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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’

enter image description here

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.

enter image description here

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.

enter image description here

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?

uhoh
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Woody
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  • @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. – Woody Dec 03 '22 at 16:19
  • What do you mean by "state vector"? – mr_e_man Dec 25 '22 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. – Ryan C Jan 02 '23 at 17:48

1 Answers1

4

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.

enter image description here

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.

enter image description here

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.

user721108
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  • 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? – Woody Dec 04 '22 at 01:31
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    @Woody this Q&A is about the spherical case – user721108 Dec 04 '22 at 01:50
  • Yes, and the answer demonstrates that the particular layout of 3 thrusters would not work to stop all types of spin. – Woody Dec 04 '22 at 02:22
  • 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. – mr_e_man Dec 23 '22 at 05:14
  • @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. – user721108 Dec 24 '22 at 07:40
  • Maybe we're talking about different things (e.g. orientation vs. angular velocity, or inertial frame vs. rotating frame). Anyway, I thought about it some more, and I think two thrusters can give complete control: Any combination of position, linear velocity, orientation, and angular velocity can be achieved (at some unspecified time). Here's one possible configuration: Put two thrusters on opposite sides of the intermediate axis, pointing in the same direction, perpendicular to that axis, but not aligned with the other axes. – mr_e_man Dec 25 '22 at 23:30
  • @mr_e_man if i get your configuration right, it's like having a cluster of two thrusters (same origin) at one end of intermediate axis, one thruster providing yawleft+pitchup, the other providing yawright+pitchdown, both together providing frwd thrust along intermediate axis. Using precession and intermediate axis and patience properties allows full control, this sounds valid to me – user721108 Dec 26 '22 at 14:01
  • That works as well, though it's not what I was trying to describe. Say the intermediate axis is y. Locate one thruster at $(0,1,0)$ and the other at $(0,-1,0)$. Point both thrusters in the direction $(1,0,1)$. – mr_e_man Dec 26 '22 at 14:41
  • @mr_e_man I think I get what you describe and your configuration also allows 100% translational efficiency since both thrustrers have parallel thrust vector. – user721108 Dec 26 '22 at 15:16
  • in the added picture your configuration is the orange one – user721108 Dec 26 '22 at 15:48
  • An intermediate axis is not necessary; the rectangle could instead be a square or a disk. – mr_e_man Dec 26 '22 at 16:18
  • Let us continue this discussion in chat. – user721108 Dec 26 '22 at 16:52