## 1 Introduction  
With the rapid development of robotics technology, the application of mobile robots has extended from indoor structured environments to unstructured operational settings [1]; typical applications include geological surveying, emergency rescue, military reconnaissance, and so on [2]. The increasing variability of external environments and the growing complexity of tasks pose higher demands on mobile robots [3]. On the one hand, robots operating in complex environments require high reliability and adaptability. Consequently, some robots have foregone traditional steering mechanisms and instead adopted differential steering, which simplifies the robot's structure while enhancing its adaptability and reliability [4]. On the other hand, the escalating demand for carrying capacity has led to the adoption of hydraulics as the power source. Hydraulic systems offer advantages such as high power-to-weight ratio, low cost, and high reliability [5]. Considering these factors, hydraulic-driven skid-steer mobile robots have garnered significant attention in recent years [5; 6; 7]. These robots can undertake a variety of tasks in high-risk environments, effectively replacing human involvement. However, trajectory tracking control for mobile robots remains a challenge due to the nonholonomic constraints inherent in their motion and the physical constraints encountered by the actual systems [8]. Furthermore, the nonlinear characteristics of hydraulic-driven systems, parameter variations caused by leaks and friction, and external environmental disturbances make the control of hydraulic-driven mobile robots more intricate compared to motor-driven counterparts, especially when operating on unstructured terrains [9]. Therefore, conducting research on trajectory tracking control for hydraulic-driven skid-steer robots holds paramount importance in the field of robotics. At present, there has been extensive research on trajectory tracking control methods for skid-steer mobile robots. Various control approaches, including PID control [10], linear quadratic regulator (LQR) [11], sliding mode control [12], robust control [13], and adaptive control [14], have been extensively studied to address the challenges posed by the nonlinearity, nonholonomic constraints, and robustness of mobile robots. However, these control methods have limitations in dealing with various physical constraints, such as actuator power constraints, velocity and acceleration limitations, and so on [15]. Neglecting these physical constraints in control strategies can potentially degrade control performance and even result in system instability [16]. Compared to these methods, Model Predictive Control (MPC) has garnered considerable interest in the field of trajectory tracking control for mobile robots. This is primarily due to its capacity to effectively address physical constraints and generate optimal control inputs [17]. Linearized MPC (LMPC) has been extensively studied in [18], demonstrating its simplicity, efficiency, and ease of solution. Nevertheless, it should be noted that LMPC lacks the capability to directly tackle the point stabilization problem. This limitation arises when the reference velocity of the robot reaches zero, rendering the robot uncontrollable [19]. In contrast, Nonlinear MPC (NMPC) can solve this issue and achieve better tracking accuracy [15]. In one study [20], the control performance of LMPC and NMPC on an articulated skid-steer unmanned vehicle was experimentally compared. Tube-based MPC was employed in [21] to solve the robust tracking control problem of skid-steer loading vehicles. A learning-based NMPC algorithm was proposed in [22] for trajectory tracking control of autonomous mobile robots, aiming to reduce tracking errors when repeatedly traversing along the reference path. Moreover, ref. [23] developed an NMPC based on the P3-AT small mobile robot, addressing the tracking control problem and estimating the robot's state using the Moving Horizon Estimation (MHE) method.