All these issues are a matter of adjusting the autopilot coefficients to specific aircraft dynamic characteristics.
In order to avoid the "spiral into waypoint" problem, you should try and implement something like this: compute a track to the waypoint from the present position, memorize it, and subsequently correct the trajectory so as to remain on that track (not heading). It's also better to correct this by using a lateral error (distance to the memorized track, not only angle), which makes these corrections independent from the distance to the waypoint. The inputs are: aircraft track (not heading, or you have to allow for a difference between held heading and desired track), desired track, lateral error. The output is a rate of turn. The "pop waypoint" condition is more efficient this way: aircraft beyond a line orthogonal to the desired track to the waypoint, and located at a distance 'd' from the waypoint (see diagram attached). In real-life IFR, if you are flying a non-RNAV aircraft, you're actually supposed to overfly the waypoints. I know, it sounded weird to me too. Of course, no one will really scold you if you manage to nicely anticipate the turn in order to find yourself just on track to the next waypoint. RNAV aircraft are supposed to anticipate every turn so as not to overshoot airways. -- Jorge Van Hemelryck
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