## 3 Simulation  
The focus of this section is to discuss the trajectory-tracking effect of the system in a simulation environment, and to verify the feasibility of the proposed control scheme and what the advantages of the proposed method are compared with other control systems in this paper. Next, the simulation results of different control systems in the face of the same disturbance will be compared to verify the control effectiveness of each system. The parameters in the system are shown in the following Table 2.  
The simulation experiments in Matlab/Simulink verified the high-precision trajectory tracking capability of the system and the stability of the pendulum in robot motion. During the simulation study, the initial position was set as \(q_{m}=[0,\ 0,\ 0]^{T}\), and the initial position of the reference trajectory was set as \(q_{mr}=[0,0,\tau/2]^{T}\), where the desired tracking velocity = 1 m/s and the angular velocity = 1 rad/s. Therefore, the trajectory of the robot should be a circle with a radius of 1 m, and the center of the circle is \((0,0)\). Therefore, the time function of the reference trajectory was chosen as \(\{x=\sin t,y=\cos t\}\). Figure 3: The control system. An external perturbation was added, as shown in the following equation. \[\left\{ \begin{array}{l}D_{1}=0.4\sin(0.4t)Nm\\ D_{2}=0.5\cos(0.5t)Nm\\ D_{3}=0.6\sin(0.6t)Nm\end{array} \right., \tag{44}\]  
To demonstrate the superiority of the proposed method in this paper, three comparative experiments were conducted under the given disturbance conditions: the first experiment involved the simulation results of the unimproved HSMC method, the second experiment involved the simulation results of the improved IHSMC method with adaptive law but without nonlinear disturbance observer, and the third experiment involved the simulation results of the proposed method in this paper (referred to as PC). First of all, by observing Figures 4-6, it can be concluded that the proposed method in this paper is better than the other two control methods in terms of both the speed of convergence of the error to the steady state and the magnitude of the fluctuation of the error after reaching the steady state when compared with the other two methods. This undoubtedly reflects the effectiveness of the method in this paper, which can track the given reference trajectory very accurately. Figures 7 and 8 give the tracking of the desired speed of the WIPR system under the three control methods. Compared with the other two methods, firstly, the control method in this paper can track the desired speed more rapidly, reaching the effect of tracking the desired speed at 0.7 s, whereas the other two methods track the desired speed in more than 1 s, which is much slower than the method in this paper, and the fluctuation frequency is high, which may affect the stability of the WIPR. As can be seen from Figure 8, the present method exceeds the other two methods in the tracking effect of rotational velocity relative to the forward velocity, for one. The convergence speed is fast, and more importantly, the proposed method is very stable after the velocity tracking reaches the steady state, which can be regarded as showing no fluctuation compared with the other two methods. Figure 5: The error of \(y\). Figure 6: The error of \(\theta\). The HSMC method with general exponential convergence law has more frequent angle oscillations, and the system is more unstable, as can be seen from the angle change of the WIPR pendulum shown in Figure 9, whereas the IHSMC improved convergence law method's pendulum has smoother oscillations after reaching stability, and the control effect is obviously stronger than that of the HSMC with general exponential convergence law. In terms of response time and maximum overshoot, the suggested method outperforms the other two ways, and it can continue to operate smoothly and without oscillations once it has reached the stabilization point. Figure 8: WIPR rotation velocity \(w\). Figure 7: WIPR forward velocity \(v\).