In this paper we consider the motion of non-closed planar polygonal curves governed by generalized crystalline curvature flow with a driving force. In the context of ”crystalline motion”, we usually restrict the curves in the special class of polygonal curves, so-called ”admissible class.” We here extend the previous results to wider class which is called ”essentially admissible class.” In such a class, there are no order-preserving structure, thus, controlling the movement of the solution curves becomes more difficult. In this paper we investigate the estimate of the movement of each facet of the solution curves in the essentially admissible class and show the global existence of V-shaped solutions. MI: Global COE Program Education-and-Research Hub for Mathematics-for-Industry グローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点」