How does desaturation of the reaction wheels work?

Question

I'm having a little trouble understanding the 'principle' behind the desaturation process. The aim of desaturation is to reduce the the speed of the Reaction Wheel. My understanding is that I can't just turn off or reduce the speed of the wheel because the momentum would merely transfer to the spacecraft. We could counter this torque on the body due to the reaction wheels, if we apply a counter torque using another actuator such as the magnetorquer. However on reading more on desaturation in books and papers, there seems to be no mention for the need to manually slow down the reaction wheel speed and instead merely applying an external torque has the effect of reducing the wheel speed. This confuses me because I thought that the wheel speed only depended on the current, which is something we supply and hence control. How does the application of an external torque such as the magnetic torque, slow down the reaction wheel ?

Answer 1

"[T]here seems to be no mention for the need to manually slow down the reaction wheel speed and instead merely applying an external torque has the effect of reducing the wheel speed. This confuses me because I thought that the wheel speed only depended on the current, which is something we supply and hence control. How does the application of an external torque such as the magnetic torque, slow down the reaction wheel?"

The unstated assumption here is that you never want your spacecraft to tumble, so you're always running an attitude control system that transfers any unwanted angular momentum from your spacecraft to the reaction wheels, spinning them up or down to keep the spacecraft pointed in the direction it should be pointing.

So when you find your reaction wheels approaching their speed limits, what you do is determine the direction of torque needed to slow them down and then (using some external source of torque, like attitude control thrusters or magnetotorquers or whatever) apply that torque to your spacecraft and let your attitude control system automatically transfer it to the wheels.

If you like, you can think of it as slowing down the wheels — which will transfer momentum from the wheels back to your spacecraft, and send it tumbling unless corrected — and then correcting that tumbling using some external torque source/sink. But it's probably more useful to instead think of it the opposite way, as first applying an appropriate external (counter)torque to your spacecraft and then letting your attitude control system adjust the reaction wheel speed to keep your spacecraft steady and, as a side effect, also slow the wheels down.

One reason to look at it this way is that, usually, whatever external attitude control mechanism you're using won't be as precise or as quickly reactive as the reaction wheels. (That being one of the major reasons for having reaction wheels in the first place.) So if you were to design your desaturation routine to first slow down the wheels by some arbitrary amount and then try to apply an external countertorque to cancel it, you'd likely find that the countertorque wouldn't exactly match the torque from slowing down the wheels and you'd end up having to adjust the wheel speeds again to correct the mismatch anyway.

Answer 2

Reaction wheels are used to control attitude of the spacecraft and they work due to conservation of momentum. Let's say we have a reaction wheel on the x-axis of the space craft, aligned with the principle axis of inertia, then the total moment of inertia is

$h = h_s + h_w$,

where $h_s$ is the moment of inertia of the spacecraft (excluding the wheel) and $h_w$ is the moment of inertia of the reaction wheel. In absence of external forces, momentum is preserved and the time derivative of the total momentum is zero:

$\dot{h} = 0 \Rightarrow \dot{h}_w + \dot{h}_s = 0 \Rightarrow \dot{h}_w = -\dot{h}_s$.

In other words: if the momentum of the reaction wheel increases, we get the opposite reaction in the change of momentum of the spacecraft. In particular, recall that $h = I \omega$, with $I$ the moment of inertia and $\omega$ the rotation speed and we see that spinning the reaction wheel in one direction results in the spacecraft spinning in the opposite direction. More precisely:

$\omega_s = - \frac{I_w}{I_s}\omega_w$.

Basically, if you want to change the attitude of the spacecraft, you spin up the reaction wheel to get the spacecraft rotating, and then spin it the other way to make it stop. At the end of the manoeuvre both the spacecraft and the reaction wheel are at zero speed again.

A saturated reaction wheel is a wheel that is spinning at its maximum speed. This can occur if there is an external disturbance that changes the momentum balance:

$\dot{h} = \tau$,

where $\tau$ is the external disturbance force. To make keep the space craft attitude, you need to counter this force (torque) with the reaction wheel. But:

$\dot{h}_s = \frac{d}{dt} I_s \omega_s = I_s \dot{\omega}_s$,

the reaction wheel must accelerate to create a counter-torque. At some point the wheel will have reached it maximum speed and is said to be saturated. It can no longer provide a counter-torque (because it can no longer accelerate) and the spacecraft attitude will be disturbed.

To get out of this, you need to desaturate the wheel, i.e. removing momentum from the wheel so it has "room" to accelerate again if needed. One way to do this is to indeed use a magnetorquer: these devices use the Earth's magnetic field to introduce a deliberate external torque, so that the reaction wheel can use this to "offload" momentum.

This confuses me because I thought that the wheel speed only depended on the current, which is something we supply and hence control. How does the application of an external torque such as the magnetic torque, slow down the reaction wheel?

The wheel acceleration depends on the current, not the speed. The application of the magnetic torque itself does not slow down the reaction wheel, but it provides some room in the momentum balance to reduce the momentum (and thus the speed) of the reaction wheel:

$\dot{h} = \tau_m$,

with $\tau_m$ the magnetic torque. If you make $\tau_m$ such that $\tau_m = -K h_w$ for some suitable value of $K$, the wheel can spin down without affecting the space craft attitude.

Answer 3

This is not a high level answer but I think I understand what it is that you are asking, and if I'm right it doesn't need much to clear up.

My understanding is that I can't just turn off or reduce the speed of the wheel because the momentum would merely transfer to the spacecraft. We could counter this torque... if we apply a counter torque using another actuator such as the magnetorquer.

That's basically right I think except for some rough wording.

However on reading more on desaturation in books and papers, there seems to be no mention for the need to manually slow down the reaction wheel speed and instead merely applying an external torque has the effect of reducing the wheel speed. This confuses me because I thought that the wheel speed only depended on the current, which is something we supply and hence control. (emphasis added)

Don't feel confused, you're still basically right, except think of it as "automatic" rather than "manual".

How does the application of an external torque such as the magnetic torque, slow down the reaction wheel ?

As mentioned in this comment:

...the key to your question is that the spacecraft is assumed to be actively maintaining its attitude with the reaction wheels, so that when a torque is applied the spacecraft senses this and automatically slows the wheels to compensate via electrical signals to the motor controller.

That you would never intentionally let your spacecraft lose attitude control and keep it running all the time (when possible) is such an intuitive thing to both the author and the reader that it wasn't mentioned explicitly.