AI algorithm to “shoot” at a target in a 2d game

Question

in my 2d game I would like to create an intelligent bot who can \"shoot\" to the player. Suppose I can pass to my bot:

actual xEnemy, yEnemy

also enemy speed a         


        

Answer 1

The solution to this problem for the "simple" case, where the gun does NOT need to rotate has been asked and answered several times. I found the best answer to be in this post (Jeffrey Hantin has a good explanation).

As Jim Mischel said, the answer to the rotating case is not even a little trivial. This is because the amount of rotation you have to do is a function of the intercept position of the target (where the collision happens). I do not know if there is a clean closed form solution to this (seems unlikely given the formulas), but I was able to solve it by working backwards and iterating to a solution.

The basic idea works like this:

Assume your "entity" will rotate from its current facing position to face the intercept position and then fire immediately.

• Pick a time of impact.
• Calculate the final position of the target at that time given it is moving at constant velocity.
• Calculate how long it would have taken the projectile to travel to that position.
• Calculate how long it would take your entity to rotate to face the intercept position.
• Calculate the difference of the impact time and the rotation/travel times.
• Adjust the time of impact up/down so that the difference gets smaller each time (e.g. binary search).
• When you get close enough, you are done. Otherwise, start the calculations again.

In the simple case, you could know from the discriminant of the quadratic if you had 0, 1, or 2 solutions and pick the best one. I don't think you can guarantee that here, but you can bound the time range you are willing to search over and how many iterations you will search. This works very well in practice.

Since I could not find the solution to this on the web, I am going to post mine. I wrote a function to handle this problem specifically. I have a blog entry on the entire calculation here. There is also a nice video showing it here.

The Code:

/* Calculate the future position of a moving target so that 
 * a turret can turn to face the position and fire a projectile.
 *
 * This algorithm works by "guessing" an intial time of impact
 * for the projectile 0.5*(tMin + tMax).  It then calculates
 * the position of the target at that time and computes what the 
 * time for the turret to rotate to that position (tRot0) and
 * the flight time of the projectile (tFlight).  The algorithms
 * drives the difference between tImpact and (tFlight + tRot) to 
 * zero using a binary search. 
 *
 * The "solution" returned by the algorithm is the impact 
 * location.  The shooter should rotate towards this 
 * position and fire immediately.
 *
 * The algorithm will fail (and return false) under the 
 * following conditions:
 * 1. The target is out of range.  It is possible that the 
 *    target is out of range only for a short time but in
 *    range the rest of the time, but this seems like an 
 *    unnecessary edge case.  The turret is assumed to 
 *    "react" by checking range first, then plot to shoot.
 * 2. The target is heading away from the shooter too fast
 *    for the projectile to reach it before tMax.
 * 3. The solution cannot be reached in the number of steps
 *    allocated to the algorithm.  This seems very unlikely
 *    since the default value is 40 steps.
 *
 *  This algorithm uses a call to sqrt and atan2, so it 
 *  should NOT be run continuously.
 *
 *  On the other hand, nominal runs show convergence usually
 *  in about 7 steps, so this may be a good 'do a step per
 *  frame' calculation target.
 *
 */
bool CalculateInterceptShotPosition(const Vec2& pShooter,
                                    const Vec2& vShooter,
                                    const Vec2& pSFacing0,
                                    const Vec2& pTarget0,
                                    const Vec2& vTarget,
                                    float64 sProjectile,
                                    float64 wShooter,
                                    float64 maxDist,
                                    Vec2& solution,
                                    float64 tMax = 4.0,
                                    float64 tMin = 0.0
                                    )
{
   cout << "----------------------------------------------" << endl;
   cout << " Starting Calculation [" << tMin << "," << tMax << "]" << endl;
   cout << "----------------------------------------------" << endl;

   float64 tImpact = (tMin + tMax)/2;
   float64 tImpactLast = tImpact;
   // Tolerance in seconds
   float64 SOLUTION_TOLERANCE_SECONDS = 0.01;
   const int MAX_STEPS = 40;
   for(int idx = 0; idx < MAX_STEPS; idx++)
   {
      // Calculate the position of the target at time tImpact.
      Vec2 pTarget = pTarget0 + tImpact*vTarget;
      // Calulate the angle between the shooter and the target
      // when the impact occurs.
      Vec2 toTarget = pTarget - pShooter;
      float64 dist = toTarget.Length();
      Vec2 pSFacing = (pTarget - pShooter);
      float64 pShootRots = pSFacing.AngleRads();
      float64 tRot = fabs(pShootRots)/wShooter;
      float64 tFlight = dist/sProjectile;
      float64 tShot = tImpact - (tRot + tFlight);
      cout << "Iteration: " << idx
      << " tMin: " << tMin
      << " tMax: " << tMax
      << " tShot: " << tShot
      << " tImpact: " << tImpact
      << " tRot: " << tRot
      << " tFlight: " << tFlight
      << " Impact: " << pTarget.ToString()
      << endl;
      if(dist >= maxDist)
      {
         cout << "FAIL:  TARGET OUT OF RANGE (" << dist << "m >= " << maxDist << "m)" << endl;
         return false;
      }
      tImpactLast = tImpact;
      if(tShot > 0.0)
      {
         tMax = tImpact;
         tImpact = (tMin + tMax)/2;
      }
      else
      {
         tMin = tImpact;
         tImpact = (tMin + tMax)/2;
      }
      if(fabs(tImpact - tImpactLast) < SOLUTION_TOLERANCE_SECONDS)
      {  // WE HAVE A WINNER!!!
         solution = pTarget;
         return true;
      }
   }
   return false;
}