A Function which measures "error"

I have an error, denoted [itex]e(x,y)[/itex] for example. It's not a norm and it isn't symmetric, that is to say: [itex]e(x,y)\neq e(y,x)[/itex].

My question is simply this: With such properties, is the choice of such a function a good one?

 

I'm not worried about showing it's bounded but if it's a good measure of error with it being non-symmetric.

 

Okay, I am trying to compare two different positions and orientations of a drill. The x and y are elements of SE(3).