How to make a horizontal center of gravity in 2d ?

I am making a “tappy flappy bird” kind of style game in 2d but basically the character goes in vertical instead of horizontal and you need to avoid the obstacles etc. The concept is that the character must stay in the center, when I move it to avoid obstacles (no matter if is right or left) it needs to get back to the center.
I would like to make a center of gravity based on the player, so it can moves on horizontal (both +x and -x) only and always get back to the center. I tried to use spring joints 2d and stuff and did a research to find something like that but didn’t got anywhere.
Any idea ??

PS: Unity 4.7.2

Create an anchor x-coordinate and have your “bird” accelerate towards it when it’s x position != the anchor.

float anchor = 0f;

private void Update()
{
    if (transform.position.x > anchor)
        // Add force towards transform.right
    else if (transform.position.x < anchor)
        // Add force towards -transform.right
    else
        // No forces applied; kill velocity
}

Another fun method might be using a Spring Joint. Personally I’ve never used it but from the documentation it looks like you can make it work. Might be easier and give it a cool effect but, again, not sure what it’ll actually produce.