Given a 3d goal point and a turn rate in radius, find max speed to reach goal

Question

I'm working on a 3d space sim with AI agents flying ships through space. I am trying to get them to follow a waypoint path nicely, which involves figuring out how fast they can be travelling and still manage to pass through the waypoint.

Given a point in 3d space and an an agent that only travels forwards and has a given turning rate in radians / second and an acceleration rate, how can I find the maximum speed the ai can travel and still manage to face the point before overshooting it?

Can any math wizards lend a hand?

public class AIPilot { //relevant fields from my class
   float turnRate_radians;
   float acceleration;
   Transform transform;

  public float FindMaxSpeedToFace(Vector3 goal) {
    //trying to fill the body of this method
  }  
}

Answer 1

Is it critical that your function be "What is the max speed"? Finding a closed-form calculation for that is a bit hairy.

It seems like a more useful/general-purpose function might be "Can I keep accelerating and still face the target point?"

You already know from your previous answer from zehelvion how to calculate whether you can reach a target given a your current velocity and max turn speed.

So, assuming each time step you are turning towards the target as fast as you can, then you can also calculate whether additional acceleration will make it impossible to face the target. If not, go ahead and accelerate.

The max speed you arrive at is your original answer, but now you maybe don't need it. :)

(I.e., using this approach your AI agent should arrive at the point as fast as possible given its physical constraints.)