# Adaptive Super-Twisting Control for Mobile Wheeled Inverted Pendulum Systems
## 1 Introduction  
Mobile robots are widely used in various fields. As a special kind of mobile robot, the mobile wheeled inverted pendulum (MWIP) has attracted more and more attention thanks to its compact size, strong mobility, and high flexibility [1]. However, the control of the MWIP system still remains a very challenging problem because of its natural instability, underactuation, nonlinearity, time variability, and strong coupling. So far, a plurality of methods have been studied based on the MWIP system. These methods are divided into two categories: model-free algorithms and model-based algorithms. The model-free ones are applicable to general systems, but the control accuracy demands of physical applications are often difficult to meet due to the lack of system information. Moreover, these algorithms, such as PID control, largely rely on well-tuned parameters, and it is difficult to guarantee the stability of the closed-loop system without the mathematical model. Another branch of the model-free approach is neural network (NN)-based approximation. The NN training method requires a large amount of data, and the corresponding parameters are adjusted through the neurons, which cause a heavy computational burden [2]. In addition, fuzzy logic control, a widely used strategy, suffers from a slow response time and a serious dependence on the experience of selecting suitable member functions and rules [3; 4]. For the model-based algorithms, we have to get the dynamic model of the system first, which might be controlled by the method of feedback linearization, traditionally. In practice, however,accurate models are often hard to come by, which requires the model-based approaches to have the capacity of handling uncertainties. Although the feedback linearization controller can be directly applied to the nonlinear dynamics without linear approximation, the controller contains high-order derivative terms and is very sensitive to noises and uncertainties. In addition, the nonlinearity of the system cannot be easily canceled out, especially for the underactuated system, whose dynamics are not invertible [5]. Sliding mode control (SMC) appears to be one of the most promising robust nonlinear control techniques for systems with parameter variations and external disturbances [6]; it can keep the system states sliding on the sliding surface and ensure the stability of the closed-loop system. Therefore, SMC has been extensively studied and applied [7; 8; 9]. However, one serious disadvantage of SMC is the \"chattering\" phenomenon, which is harmful to the mechanical systems. There are two main causes for the occurrence of the chattering phenomenon. One is the discontinuous switching function in the controller. The other one stems from the switching gain of the SMC, which must be designed to be greater than the bound of the disturbance [10]. Therefore, many researchers have tried to reduce the chattering in the conventional SMC for MWIP systems. In [11], [PERSON] replaced the sign function with the saturation function on two-wheeled self-balancing mobile robots, which is the most direct way to reduce the chattering. However, the chattering-reduction effect of this method is limited. The saturation function only transforms the discontinuous switching function into a continuous proportional function in the saturated region; therefore, the controller will fail once it leaves the boundary layer. Furthermore, the determination of the boundary layer also remains an ambiguous problem. The chattering can also be reduced by combining the disturbance observer (DO) and the SMC [12; 13; 14]. [PERSON] has successfully designed a systematic method of a high-order disturbance observer-based sliding mode control for a class of underactuated robotic systems [15]. The estimation of the disturbance can be used as a feed-forward compensation to the controller such that the switching gain of SMC is only required to be designed greater than the bound of the disturbance estimation error.