How to make something independent of the coordinate frame?

[glmhd]

TL;DR Summary

what become independent of coordinate frame when using moving trihedral

In page 49, chap 8 of the book "classical mechanics point particles and relativity" of Greiner, there is the following sentence:
"In order to become independent of the coordinate frame, a set of orthogonal unit vectors is put at the point of the trajectory of the mass point given by $s$."
Here, what become independent of the coordinate frame? And how using moving trihedral make some quantity independence of the coordinate frame?
A simple, concrete example to illustrate is welcome, like consider the orbit of mass point when throw it hozirontally

 

[olgerm]

I think it means that they choose new basevectors ($\vec{T}$,$\vec{N}$,$\vec{B}$) and new coordinatesystem to make components of positionvector indebendent of old coordinate system.
By putting by basevectors they mean defining new coordinate system.
Word trihedral indicates that there are 3 basevectors in new coordinatesystem.
Basevectors are not moving, but are not equal in every point of space. I do not know how these basevectors are defined outside trajectory, but the figure on page 50 and this illustration shows how these basevectors are on trajectory.