are great for estimating a system's state and handling uncertainty and noise in the feedback loop of sensors connected to the system. To do so, they incorporate prior knowledge of the system and update that knowledge based on sensor inputs. More importantly, they try to accurately predict the current state of the system and use sensor feedback to adjust their understanding. More advanced versions of Kalman filters, such as the extended Kalman filter and the uniquely named Unscented Kalman Filter, provide added adaptability for non-linear systems.

Non-linear complexity

Satellites with flexible appendages and ever-depleting fuel tanks are undoubtedly non-linear systems. Everything from solar wind to minor gravitational fluctuations can affect how the tip of a meters-long flexible solar panel moves. That movement, in turn, can change the inertia of the entire system and, if not accounted for and corrected, quickly lead to a runaway condition, leaving the satellite spinning out of control.

AOCS are designed to prevent this, but they are only as effective as their control architecture. Simple Kalman filters don’t represent non-linear systems well, and even more advanced versions, like the extended Kalman filter, struggle to accurately predict non-linear systems. This is because they are based on a technique called “linearization” — essentially, they simplify much of the system’s complexity by attempting to make it linear. While this works in some cases and is computationally less intensive, it becomes less accurate when the system is highly non-linear, as in the case of many forces acting on the tip of a flexible boom.

An Unscented Kalman Filter (UKF), on the other hand, attempts to define a variable — such as the inertia of the satellite system — by finding its mean and covariance. The mean is simply an average, while covariance describes how two variables change in relation to each other. For example, how does the movement of the satellite relate to the amount of fuel left in the tank?

To estimate these two values, the UKF uses a selection of points in a data set known as "sigma points" that best represent the mean and covariance of the variable it aims to model. This is then run through a non-linear function, and the outcome is used to determine the mean and covariance of the variable.

Modeling uncertainty

The authors of the paper are interested in finding two sets of variables: the system’s "parameters," such as its overall weight, and its "state," such as its orientation. Rather than using a single UKF, they utilize a system called a Dual Unscented Kalman Filter (DUKF). While this is more computationally intensive, the authors found it to be more accurate, which is crucial when controlling something as valuable as a satellite.

To prove their DUKF system worked, they ran simulations that confirmed the algorithm would settle on a stable answer to a complex set of system states and parameters. This gave the authors confidence that their method could be implemented in an actual mission. However, they noted that this stability only occurred if an external force was applied to some of the sensors, serving as a forcing function to help the math along.

Although the likelihood of such an external force occurring in orbit is very low, the research remains valuable and will contribute to the growing field of satellite control, especially in the area of flexible appendages. While the system was implemented in software for the purposes of the paper, DUKFs could, in theory, also be implemented in hardware, such as an ASIC or FPGA. While no such system has been attempted yet, as the control of increasingly flexible satellites becomes more difficult, it’s likely that more complex control systems will be developed to address these challenges.

Conclusion

Managing satellites with flexible appendages remains a challenge. While having flexible appendages that can extend beyond the satellite body is undeniably advantageous — especially for small satellites — ensuring that these appendages, along with their associated imperfections and uncertainties, don’t lead to catastrophic loss of control remains a difficult problem. As sensors, propulsion systems, and algorithms continue to improve, though, it’s only a matter of time before satellites can flex their antennas, solar panels, or robotic arms with confidence.