My problem is modeled by an edge-based graph (where each edge in the original graph is transformed into a vertex and each edge to edge "turn" is transformed into an edge).
Each edge in the original graph has multiple grades (in my specific case just two) and I have graded the turns as well. I want to minimize the cost of the solution path over one of the two grades of the edges. In addition, the other grade is a cumulative one and when it reaches a certain threshold it need to reset itself to 0 and this is achieved by taking certain "turns".
I have been searching for a shortest-path algorithm with constraints, but the problem is that the cumulative grade is based not only on the future vertices in the path, but as well in the past ones.