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This concise answer to What exactly is rhumb-line control in the context of a launch trajectory? explains the concept in the context of the SS-520 nanosatellite launch vehicle (which "doesn't exist"). The answer starts with:

"Sailing a rhumb line” means holding a constant compass bearing.

The SS-520 is a modified sounding rocket using passive spin stabilization through much of its initial launch trajectory.

Question: How do small, spin stabilized launchers follow a rhumb line? Do they use a real compass or something else? Without thrust vectoring (hard to use on a rapidly spinning rocket) how to these vehicles control the direction of flight?

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In theory, a rocket could "follow a rhumb line" starting with any kind of directional information. But there seem to be two use cases:

The very-early Pegasus rockets used a directional gyro derived from an aircraft gyrocompass, perhaps due to their aircraft parentage. This was later updated to a full IMU, eventually augmented to GPS.

Another approach is to use horizon and Sun sensors to derive pitch and yaw angles. Although this adds some launch constraints (the right time to have the Sun in the right place; clear skies), it's also really easy on a highly spinning rocket because the spin will do half of the scanning for you. The SS-520-5 is an example of this (although it also carries an IMU for upper stage guidance). That's meant to be a low-cost rocket with big weight challenges, so a simple system is desirable.

On a highly spinning rocket, the spin turns a +pitch thruster into a +yaw, -pitch, and -yaw thruster as the rocket turns. So long as the thruster is fast enough to fire during just part of a single turn, you don't even need to have separate ones for the two different directions.

Your control equations do need to take the rocket's angular momentum into account; in the limit of very high spin, the rocket's a gyroscope, so you have to apply torque 90 degrees before the angular change you want. In that limit you also get some self-stabilization, but applying pulses of torque also leads to nutation. Although small amounts of nutation average out in the final trajectory, the control equations have to keep it in bounds.

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