## 1 Introduction  
Legged robots face challenges in controlling joint displacement and foot force. Owing to the broad scope of control involved, this study reviews traditional intelligent control literature, focusing on the specific issues addressed in the study. Research on single-leg motion control for legged robots has revealed a robust algorithm [1] based on centroid momentum that demonstrates superior simulation performance. Reference [2] offers a simple and effective adaptive control method for regulating the stride length of a single-legged hydraulic robot that does not require force feedback as indicated by simulation results. Reference [3] presented a vertical jump trajectory planning method and landing buffering control for a hydraulically driven single-leg robot to achieve load-bearing takeoff and stable landing. [PERSON] et al. [4] introduced an online foot position compensator (FPC) to minimize interference for humanoid robots. In [5], a strategy using a computational torque controller (CTC) and second-order sliding mode mitigates the position deviation and system vibration. In [6], the proposed method for a whole-body control frame-based biped robot ensured a stable sliding posture by optimizing the contact force, considering dynamics, friction, and constraints. The literature above highlights the control of leg movement, posture changes, and jumping in robots, focusing on foot-end force and joint displacement control. However, there is no good solution to the problem of interference delay in the tracking process. [PERSON] et al. [7; 8; 9] investigated various control strategies, including hybrid control, SMC, ADRC, and decoupled ADRC, to enhance the efficiency of rehabilitation devices. They proposed novel control methods for knee joint rehabilitation to improve the control effectiveness of exoskeletons. Their research verifies the effectiveness of the proposed controller through numerical simulation or simulation and proves the effectiveness of the control method by considering interference. It is well known that MPC utilizes predictions of the system's future states to enhance control performance. On the other hand, SMC demonstrates robustness when facing uncertainties and external disturbances, while LADRC effectively suppresses potential oscillations in the system under disturbances. These methodologies are succinctly categorized for a comprehensive analysis and discussion of the disturbance delay and motor saturation issues pertinent to this study. Additionally, SMC effectively mitigates the nonlinearities and time-varying characteristics inherent in traditional control methods, offering extensive applicability and advantages in control engineering. The control approach in a single leg is similar to that in a multijoint manipulator arm. Studies on controlling contact force and joint angles through MPC, SMC, and LADRC are summarized below. The pertinent MPC for improving the dynamic tracking performance of a single-legged robot system can be summarized as follows: A slope-adaptive MPC algorithm was introduced in [10], which enables automatic transitions between flat terrain and unknown slopes. Meanwhile, [11] presented a trajectory-planning module and MPC based on trajectory-optimization algorithms. Numerical simulations demonstrated the system's efficiency in enabling the manipulator arm to capture the orbital debris. Reference [12] improved the performance of a flexible-joint robotic system against disturbances using an MPC with rolling time-domain control. It effectively handles Linear Matrix Inequality (LMI) optimization problems and demonstrates its effectiveness through simulations. In [13], feedback linearization MPC was used to control a two-link robotic arm, outperforming linear quadratic control using the same feedback linearization method. In summary, a single-legged robot using MPC is similar to that of a robotic arm using MPC.