## 1 Introduction  
With the rapid development of technology, human society has simultaneously achieved increased convenience and comfort [1]. In today's factory warehouses and production lines, a variety of robots add to the possibilities of Industry 4.0. In the warehouses of more advanced companies, transport robots can be found everywhere, replacing traditional manpower and eliminating the need for workers to carry out repetitive lifting and carrying. These \"smart\" robots can accomplish a task as long as they can follow the requirements of a given transport trajectory. This paper studies the trajectory tracking of a mobile wheeled inverted pendulum on a given reference trajectory to achieve perfect tracking of the ideal motion trajectory of the robots, meeting the requirements of factory transport robots and providing a powerful source of assistance to realizing smart factories [2; 3; 4; 5]. Mobile wheeled inverted pendulum models, such as WIPRs, have attracted much attention because of their special advantages, such as compactness, mobility, and human-like functions. WIPRs are widely used to verify the effectiveness of nonlinear underactuated control methods, and compared with the traditional inverted pendulum, WIPRs have more applications than traditional inverted pendulum vehicles, especially in unknown, dynamic, and nonlinear environments, and are commonly used in logistics transportation, commuting, and navigation, as well as in the aforementioned application in the environment of factory transportation. However, a WIPR is classified as a typical model of nonlinear underactuated systems with two input torques driving two wheels and three degrees of freedom (forward, rotation, and tilt angle of the pendulum), and achieving its high-performance motion control is still a challenging task for the control community [6; 7; 8; 9]. On the one hand, when a WIPR moves, it is always assumed that the ground can provide enough friction to prevent the robot from side-slipping and wheel-sliding (i.e., the robot is guaranteed to move with purely rolling wheels without skidding phenomena), which is a non-complete constraint at this point. On the other hand, consider that the underdriven inverted pendulum body needs to use the input torque of two driving wheels to control the three degrees of freedom of WIPR forward movement, rotation, and the angle of the inverted pendulum. If we want to use MWIPR to track the trajectory, we need to drive a WIPR in real time to control the three form variables with two input variables, which is a typical underactuated problem. Finally, in the real world, WIPRs operate in factories or other similar environments and always encounter various unknown disturbances that interfere with the system. Therefore, the three problems of incomplete constraints and underactuated and unknown perturbations are the main challenges faced by this particular mobile robot for trajectory tracking control [10; 11; 12; 13]. The three issues mentioned above are of importance for the following reasons. First, the incomplete constraint will lead to the WIPR being unable to follow any trajectory movement, especially in the case of high-speed heavy load; if the robot's incomplete constraints are not considered in motion planning, this is likely to lead to untimely obstacle avoidance and unreachable trajectory. Second, underdriven robots often have excellent dynamic performance or price advantages in terms of drive cost, but their biggest problem is the higher requirements in controller design. Finally, unknown disturbances will affect the control accuracy of the system to a certain extent, and more seriously, will affect the stability of the control system [14; 15]. Many researchers and practitioners have proposed several control algorithms to overcome the difficulties faced by the problems associated with WIPR systems. One of the widely used methods is fuzzy control, which is an empirical, rule-based control technique that can effectively control nonlinear systems.