What is the minimum number of RCS thrusters capable of stabilizing a satellite against an arbitrary rotation?

In case of a very specific rotation, even one suffices, but it must be located just right for that specific axis of rotation, and if you want to rotate the satellite, it can do so in that specific axis only, and one direction only.

So, first, assumptions:

• RCS is the only method of rotational stabilization. If there are any gyroscopes, magnetorquers etc, we don't take them into account.
• we don't care about translation.
• the thrusters are not gimballed, vectorable or anything like that - each provides thrust in one direction only.
• but the moment and power at which they are fired can be very precisely controlled. No negative thrust though, that requires a second thruster.
• we also have a good information about the attitude changes from the sensors, but the spin/tumble we have entered is arbitrary (within reason, not destructive).
• for simplicity let's assume a spherical satellite with all mass concentrated in the center, massless thrusters, they may be installed on (massless) trusses extending from the satellite, if need be.

The cluster of three thrusters can stop any spin/tumble except one going straight through their axis of symmetry (the blue axis). This can be negated by the two thrusters on the "equator".

I have a strong suspicion this can be done with four thrusters though - for example, offsetting the three away from the "pole", that they create a "turbine" pattern, always creating a "polar" spin next to their normal activity, and remove one of the "equatorial" ones.

I wonder if it can be done with even less, or what other layouts would be beneficial.

• I suspect four might work too - put them at three equidistant positions on the "45 degree latitude" position. Two will be firing South, and the third position will have two thrusters, offset by an angle so they fire South East and South West. Can't prove this though. (Still thinking very hard about just three; I haven't the mathematical Nouse to prove it.) – Andy Mar 11 '16 at 15:37
• Also see this (MSL entry module has only four thruster directions - though not entirely clear if this allows full three axis control): solarsystem.nasa.gov/docs/p453.pdf – Andy Mar 11 '16 at 16:23
• @Andy: That looks nice. The setup would be awfully weak against a rotation in this axis (marked pink) but that could be still overcome by offsetting the thrusters farther outwards. Plus this obviously is self-stabilizing in the atmosphere, and suffers heatshield-related compromises. – SF. Mar 11 '16 at 17:38

Your suspicion that it can be done with 4 is correct, and the method you suggest will work.

More formally, to prove that a set of thrusters allows arbitrary torque is equivalent to showing that you can produce torque in each direction about the 3 orthogonal axes shown.

The ones drawn allow +blue and -blue (the two equatorial thrusters), +green (the one at the top pointing opposite to the red direction) and -green (firing the other two at the top equally). We can then get +red or -red by firing one of these two together with enough of the first one to counteract the -green torque that they also deliver.

In other words the torques from the thrusters are:

• +B
• -B
• +G
• -aG + bR
• -aG - bR

(If the thrusters are identical, symmetrically arranged and equidistant from the centre of gravity then a and b are 1/2 and $\sqrt3/2$, but the proof works as long as the two thrusters are symmetric, whatever the values of a and b are.)

If you offset each of the top set by moving each one to the right, each one will also create some +blue torque:

• +G + cB
• -aG + bR + cB
• -aG - bR + cB

Those 3 are still enough to produce any torque you like about the G or R axis (in either direction) but now they will also produce an uncontrolled amount of +blue torque. You can always produce more +blue torque (just increase the thrust of all 3 equally) but never reduce it just by firing these three. However, we now only need one more thruster, -B, to get arbitrary torques in all directions.

• I have a feeling that a tetrahedral arrangement might be optimal. Imagine a thruster at each of the four corners, and each aimed parallel to a unique adjacent face. – Caleb Hines Mar 12 '16 at 2:01
• Is a simple drawing or sketch possible? I can't picture the 4 thruster solution in the question. Thanks! – uhoh Mar 12 '16 at 4:49
• Yes, I agree that a tetrahedral arrangement would probably be most efficient, at least for providing torque about any arbitrary direction. – djr Mar 12 '16 at 12:27