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Grotius wrote:

Then x~``traj(s0) for example :
- This is second derivate = curvature, given the ``
 

The second derivative is not the curvature, the first one is.

Grotius wrote:

- Is already transfomed, given the ~ above the x
 

This means you need to enter the value of the already transformed trajectory. If we want to use eq. 30 for this we need to transform it first because there it is given before the transformation.

Grotius wrote:

- Is for startpoint s0.
- Is for x coordinate.
 

yes

Grotius wrote:

- It is the curvature kappa value for the transformed startpos?
 

No, it is the second derivative with respect to curve length s, this is not the same thing as curvature (would be first derivative)

Grotius wrote:

- I now get it, that a line has no curvature, so value is : 0
- So for a arc, we just put in the curvature value like : "1/Radius" normally.  But they use eq.30 :   -1/r*cos(s/r).....etc.... Why so difficult?
 

For the line all derivatives are 0. For the arc see above, ti is not the curvature but the second derivative, not sure if it has a name, "curvature of the curvature" basically.

Back at the computer. From what I can see you should get rid of that when you select G53 for the safe retracts