Hinge joints not reaching target position

I have a hinge that has the following settings:

Rigidbody:

Mass = 1 Ang Drag = 0 Constraints: frozen on all but Y-axis

Hinge Joint:

Spring = 1 Damper = 0.001 Target Position = 0

...

The problem being the hinge joint fluctuates around the target position yet does not stop at 0, but instead +- ~6 degrees. Turning up the spring helps, but fails aesthetically.

Also, in physics I turned sleep angular velocity to zero, which did not help.

I also tried switching the damper to 0 and angular drag to 0.05 (the default), which worked no better.

I wanted to script the hinge myself until I read changing a rigidbody's angular velocity can yield unrealistic results.

[edit:]

Initially, I planned not to share my solution script, but after having to rewrite it, I decided to share it. This script is meant for a western style saloon door, which reacts to the player's movement. Increasing the spring in the HingeJoint and the script will make the door swing faster (and vice-versa) and increasing the damper will make it slow down faster. (I put damper at .1 and spring at 1 for a nice slow western feel). Furthermore, the mass of the player should be at least a magnitude of 3 (ie 1000x) greater in order for the door to perform optimally without getting in the way of gameplay.

 #pragma strict
 
 //next 3 are cached variables to increase performance
 var Trans : Transform;
 var Rigid : Rigidbody;
 var Hing : HingeJoint;
 
 //resting position in Eulers.y for the (saloon) door
 //ie where it is initially placed in the scene, called at start
 var rest : float;
 rest = Trans.eulerAngles.y;
 
 function Update ()
     {
     //this line makes sure we aren't checking when the door is at rest (optimization)
     if(Hing.spring.spring < 2)
         {
         //when the door is barely moving
         if(Mathf.Approximately(0,Rigid.angularVelocity.y))
             {
             //make the door swing faster, also stops sleep
             //(we don't necessarily want the door to sleep)
             Hing.spring.spring += Time.deltaTime;
             //if the door is approximately at resting position
             if(Mathf.Approximately(rest,Trans.eulerAngles.y))
                 {
                 //make the door stop all angular velocity
                 Rigid.angularVelocity.y = 0;
                 //manually place the door at the resting position
                 Trans.eulerAngles.y = rest;
                 //manually force the door to sleep
                 Rigid.Sleep();
                 }
             }
         }
     }
 
 function OnCollisionEnter ()
     {
     //THIS IS IMPORTANT
     //if this weren't here, the door spring would increment to infinity
     //making it swing super fast in the future, compounded each time the door is hit
     //AND continue to reach approximately plus or minus 6 degrees, not perfect equilibrium position
     Hing.spring.spring = 1;
     }