## 1 Introduction  
Mobile robots can be divided into three types: legged-foot, caterpillar, and wheeled. Compared with caterpillar and wheeled robots, legged-foot robots have discrete landing locations and can choose the best landing locations for support. Therefore, legged-foot robots have more advantages in terrain adaptability, motion flexibility, and load capacity. They have broader application prospects in the fields of material transportation, military investigation, field exploration, rescue, and disaster relief. The first real quadruped robot research began in the 1960s. After more than half a century of development, various styles of quadruped robots emerged. At present, one of the most representative quadruped robots is the BigDog series developed by Boston Dynamics [1, 2]. They have good topographic adaptability, load capacity, impact resistance. They can walk and trot under more complex terrain such as gravel, grassland, snow, etc. Starting in 2015, Boston Dynamics introduced the Spot series of quadruped robots, which are small, powerful, and flexible [3, 4]. ANYmal, a quadruped robot developed by the Swiss Federal Institute of Technology Zurich (ETH Zurich), has improved the series elastic driver to achieve more flexible and precise moment control. It has a laser sensor and a camera installed on its body to sense the terrain of the environment for map building and self-positioning [5]. The Cheetah series quadruped robots, developed by the Massachusetts Institute of Technology (MIT), mimic the body structure of a cheetah. It can run at high speed, up to 6 m/s. In 2019, MIT launched Mini Cheetah, the first quadruped robot that can flip backward. It can move horizontally, jump, climb from a fall, and flip backward without visual aids. It has epochal significance in robotics. Balance control of quadruped robots is one of the research hotspots in the field of motion control of quadruped robots. Currently, there are many ways to control robot body balance. The classical control methods include Zero Moment Point (ZMP) [6; 7; 8], Virtual Model Control (VMC) [9; 10; 11], Central Pattern Generator (CPG) [12; 13], and Spring Loaded Inverted Pendulum (SLIP) [14; 15; 16]. ZMP is mostly used in static gait, which requires calculating the ZMP position of the quadruped robot so that the ZMP point is always inside the polygon formed by the support point at the foot of the robot to ensure that the robot has a static balance state [17]. In [7], an LIP model of the robot was combined with the ZMP calculation to derive joint space control action based on a PD controller. The anti-interference ability of the robot is enhanced and the stability is improved. In [8], a gait planning method and a posture adjustment strategy were designed based on the ZMP stability criterion, which can improve the stability and obstacle crossing ability of the quadruped robot effectively. However, ZMP is mostly suitable for the balance control of static gait and cannot meet the needs of the fast motion of the robot. The core idea of VMC is to drive the robot to produce the desired action by generating corresponding virtual force from virtual components. In [10], an improved VMC control method was proposed, which can realize stable walking of quadruped robot with dynamic gait and have a low computational cost. In [11], a fractional-order VMC (FOVMC) for quadruped robot trotting motion was proposed. The fractional-order virtual damper was integrated with the classical VMC, the robustness and accuracy of VMC were improved effectively. However, the main problem of VMC is still high computing cost and response lag. In [14], a new template model called Spring Loaded Inverted Pendulum with Swing Legs (SLIP-SL) was proposed, which was improved on the popular SLIP model by adding passive swing leg dynamics. Based on the SLIP model, quadruped robots such as Spot, SpotMini, etc. show excellent motion performance. In previous studies, the criteria to judge the stability of quadruped robot walking are mostly applicable to static gait.