# Stability Criterion for Dynamic Gaits of Quadruped Robot
## 1 Introduction
Other researchers have proposed some energy-based stability criteria [21; 22], e.g., the normalized dynamic energy stability margin (NDESM). These methods evaluate the stability of the robot by quantifying the maximum impact energy that the robot can absorb without losing its stability. The limitation of the above stability criteria is that they are all suitable to the situation in which a support polygon exists when the legged robots are moving. They can be used to analyze the stability of walking gait (when the supporting triangle exists), but are not suitable for dynamic gaits, which only have a support line in most cases, such as trotting, pacing, and bounding. There have been few studies concerning the stability criteria for quadruped robots with dynamic gaits, all of which focus on trotting. Most researchers believe that a robot with a trot gait is stable only when the current _ZMP_ is located on the support line, because there is no tumbling moment around the support line. The distance between _ZMP_ and the support line (substantially equivalent to the tumbling moment around the support line) is used to guide the motion control of the robot [14; 23; 24]. Such analytical methods do not consider the effect of the dynamic process of support-leg alternation on the robot's motion stability. In practice, even if _ZMP_ is not located on the current support line, the robot will not necessarily lose its balance and fall. Hence, it is difficult to correctly evaluate the stability of the robot by directly using the distance between _ZMP_ and the current support line. Therefore, the dynamic-stability criterion requires further consideration. [PERSON] et al. [25] proposed a stability analysis method based on Lyapunov theory of switching system. They built the dynamic model of the trotting robot as a nonlinear switching system, and designed a Lyapunov function to prove that the switching system is uniform asymptotic stable. This method is used to choose the landing position of the swing feet to make the position tracking errors of the robot's body converge to zero. The computational complexity of this method is too high, and it cannot evaluate the stability of the robot's motion. Another stability measure is called Landing Accordance Ratio (LAR) [26]. This method considers the time difference between the moments of the feet in the same phase touch down. If the discordance amount is zero, there is no unexpected moment, and the robot moves stably. There are three major shortcomings of this method: (1) it cannot evaluate the stability of the robot in time, because LAR is updated when the two feet in the same phase both touch the ground; (2) it is not suitable for the situation where the ground-reaction forces at current support feet cannot prevent the robot from tumbling; and (3) it is hard to decide when to update the value of LAR if the robot tips over, because one of the feet will never touch the ground. The good stability criterion and evaluation method should have the following characteristics: (1) it should determine the stability state of the robot efficiently and correctly; (2) it should evaluate the robot's stability accurately; and (3) it should provide guidance for the robot's motion planning and balance recovery. The purpose of this paper is to present a new stability criterion for the quadruped robots with dynamic gaits over irregular terrain. When animals experience impacts in nature, they quickly adjust the position of their next set of support feet to maintain stability. This indicates that dynamic stability depends not only on their current motion and support state but also on the positions of the next set of support points. Inspired by quadruped animal moving gesture and referring to the previous dynamic-stability criteria, this paper assumes that a robot is stable if the current or next set of support feet can provide the moment necessary to prevent the robot from tipping in any direction.