I have been trying to build a 2D skeletal animation system for a while, and I believe that I'm fairly close to finishing. Currently, I have the following data structures:

struct Bone {
    Bone *parent;
    int child_count;
    Bone **children;
    double x, y;
};

struct Vertex {
    double x, y;
    int bone_count;
    Bone **bones;
    double *weights;
};

struct Mesh {
    int vertex_count;
    Vertex **vertices;
    Vertex **tex_coords;
}

Bone->x and Bone->y are the coordinates of the end point of the Bone. The starting point is given by (bone->parent->x, bone->parent->y) or (0, 0). Each entity in the game has a Mesh, and Mesh->vertices is used as the bounding area for the entity. Mesh->tex_coords are texture coordinates. In the entity's update function, the position of the Bone is used to change the coordinates of the Vertices that are bound to it. Currently what I have is:

void Mesh_update(Mesh *mesh) {
    int i, j;
    double sx, sy;

    for (i = 0; i < vertex_count; i++) {
        if (mesh->vertices[i]->bone_count == 0) {
            continue;
        }

        sx, sy = 0;
        for (j = 0; j < mesh->vertices[i]->bone_count; j++) {
            sx += (/* ??? */) * mesh->vertices[i]->weights[j];
            sy += (/* ??? */) * mesh->vertices[i]->weights[j];
        }

        mesh->vertices[i]->x = sx;
        mesh->vertices[i]->y = sy;
    }
}

I think I have everything I need, I just don't know how to apply the transformations to the final mesh coordinates. What tranformations do I need here? Or is my approach just completely wrong?