What is the minimum number of thrusters required to control attitude, propulsion and spin?

What is the minimum number of thrusters required to control attitude, propulsion and spin?

The first part of this problem (attitude) was asked in What is the minimum number of RCS thrusters capable of stabilizing a satellite against an arbitrary rotation? The OP thought it was 5. One answer suggested 4. But attitude control can be acheived with 3

Some definitions:

• Thrusters are fixed
• Spacecraft is rigid
• “Attitude” means direction of the state vector, absence of angular momentum, and disregarding imparted translation
• “Propulsion” is a change in the length of the state vector
• “Spin” or "tumbling" means significant rotation around any given axis, as in insufficient de-spin following spin stablization.

The sketch below shows a spacecraft with a single cluster of 3 thrusters. Opposite sides of the craft have the same colors, but are distinguished by A,A’ B,B’ and C,C’

By using combinations of one or two thrusters, rotation can be achieved around the blue faces or the pink faces (B-B’ and C-C’ axis).

Rotation around the green faces is trickier, but can be done similar to the way a cat lands on its feet. Below, using 3 steps, the blue face B is rotated from the side the the zenith.

The 3 thrusters could be angled into a tetrahedral shape, as in the sketch below. They retain the ability to change attitude as before. But now, if fired together, they (inefficiently) generate thrust through the COM and can therefore produce translational thrust as well as attitude control.

So, the tetrahedral arrangement of 3 thrusters can provide attitude control and propulsion. But what about control of tumbling (spin)? If the axis of rotation is around the pink or blue faces (the B-B’ and C-C’ axis), the thrusters could handle it. But if the craft tumbles around the green faces (axis A-A’), it will have problems. With any attempt to change the orientation of the spin axis, precession will keep the A-A’ faces aligned with the spin axis. It is similar to holding a spinning bicycle wheel by the axle. No matter how you twist, you can’t slow the spin.

So, what is the minimum number of thrusters needed to control all axes of spin as well as attitude and propulsion?

• @qqjkztd ... low speed yaw, pitch and roll don't need to deal with gyroscopic precession. It's like the difference between a stationary bicycle and one which is moving. Try the spinning-bicycle-wheel-in-your-hands thing. The difference is apparent. Dec 3, 2022 at 16:19
• What do you mean by "state vector"? Dec 25, 2022 at 23:02
• As a general design criterion, you want to launch a spacecraft with lots more than the minimum number, so that when they start to fail on orbit, you still have enough left working to stay in operation for a while. Thus I find the more useful and interesting question to be, for some chosen margin of safety, what is the optimal arrangement of N thrusters such that if any K of them cease to function, the remaining ones are enough to keep control. Jan 2 at 17:48

There is indeed no control about AA' axis (let's call it roll axis since it's aligned with thrust vector).

The cat technique works well, but still cannot start or stop roll relative to internal frame of the craft. (Yet it can, as visible in your illustration, transform one roll attitude into another relative to distant stars.)

Using precession to change attitude works for pitch and yaw with adequate 90° anticipation, but won't slow down or accelerate roll rate easily.

One way to go around this could be to hinder the spacecraft ability to roll in the first place, using the intermediate axis theorem by aligning your thrust vector with the intermediate axis of rotation,

so that any roll perturbation always decomposes by itself into yaw and pitch, that the thrusters can handle.

This way attitude control and propulsion can be achieved using a cluster of three thrusters in tetrahedral arrangement, if the spacecraft looks like a Kubrick's monolith, and the cluster is placed on the correct face.

Having control on two axis only is enough to steer in any direction in space, but the third axis has to be somehow stable or dampened so that perturbations on this axis wont spin the craft forever.

Following @mr_e_man comment, two thrusters options could work using both precession and intermediate axis effects at its advantage to ultimately have full attitude control.

• Nice point about the intermediate axis. But this answer requires restriction on the structure of the craft. How about for the general case, a roughly spherical craft? Dec 4, 2022 at 1:31
• @Woody this Q&A is about the spherical case Dec 4, 2022 at 1:50
• Yes, and the answer demonstrates that the particular layout of 3 thrusters would not work to stop all types of spin. Dec 4, 2022 at 2:22
• Last paragraph - Having control on two axes can give indirect/delayed control on the third axis, by precession. It's possible to stop any spin using just two thrusters (as shown in my answer that you linked). But it's not clear to me whether spin and attitude can both be controlled with two thrusters. Dec 23, 2022 at 5:14
• @mr_e_man control on two axis gives indirect control on the third by decomposition, (+yaw, +pitch, -yaw = change in roll). Spin cannot develop around intermediate axis, so thrusters don't need to be able to counter it. Dec 24, 2022 at 7:40