# Smooth Fractional Order Sliding Mode Controller for Spherical Robots with Input Saturation
## 1 Introduction  
Spherical robots are a new type of robot with a ball-shaped exterior shell. Their advantages, such as the lower energy consumption and enhanced locomotion compared to traditional wheeled or legged mobile robots have motivated many researchers to develop them with different structures in recent decades [1; 2; 3; 4; 5; 6]. The inner parts are hermetically sealed inside the spherical shell, resulting in increased reliability in hostile environments, making them suitable for security and exploratory tasks [7; 8]. Among the current structures of spherical robots, the pendulum-driven type is popular in industry and academia owing to its easy implementation and strong driving torque. Linear motion is the most important motion mode for the execution of a task. However, a pendulum-driven spherical robot in a state of linear movement has underactuated and strong nonlinearity, which is similar to a class of underactuated systems such as the ball-beam system and the pendulum system. In order to overcome these problems in spherical robots or similar underacted systems, many control methods have been implemented such as adaptive control, feedback linearization, back-stepping, sliding mode control, trajectory planning, and the intelligent control method [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. In all these methods, the sliding mode control approach is suitable for motion control systems owing to its unparalleled merits, such as fast response, robustness to parametric uncertainties, and resistance to disturbances [20]. To cope with the control problem of this underactuated system, the hierarchical sliding mode control method was developed [21]. This method designs the first layer sliding surface for each subsystem and then achieves the second surface by a linear or other combination of all first-layer sliding surfaces. This method demonstrates excellent convenience in the design of sliding mode controllers for underactuated systems, and it ensures the stability of the entire system, as well as of each subsystem [22; 23]. [PERSON] et al. proposed a hierarchical sliding mode controller for the velocity control of a pendulum-driven spherical robot in [24; 25]. However, the traditional integral sliding surface requiresa long time to track the desired velocity and has a significant overshoot. It is worth noting that none of this research on spherical robot control systems considered input saturation. The driving capacity is relatively limited in actual spherical robots, and input saturation will be an inevitable phenomenon. Without considering the input saturation in the controller design, the control performance of a spherical robot may be weak or instability may occur in the whole system. The above research also neglects the movement of the shell in the opposite direction at the start-up stage due to the reaction force acting on the robot. As can be seen from aforementioned works, this phenomenon is caused by the conservation of momentum. Although this situation may be improved when values of control targets are higher, the problem cannot be wholly solved. In recent years, fractional order calculus has been widely used in science and engineering. Many fractional-order controllers have been developed to enhance control performance [26; 27; 28; 29; 30]. For example, [PERSON] proposed a fractional PI\({}^{\lambda}\)D\({}^{\mu}\) controller in [26]. [PERSON] et al. presented a fractional sliding mode controller and showed its benefits over integral controllers [27]. [PERSON] et al. developed a fuzzy fractional sliding surface for a nonlinear system and optimized the controller parameters using the genetic algorithm, achieving better performance than with integral sliding mode controllers [28]. [PERSON] et al. put forward a fractional integral sliding mode controller for a wind power generation system to stabilize and enhance the robustness [29]. [PERSON] et al. studied a fractional sliding mode controller for a permanent magnet synchronous motor [30], while [PERSON] et al.