# Adaptive Super-Twisting Control for Mobile Wheeled Inverted Pendulum Systems
## 4 Experimental Studies  
### Experimental Setup  
The physical MWIP system of the experiment is shown in Figure 3a. The hardware of the system is comprised of a main control circuit board (LM3S2965, Texas Instruments, Austin, TX, USA), an accelerometer, a three-axis gyro, and two encoders. The angle information is provided by the accelerometer and gyroscope, and the speed information is provided by the encoder. All the information is sent to the main control board, in which the information is processed, then the control algorithm is completed to generate the control signal. The control signal modulates a pulse signal of a certain width to control the motor speed after each sampling. The sampling period is set as 5 ms.  
Figure 2: The diagram of the control strategy. STA—super-twisting algorithm.  
then maintain the upright state on flat ground. This means that the desired inclination angle, speed, and yaw angle should be set as zero. Thus, the sliding surface can be simplified and rewritten as  
\[\sigma=\left[\begin{array}{c}\sigma_{1}\\ \sigma_{2}\end{array}\right]=\left[\begin{array}{c}q_{2}+\lambda_{1}q_{2}+ \lambda_{2}q_{1}\\ q_{3}+\lambda_{3}q_{3}\end{array}\right]. \tag{30}\]  
The acceptable initial error of inclination angle is discussed in Remark 3.  
The whole experiment is divided into two cases, one of which is the nominal case, and the other adds uncertainties to the system. The corresponding MWIP systems under these two cases are shown in Figure 3a,b, respectively.  
**Remark 3**.: _The MWIP system is a peculiar and complex underactuated system. Due to its nonlinear coupling, as depicted in (22), we cannot get a feasible set of initial errors. However, we only need to consider a relatively small initial error of angle. This makes sense in physical applications, since too large an initial error may lead to overshoot, thereby causing unsafe factors. Moreover, in other studies on MWIP systems, the significant initial errors are not taken into account [1, 24, 25, 26]. Especially, in [26], there exists a support frame to guarantee a small initial error, which ensures a smooth start for the MWIP._  
_In addition, we conducted simulations with various initial errors, which prove that the MWIP system is under control as long as the controller is non-singular. The feasible range of the inclination angle is (\(-84.27^{\circ}\), \(-84.27^{\circ}\)). As for the experiment, the feasible initial error is limited by many other physical factors, such as the modeling error and external disturbances. We cannot expect to obtain a specific acceptable initial error range. Through a number of experimental trials, we get a range of (\(-56.17^{\circ}\), \(55.28^{\circ}\)) as the initial error of the inclination angle, and this range is adequate for practical applications._  
For the sake of fairness, we preset the same initial conditions for each experimental scenario. The initial condition is set as  
\[\left\{\begin{array}{c}q_{1}\left(0\right)=0,q_{1}\left(0\right)=0\\ q_{2}\left(0\right)=-\pi/18\left(rad\right),q_{2}\left(0\right)=0\\ q_{3}\left(0\right)=0,q_{3}\left(0\right)=0\end{array}\right..\]  
In addition, the two control techniques, adaptive SMC and adaptive STA, are based on the same sliding surface and control parameters. They are selected as follows:  
\[\lambda_{1}=10,\lambda_{2}=0.3,\lambda_{3}=3, \tag{31}\]  
\[K_{1}=10,K_{2}=10,\Gamma_{1}=5,\Gamma_{2}=5. \tag{32}\]  
According to the values defined in (31) and Table 2, the denominator term \(g_{2}+\lambda_{2}g_{1}\) will always be non-negative during the experimental process, which means the design of the controller is feasible.