Geared DC servo hunts back and forth around the target position: backlash, stiction or tuning?

That pattern is a limit cycle caused by a nonlinearity inside the loop, and no set of linear PID gains removes it. With the encoder on the output, the backlash sits between the motor and the sensor. While the gear teeth cross the gap the motor turns but the arm does not move, so the encoder reports no change and the integrator keeps ramping. When the teeth finally meet, the accumulated command throws the arm past the target, the error changes sign and the same thing happens in the other direction.

The clue is exactly what you observed: gain changes alter the frequency and amplitude, but the oscillation persists as long as the integral term is active.

Static friction gives an almost identical symptom, so separate the two before fixing anything. With stiction, the integrator ramps until the torque breaks the arm free, friction then drops to its lower sliding value and the arm jumps past the target.

Two quick tests: hold the motor shaft still and rock the arm by hand to feel the free play, which is backlash. Then command a very slow ramp and log the position; a staircase of stops and jumps points to stick-slip. If you can log the controller output, a sawtooth in the command while the position sits still confirms that the integrator is winding against something that is not moving.

On the control side, the usual fix is to stop the integrator from working inside the dead zone. Freeze the integral term, or the whole output, when the error magnitude is below a small band, for example 0.3° for your 1° of backlash, and only integrate outside it. You accept a small resting error in exchange for a quiet axis. A slow leak on the integrator has a similar effect.

Also check the sensor resolution. A 10-bit absolute encoder gives 360 / 1024 = 0.35° per count, so the loop cannot hold anything finer than that, and a derivative term acting on single-count flicker will kick the motor at rest.

The more robust structure is dual-loop feedback: close a velocity loop on a sensor at the motor shaft, where there is no backlash between actuator and sensor, and close only the outer position loop on the output encoder with proportional gain and little or no integral. The inner loop gives the damping, the outer loop gives the accuracy, and the backlash is no longer inside a high-gain path. Many motor drives support this directly as dual-encoder or load-feedback mode.

If only one sensor is possible, putting it on the motor makes the loop stable but leaves the backlash as an uncorrected position error at the arm.

Mechanical fixes remove the cause instead of hiding it. A constant bias on the arm, from gravity, a spring or a small counterweight, keeps the teeth loaded on one flank so the gap is never crossed near the target. Anti-backlash gears, a belt stage or a low-backlash planetary or harmonic gearbox do the same job at higher cost.

Approaching every target from the same direction is a cheap trick used on positioning stages: overshoot the command slightly on moves in one direction and come back, so the final approach always takes up the play the same way.