Benefiting from its hyper-redundant structure, the biomimetic snake-like manipulator retains its remarkable flexibility even within confined spaces. However, its motion planning and control pose significant challenges. This paper imitates the winding uncoiling behavior of snakes to achieve controllable constrained path following. Firstly, based on control points, a recursive computational model and an equivalent planning angle model are established, enabling efficient and analytical determination of joint positions, collision regions, and motion parameters during the path following. Subsequently, the sliding control point algorithm and motion smoothing restriction algorithm are designed. The former ensures that the remaining segments during following strictly remain within the collision-free regions defined by the base and path controls, while the latter smooths the control parameters based on velocity and acceleration limitations. Finally, simulation and practical experiments demonstrate the feasibility of the proposed methods. The prototype that applied our method can reach targets and accomplish tasks, further validating the applicability of the snake-like manipulator.