title: physics game programming box2d - orientating a turret-like object using torques
body: <p>This is a problem I hit when trying to implement a game using the <a href="http://love2d.org" rel="noreferrer">LÖVE</a> engine, which covers <a href="http://www.box2d.org/" rel="noreferrer">box2d</a> with Lua scripting.</p>
  
  <p>The objective is simple: A turret-like object (seen from the top, on a 2D environment) needs to orientate itself so it points to a target.</p>
  
  <p>The turret is on the x,y coordinates, and the target is on tx, ty. We can consider that x,y are fixed, but tx, ty tend to vary from one instant to the other (i.e. they would be the mouse cursor).</p>
  
  <p>The turret has a rotor that can apply a rotational force (torque) on any given moment, clockwise or counter-clockwise. The magnitude of that force has an upper limit called maxTorque.</p>
  
  <p>The turret also has certain rotational inertia, which acts for angular movement the same way mass acts for linear movement. There's no friction of any kind, so the turret will keep spinning if it has an angular velocity.</p>
  
  <p>The turret has a small AI function that re-evaluates its orientation to verify that it points to the right direction, and activates the rotator. This happens every dt (~60 times per second). It looks like this right now:</p>
  
  <pre><code>function Turret:update(dt)
    local x,y = self:getPositon()
    local tx,ty = self:getTarget()
    local maxTorque = self:getMaxTorque() -- max force of the turret rotor
    local inertia = self:getInertia() -- the rotational inertia
    local w = self:getAngularVelocity() -- current angular velocity of the turret
    local angle = self:getAngle() -- the angle the turret is facing currently
  
    -- the angle of the like that links the turret center with the target
    local targetAngle = math.atan2(oy-y,ox-x)
  
    local differenceAngle = _normalizeAngle(targetAngle - angle)
  
    if(differenceAngle &lt;= math.pi) then -- counter-clockwise is the shortest path
      self:applyTorque(maxTorque)
    else -- clockwise is the shortest path
      self:applyTorque(-maxTorque)
    end
  end
  </code></pre>
  
  <p>... it fails. Let me explain with two illustrative situations: </p>
  
  <ul>
  <li>The turret "oscillates" around the targetAngle.</li>
  <li>If the target is "right behind the turret, just a little clock-wise", the turret will start applying clockwise torques, and keep applying them until the instant in which it surpasses the target angle. At that moment it will start applying torques on the opposite direction. But it will have gained a significant angular velocity, so it will keep going clockwise for some time... until the target will be "just behind, but a bit counter-clockwise". And it will start again. So the turret will oscillate or even go in round circles.</li>
  </ul>
  
  <p>I think that my turret should start applying torques in the "opposite direction of the shortest path" before it reaches the target angle (like a car braking before stopping).</p>
  
  <p>Intuitively, I think the turret should "start applying torques on the opposite direction of the shortest path when it is about half-way to the target objective". My intuition tells me that it has something to do with the angular velocity. And then there's the fact that the target is mobile - I don't know if I should take that into account somehow or just ignore it.</p>
  
  <p>How do I calculate when the turret must "start braking"?</p>