This paper presents a novel dynamical path following feedback control method for a snake-like robot. The snake-like robot is an undulatory locomotor transforming its periodic changes in shape into its displacement. To achieve work involving movement, e.g., an inspection task, in its working environment, it is necessary to specify its motion quantitatively. Since each of the links of the locomotor has passive wheels at its midpoint, it is not simple to design a feedback control system which enables the locomotor to follow desired paths (or to track desired trajectories) compared with wheel drive vehicles. Especially, the locomotor has singular attitudes in which the driving of its joints cannot be transformed into its movement. In this control method, a virtual link with a virtual steering system at its tip and a virtual axle at its midpoint connected to a top link through a virtual joint at its tip is caused to follow winding paths, which makes it possible to transform the driving torque of its joints into its propulsion force. The kinematical equations of the locomotor are formulated in a curvilinear coordinate system in which a winding path is one coordinate axis. A part of the kinematical equations describing the motion of the virtual mechanical elements only are converted into a chained form which facilitates to design a kinematical path following feedback control system. By a backstepping method, the acceleration required to realize the desired velocity in the kinematical control system is calculated. Based on the dynamical equations of the locomotor derived in Lagrangian mechanics, the torque of the joints of the locomotor is designed so as to realize the acceleration calculated by the backstepping method. The validity of this dynamical control method is verified by simulations in which the locomotor climbs a slope while following a serpenoid curve path.