The study of touchdown angle's impact on quadrupeds such as gait, stability, speed, height is valuable. This paper unifies two common models for quadrupeds-SLIP and planar model, gets a feasible and efficient fixed point search strategy, and proposes a gait classification method based on the event sequence. On this basis, we study the influence of touchdown angle on quadruped robot in a variety of energy levels, and find that the change of touchdown angle would cause cycle bifurcation and chaos phenomena. Meanwhile, we notice in certain energy level and reasonable parameters configuration, different touchdown angles mean different gaits and only change of touchdown angle can make gait transition. In a fixed energy level, the impact of touchdown angle on horizontal speed, apex height and pitch angle is not obvious, but for pitch angular velocity it has obvious effect, and we reveal the importance of pitch angle and related variables such as inertia, mass distribution for quadruped control. Under the meaning of stable fixed points, speed adjust by changing touchdown angle is of little effect; however energy control is a feasible method.