## 2 Materials and Methods
### The Design of Improved Slide Mode Control  
The sliding mode control algorithm consists of two key elements: (i) the design of the sliding mode surface; and (ii) the design of the convergence rate. The design of the sliding mode surface is mainly based on the system structure as well as the control objective. As for the design of convergence law, there are four different convergence laws: the isokinetic convergence law, exponential convergence law, power convergence law, and general convergence law. In this paper, based on the optimal control objective of WIPPR to cope with nonlinearity and underdrive, as well as unknown disturbances, the traditional exponential convergence law is improved by introducing an adaptive control function, and an improved sliding mode control based on the adaptive exponential convergence law is proposed, which can weaken the system jitter while speeding up the system response, making the sliding mode control more suitable for tracking the reference trajectory of the WIPR system under the action of unknown disturbances [36]. The design for the traditional exponential convergence law is shown in Equation (20).  
\[\dot{s}=-\eta\text{sgn}(s)-\mu s, \tag{20}\]  
where: \(s\) denotes the slip surface function; the parameters \(\eta\), \(\mu\) denote the convergence coefficient; and \(\text{sgn}(s)\) denotes the sign function.  
In the traditional exponential convergence law, the isokinetic term is denoted \(-\eta\text{sgn}(s)\), and the exponential term is denoted \(-\mu s\). When the state of the system is far from the slip surface, the exponential term and the isokinetic term in the convergence law act simultaneously to help the system move toward the slip surface, and the magnitude of the isokinetic term and the exponential term are mainly determined by the reference \(\eta\), \(\mu\). The exponential term is small, and the isokinetic term acts mainly when the system is moving close to the surface.  
This paper makes a corresponding improvement based on the traditional exponential convergence law and introduces the adaptive \(o(s)\) function to adjust the convergence law in accordance with the control state of the system, as shown in Figure 2, which can accelerate the convergence speed of the sliding mode and weaken the overshoot phenomenon. This allows the sliding mode control to reduce the jitter phenomenon of the system. Following the inclusion of the adaptive function \(o(s)\), the new exponential convergence law is as follows:  
\[\left\{\begin{array}{l}\dot{s}=-\eta o(s)\text{sgn}(s)-\mu s\\ o(s)=\frac{(|s|+1)|\log_{\text{s}}(|s|+a)|}{|s|+\log_{\text{s}}(|s|+a)+b}\end{array} \right., \tag{21}\]  
where \(a>0\), \(b>0\), \(\eta>0\), \(\mu>0\).  
Through the analysis, relative to not adding the adaptive function (i.e., o(s) = 1), it can be found that when the system motion point is far away from the sliding surface (namely, when s is far away from the origin 0), the adaptive function \(o(s)\) will increase the convergence law, which will speed up the system convergence speed, shorten the system state convergence time to the target state, and reduce the control time; when the system motion point is close to the sliding surface, \(|s|\) will converge to 0 and \(o(s)\) will be less than 1. The role of \(o(s)\) here is to suppress the jitter amplitude and weaken the state variable fluctuation problem after the system is stabilized, and the suppression effect will be more obvious as the parameter \(b\) increases. To further weaken the jitter problem of the stabilized system, the smoothing process is carried out for the symbolic function \(\text{sgn}(s)\) in this paper, which is known as the traditional symbolic function [37], as shown in the following equation.  
\[\text{sgn}(s)=\left\{\begin{array}{ll}1&s>0\\ 0&s=0\\ -1&s<0\end{array}\right., \tag{22}\]  
Figure 2: The adaptive function \(o(s)\).  
The symbolic function after the smoothing process is shown below.  
\[\text{sgn}(s)=\frac{s}{|s|+\beta},\ \beta=0.01, \tag{23}\]