mathematics - Drone targeting

Imagine a "drone" and a target point on a 2d plane. There are eight parameters:

P = my position
Q = target position
V = my velocity
I = my moment of inertia
w = my angular velocity

s = my angular position
T = max thrust
U = max torque

(we'll just say the target is stationary)

The drone's job is to get to the target as fast as possible, obeying max torque and max thrust. There are only two ways to apply the torque, since this is only in a 2d plane. Thrust is restricted to only go in one direction relative to the orientation of the craft, and cannot be aimed without rotating the drone. Neglect any resistance, you can just pretend it is floating around in 2d outer space. Let's say the drone checks an equation at time interval t (maybe something like every .01 seconds), plugs in the parameters, and adjusts its torque and thrust accordingly. What should the equations for thrust and torque be?

Answer

In light of your question's context, http://nodewar.com/, there are a couple specific considerations for your solution:

1. You have a (low) maximum angular velocity, and enough maximum torque to reach it in very short time.



2. Your drone and target each have velocity and external acceleration unrelated to thrust (gravitation abounds).


3. Your desired target changes so frequently that trying to aim perfectly would be a waste. You should try to get close, and correct it every frame.

These methods are what I determined to work for reaching a desired acceleration.

Acceleration, not velocity

Because you have a given velocity already, and your target is moving, you don't need thrust toward a point. You need thrust to change your velocity to what it should be. This means that your ship needs to point not toward where it is going, but in the direction in which it should accelerate.

// My target velocity is for maintaining a circular orbit.  Yours may differ.
// Earlier, I calculated total gravity and the perpendicular direction.
// You may wish to subtract gravity from your total, rather than match it.
var targetVel = o.lib.vec.times(lateralDir, targetVelMag);


var targetAccel = lv.sum(
  o.lib.vec.diff(targetVel, o.me.vel), 
  o.lib.vec.times(gravity, 1 / o.me.mass)  
);

Steering toward the right heading

You have an acceleration vector, now you want to apply it. Determine how far you need to rotate. I probably used more steps than required here, but rotational coordinates confound me, and I think the uncapped ship rotation value is a bug in the API anyway.

// convert acceleration to an angle
var polar = o.lib.vec.toPolar(targetAccel);

var traj = polar[1];

// constrain the angle to +/-2PI, because the ship's rotation is not limited 
// by default
var fixed_rot = o.lib.ang.rescale(o.me.rot);

// limit the correction to be +/-1PI
var traj_correction = traj - fixed_rot;
if (traj_correction > (Math.PI)){
  traj_correction = (2 * Math.PI) - traj_correction;

} else if (traj_correction < (-1 * Math.PI)){
  traj_correction = (2 * Math.PI) + traj_correction;
}

A simple formula. There is no harm in turning all the time, so don't bother applying partial torque values. If you need a small correction in angular velocity, you get to make this determination many times per second, anyway.

if (traj_correction > 0){
  torque = 1;
} else if (traj_correction < 0){
  torque = -1;
}

A less simple formula. There will come a point where you don't want to continue turning, because you eventually want to stop. Fortunately, that angular velocity cap means that you can rapidly slow down from max angular speed to zero. You only need to calculate when to do so.

var max_a_accel = c.MAX_TORQUE / o.me.m_i;
var a_deccel_time = Math.abs(o.me.a_vel) / max_a_accel;
// the same math as linear acceleration, now in angles.
var stopping_angle = 0.5 * max_a_accel * a_deccel_time * a_deccel_time;


if (stopping_angle >= Math.abs(traj_correction)){
  // slowdown required.  Reverse torque

  torque *= -1;
}

After tweaking the code above to suit your needs, your ship should rapidly and precisely rotate to whatever angle you gave it to target.

Ramming speed

So, when to thrust? Again, the rapid change of the target and other factors create great difficulty in resolving an exact solution. Don't try to.

// if the heading is close to the final value, thrust.
if (Math.abs(traj_correction ) < 0.02) {  // about 1 degree
  if (true 
      // some logical test, in case you don't want to accelerate past

      // a maximum speed, or some such.  Not required for your stated purpose.
     ){
    thrust = 1;
  } 
}

For those cases where you need partial thrust, you can again rely on the fact that you can choose between 0 and 1 thrust many times per second. This gives you an effective partial thrust without varying the actual value.

Good luck!