The relative spatial position of each chain link and the phase angle of engaged tooth are not concerned in traditional sprocket designs. As such, the chain shifting might fail regularly. In this study, an optimized path of one chain for up-shifting is derived. Such path is formed by several bended chain links sustain yaw and roll. Empirically, appropriate bending, yaw and torsion angles for tooth chamfer, could be shorten the up-shifting distance and phase angle required, eventually facilitate not only more ascending points compactly arranged on sprocket but larger chain laterodeviation which curtails sprockets clearance and integrates the sprocket unit. Then again, the optimized shifting path promotes efficiency and decreases required movement yielded by derailleur tappet pressure.
