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
- 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?