# Why don't reaction wheels destabilize spacecraft over time?

Let's consider a hypothetical situation where a communications satellite is launched into orbit. However, no launch is perfect so after burnout the satellite has picked up some roll. In order to correct this roll, the reaction wheels that the satellite is equipped with are used to apply the appropriate torque necessary to negate the rotation of the spacecraft. This is done by rotating the reaction wheels in the opposite direction to the rotation of the spacecraft, until enough angular momentum is applied that the spacecraft's rotation is nullified.

However, in this situation the reaction wheel would still be spinning even when the angular velocity of the craft is corrected. There are a couple problems here...

# If we assume friction is negligible...

If we assume friction is negligible or non-existent, then the reaction wheel which is now spinning will continue to spin until the reaction wheel is powered up again (as per Newton's first law of motion). Now lets say that due to external factors the spacecraft picks up more rotation in the same direction as before, and the reaction wheel must be sped up further in order to counter this. Eventually, this cumulative application of angular force to the reaction wheel will speed up the reaction wheel so fast that it goes beyond its limits of structural integrity - which is bad.

# If we consider friction...

If we consider friction to be a factor in the aforementioned hypothetical situation, then the reaction wheel will begin to slow down as the force of friction is applied on it. However, this force of friction would not only be applied to the wheel, it's equal and opposite reaction would also apply a rotation to the spacecraft and destabilize it, making it necessary to use the reaction wheels even more...

Is it true that reaction wheels are not, on their own, capable of stabilizing a spacecraft? If not, why not?

• They would. That's why momentum is desaturated with something like torque rods or thrusters. Commented Dec 2, 2015 at 6:14

To answer your question "Is it true that..." then it is best to understand the context.

The reaction wheel will be in a loop with a sensor that detects one or more dynamic properties of the satellite, such as an angle through a Earth or star sensor or a rate from a gyroscope. There are also likely to be some "external" actuators in a related control loop, such as thrusters or magnetorquers.

As you pointed out if a disturbance torque is continually applied then the loop will command the reaction wheel to run faster until its speed saturates. There is a choice, long before this point is reached, to prevent saturation by off-loading the momentum stored in the wheels via one of the external actuators.

In the case of an Earth pointing mission some external torques are periodic over the satellite orbit, some not. Depending upon the application, the wheels and control-loop may be sized so as to take up and give back the periodic disturbance in time with its periodicity (i.e. net speed change over a cycle = 0) and to only go to the bother of off-loading with a thruster where the buildup is continual.

To address your point about friction, there is some and it occurs between the satellite structure and the wheel. Putting a little energy into keeping the wheel speed constant has just that affect, by definition using "the reaction wheels even more..." just keeps them at their commanded speed.

• The main point of my question that perhaps didn't come across is "where does the energy go?". The application of force from the reaction wheels results in the stabilization of the spacecraft, but also the continual motion of the reaction wheels. So does this mean that reaction wheels are constantly spinning at all times, and if so wouldn't this lead to problems as described in my answer about buildup of rotation and eventually the RW exceeding its structural limitations? Commented Dec 2, 2015 at 1:54
• @kimholder both are valid points - however I know that it is not practical for thrusters to be used for general attitude control on their own, since the use of finite fuel is at a big cost and wouldn't be used for regular attitude control. Also, torque applied by things such as radiation pressure, gravitational gradient, atmospheric drag, et cetera wouldn't be controllable enough to fully facilitate attitude control (except in Kepler's case). What I was trying to figure out is how RWs are practical despite the issues I wrote about, and those solutions aren't really practical on their own Commented Dec 2, 2015 at 2:22
• The ISS uses control moment gyros but when they get saturated they must be unloaded by firing thrusters. This is a practical solution that is used in the real world. Commented Dec 2, 2015 at 5:16
• The basic shortcoming of the thrusters is that their thrust isn't fully controllable. There are ignition/flameout inaccuracies, there is flow inaccuracy and so on. They can't be used for fine attitude control. But they are still good enough to desaturate the wheel and bring it down from a few thousand RPM down to a couple RPM. Then you can use the reaction wheel to fine-tune the craft's attitude with precision impossible with RCS thrusters alone. And yes, fuel for RCS is one of limiting factors of lifetime of a probe.
– SF.
Commented Dec 2, 2015 at 8:01
• It's a bit easier with satellites: magnetorquers, while much weaker, don't run out of fuel but they need Earth magnetic field to work. So, they can get the reaction wheel down to flat 0 speed with the satellite stabilized. And once stabilized, the satellite tends to remain stable, Earth's tidal forces acting in a stabilizing way. Probes have it worse without magnetic field to push against or tidal forces to stabilize them - they must depend on the RCS thrusters and their fuel reserve to desaturate RWs.
– SF.
Commented Dec 2, 2015 at 8:05

I am just an enthusiast but I did work for a time as an intern at an aerospace company and had some exposure to satellite designs, mostly from the attitude control system programming viewpoint.

That particular satellite had reaction wheels, magnetic bars and thrusters.

The magnets were very useful to press against the Earth's magnetic field and allow the reaction wheels to slow down.

Thrusters were used for extreme situations when the rotation rate is very high. That could be a bad separation from the bus or a micrometeor/debris strike or even a bearing freeze on a wheel. This was aerospace engineering: every possible failure was on a checklist. At any rate, thrusters were the last possible choice since every use was a big hit to mission time. There's nowhere to get more fuel out there.

Depending on the spacecraft and its orientation requirements, reaction wheels can be used together to slow down and spread the rotation rate of the wheels. The spacecraft can spin around so that the wheel can begin spinning the other direction. Also, wheels have gyroscopic effects which have to be managed, and these can also be used to counteract spin of another wheel.

• You are referring to a magnetorquer at the beginning there - en.wikipedia.org/wiki/Magnetorquer . you could add the link as a reference ;). Commented Dec 2, 2015 at 2:15
• Very interesting point about how by changing the attitude of the craft the reaction wheels can be used in reverse to counter rotation, and the bit about the gyroscopic effects of the wheels - this is exactly what I was looking for Commented Dec 2, 2015 at 2:16

Reaction wheels can store angular momentum up to a certain limit. If you Fourier analyze the torques on the satellite, some are constant and some are sinusoidal. The reaction wheels can store the sinusoidal part and deliver it back when the torque is the other direction. They don't help (in the long run) for the constant torques. For that, you need either thrusters (which use fuel) or magnetic torquers (which react against the earth's field). Both of these can deliver torque to the satellite that does not need to be stored. You can generally assume the friction on the wheels is negligible-you overcome that with a small amount of electrical power. Generally you size the reaction wheels to cover the worst case sinusoidal torque, then deal with the secular torque some other way. If you use thrusters, this is one entry in the propellant budget, which can be the limit to the lifetime of the satellite.

Reaction wheels are fully capable of stabilizing the spacecraft from small perturbations such as gravity gradient torque, atmospheric drag and solar radiation pressure, however coming to your initial scenario after the orbital insertion, it is quite difficult to use reaction wheels to stabilize. For that we use passive control such as yo-yo despinning or libration damper.

Reaction wheels having a zero initial angular momentum will take relatively more time to destabilize that, thus have a less preference. An alternate to them would be to use Inertia wheels that have some initial angular momentum which can be pre-adjusted before launch to have maximum effect.

Let's talk about the case of no-friction

You had a small disturbance which activated the reaction wheels, once the satellite is stabilized and the wheels are turned off they will keep rotating infinitely, but there is a limit to the maximum angular momentum that can be reached after which they are no longer effective thus not giving any output torque and this would require some momentum dumping mechanism to operate. This is where thrusts come into picture. Once they are fired in the opposite direction, the wheels gets desaturated and are now back in action, but since they are still rotating in the same direction, this will cause the satellite to rotate along with it, which to the system is fed as a perturbation in the opposite direction as before thus sending the signal to the controller to rotate the wheels in the opposite direction. Thus a satellite will never be at zero position but will move about an equilibrium point with a prescribed amplitude and frequency that is predetermined considering the case you mentioned.

In your scenario after the momentum dumping if there is another perturbation in the same direction as before that would activate the wheels again but the system is aware of its current angular velocity and the velocity needed to counter that perturbation so it will not necessarily accelerate the wheels, rather it might slow them down instead, because there is also a limit to the maximum ang. vel. that can be achieved.

There is an alternate approach to this which involves configuring the wheels not along the axes but like a pyramid with 4RWs such that each wheel corresponds to some contribution in control, thus further reducing the harmonic motion.

As a control engineer it's your job to minimize this harmonic motion. And thrusters are the worst ones requiring a robust non-linear controller.