## 1 Introduction
By establishing a dynamic model of the WIPR, a fuzzy-logic-based controller can be designed to take the position and angle information of the WIPR as the input and output control signals to control its motion state. For instance, [PERSON], in [16], proposed an Integral Interval Type 2 Fuzzy Logic (IT2 FL) method that can maintain the MTWIP equilibrium while obtaining the desired position and orientation to make it work in an uncertain environment. However, the disadvantages of fuzzy control include low control accuracy, strong dependence on control rules, and difficulty in designing control rules. The second control algorithm type is neural network control. [PERSON] [17] decomposed the underdriven WIPR model into two subsystems. The approximation characteristics of the neural network were used for motion control of the fully driven subsystem, and the sub-fully driven system was used to indirectly control the tilt angle motion of the pendulum. However, the method requires a large number of wavelet coefficient vectors, making the neural network computationally intensive. Finally, sliding mode control, as the most typical robust control method, shows good tracking performance and strong robustness, which support its wide use in linear and nonlinear systems. For underactuated systems, various sliding mode control methods have been proposed by researchers to achieve different control effects, such as integral sliding mode control, terminal sliding mode control, and hierarchical sliding mode control [18; 19; 20]. Among them, the application of hierarchical sliding mode control in practical underdriven systems is receiving more and more attention, such as balancing control of a double-inverted pendulum and trajectory tracking control of a wheeled inverted pendulum. [PERSON], in [21], proposed an integral inverse-step sliding mode controller for underdriven system control. A feedback control law was designed based on the backpropagation method, and a sliding surface was introduced in the final stage of the algorithm. [PERSON], in [22], designed two terminal sliding mode controllers to control the speed and braking of a UW-Car based on the dynamic model and the terminal sliding mode control method. [PERSON] [23] proposed a hierarchical sliding mode controller (HSMC) developed to simultaneously perform speed control and balance control of a two-wheeled self-balancing vehicle (TWSBV). Hierarchical sliding mode control is a control strategy based on sliding mode control, which divides the sliding surface into two layers. In the first layer, a high-speed sliding surface is introduced, and the control system approaches the desired state quickly. In the second layer, a low-speed sliding surface is introduced, and the control system stabilizes near the desired state. The layered sliding mode control can improve the control accuracy and stability and also has a good effect on the response speed and robustness of the system. Therefore, hierarchical sliding mode control has the same drawback in that it is insensitive to disturbances, which can easily cause the \"jitter\" phenomenon of the system. To address the shortcomings of sliding mode control, this paper proposes an improved hierarchical sliding mode control method with adaptive exponential convergence law, which can adaptively adjust the control convergence law according to the control state and smooth the sign function, thus effectively improving the problem of the strong jitter of the traditional sliding mode control, and combining the nonlinear disturbance observer (NDO), which is the most powerful method for the control of sliding mode. The NDO can effectively solve the negative impact caused by the unknown disturbance and make the system more robust, and achieve an ideal control effect on the trajectory tracking ability of the WIPR system [24; 25; 26; 27; 28; 29; 30].