## 1 Research Background
### Existing Control Methods and Limitations  
To achieve efficient motion control of quadruped robots in complex terrains, many researchers have proposed various methods and strategies. These studies mainly focus on traditional control methods, bio-inspired methods, and the emerging deep reinforcement learning methods in recent years:  
Model Predictive Control:  
MPC is an advanced control method that achieves precise control of the system by predicting a series of future states and optimizing the current control input. MPC has been widely used in robot control, especially for the motion control of quadruped robots in complex terrains.  
MPC works by predicting the system's state changes over the next few steps. It creates a dynamic model and adjusts the control input at each step to match the desired state [8]. The advantage of MPC is that it can achieve optimal control under multiple constraints and multiple objectives. However, MPC relies heavily on the accuracy of the model, and inaccurate models can lead to poor control effects [9]. [PERSON] and others proposed a cascaded MPC framework based on a disturbance observer for the walking of humanoid robots, which integrates ankle, step, hip, and height change strategies to achieve high robustness [10]. [PERSON] and others developed a variable horizon MPC for bipedal walking control. This allows the robot to adjust its prediction horizon in real time. This improves flexibility and adaptability during walking [11].  
MPC has shown significant advantages in the control of quadruped robots, especially in known and static terrains. However, when dealing with highly dynamic and unknown complex terrains, the performance of MPC is still limited by the accuracy of the model and the real-time computing [12].  
Zero Moment Point (ZMP) Control:  
ZMP control is a classic method for robot stability control, which ensures the balance of the robot by maintaining the position of the zero moment point. The ZMP control method is widely applied in biped and quadruped robots, especially playing a crucial role in ensuring stable walking in robots [13].  
The concept of ZMP refers to a point on the robot's supporting surface where the resultant moment of all supporting forces is zero. Controlling the position of the ZMP can ensure that the robot does not tip over during walking and thus achieves stable walking. The advantage of the ZMP control method lies in its simplicity and effectiveness. The precise ZMP position can be obtained through calculation, and the robot's posture can be adjusted in real time to maintain balance [14]. [PERSON] and others presented a method for generating stable walking patterns for biped robots using ZMP preview control, allowing real-time adjustment and stability during locomotion [15]. [PERSON] and others discussed a method for online walking motion generation that uses ZMP to automatically adjust foot placement, improving the robot's ability to navigate unpredictable terrain [16].  
However, when dealing with dynamic changes and complex terrains, ZMP control shows certain limitations. The ZMP control method assumes that the contact between the robot and the ground is completely known and controllable. When the terrain is complex and irregular, this assumption is often difficult to meet [17]. For example, in rubble or crushed stone terrains, the contact points between the robot's foot and the ground keep changing, making it difficult to accurately calculate the ZMP position, resulting in poor control effects. In addition, the ZMP control method mainly focuses on balance and ignores the issues of walking speed and efficiency. In some application scenarios, such as disaster relief and emergency response, robots are required to walk stably and quickly [18].  
The ZMP control method has significant advantages in ensuring stable walking in robots, especially in known and static terrains. But, on dynamic and complex terrains, ZMP performance is limited. It needs improvement with other methods.