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

Question

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?

Answer 1

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.