## 1 Introduction
Drawing inspiration from the above MPC, a force-position MPC controller was designed and implemented to address the challenge of tracking the foot force and position in the presence of interference, and its performance was compared with that of the SMC. The relevant SMC algorithms are summarized as follows to compare the dynamic tracking performance of the different algorithms in a single-legged robot system. Reference [14] utilized cross-coupling error control and SMC to track the trajectories of synchronous moving arms and demonstrated their efficacy through simulations and comparisons with alternative controllers. Reference [15] proposed an adaptive SMC method for an end gripper that accurately predicts forces at a distance using a transmission model and a force-control methodology. In [16], an SMC technique was proposed to separate position and force. The simulation results demonstrate that it can effectively execute force-control tasks with a three-link manipulator even under unknown uncertainty. Reference [17] proposed a hybrid SMC impedance control method for precise constant contact-force tracking without force/torque sensors. The simulation experiments confirmed its effectiveness. In [18], a new controller design based on fractional-order SMC was proposed. Simulation results show that the FSMC reduces the fluctuation amplitude in the end section of the multi-loop robot, leading to improved stability with a fivefold reduction. In light of the insights drawn from several sliding mode control studies, we devised an SMC tailored to the subject of our research. Furthermore, a comparative analysis was conducted between the performance of the SMC and LADRC-SMC schemes to determine their relative effectiveness in addressing force-position control. To achieve outstanding dynamic tracking performance in a single-leg system, the relevant ADRC algorithms are outlined as follows, even in terms of interference and delays. The literature [19] suggests a LADRC method to achieve trajectory tracking for a 2-DOF manipulator system. In [20], a six-axis serial manipulator control system based on an ADRC strategy did not require an exact dynamic model. Literature [21] confirms the excellent performance of ADRC systems. The payload and disturbance controlled the new 6-DOF parallel robot. Reference [22] proposed an error-based custom ADRC method that does not require a time derivative of the reference trajectory. The experimental results validate its effectiveness in trajectory tracking and interference suppression. The theoretical applications of the ADRC algorithms demonstrate their exceptional trajectory-tracking performance in the presence of interference. In the research on tracking control under interference delay, the previous solutions mainly were single MPC, SMC, and LADRC schemes. The integration of multiple control schemes can give full play to their respective control advantages. Subsequently, we elaborate on the performance of the combined LADRC-SMC approach. The literature [23] recommends a linear extended-state observer to estimate the total disturbance. An ADRC-SMC scheme was proposed to increase the robustness of the system. In [24], an ADRC-SMC approach was proposed and validated through simulations for a tiltrotor aircraft during conversion flight. The literature [25] implemented an ADRC nonsingular terminal SMC to mitigate disturbances in a reconnaissance robot, and experimental results confirmed its effectiveness. Additionally, Reference [26] introduced an improved ADRC-SM method for tracking uncertain objects, demonstrating its efficacy in reducing buffeting with minimal overshoot, low steady-state error, and strong anti-interference capability. The algorithm for our research topic was designed based on the LADRC-SMC literature mentioned above to enhance the contact control effect. However, disturbances exist in the application of control systems, and there may also be situations involving system delays.