Transform between cartesian and frenet coordinate system

Hi @fjp , I'm working on my own version of this algorithm to generate a local path to avoid obstacles.
Currently I have a problem which I have no idea about how to fix. Here is the image to show my problem, the pink path is the original reference path, the left blue path is the generated path to avoid the obstacle lies on the original path, and the red circle marks my problem. When I try to convert my path from Frenet coordinate (s, d) to Cartesian coordinate (x, y), sometimes the path will suddenly turn back like the red circle part shows.

I use the spline function to map between s-to-x and s-to-y according to the original reference path. And I calculate the new path point which has d distance away from the original reference path and is perpendicular to the tangent direction of the relative point on the original path. I think the problem is coming from the curvature of the original path, but I have no idea about how to fix it. Can you give me some suggestion?