# Smooth Fractional Order Sliding Mode Controller for Spherical Robots with Input Saturation
## 1 Introduction
combined the neural network and fractional order sliding mode control and presented a peristaltic motion controller for a bionic peristaltic robot [31]. [PERSON] et al. studied a fractional sliding control with a PI\({}^{\lambda}\)-D\({}^{\mu}\) sliding surface for a two-tank hybrid system and achieved better performance than that of a controller with a PD\({}^{\mu}\) sliding surface [32]. In [33], a fractional PI\({}^{\lambda}\)D\({}^{\lambda}\) sliding controller is proposed; the PSO algorithm was used to tune the controller parameters. Furthermore, the benefits of a fractional PI\({}^{\lambda}\)D\({}^{\lambda}\) sliding surface are shown in comparison the PD\({}^{\mu}\) sliding surface in [34]. [PERSON] et al. combined fractional calculus and second-order sliding mode control to give a fractional PDD\({}^{\mu}\) sliding controller for a class of nonlinear systems [35]. [PERSON] et al. proposed a fractional order sliding mode controller for the finite-time stabilization of a nonautonomous fractional order underactuated system with model uncertainties and external noise [36]. [PERSON] et al. focused on the finite-time convergence of a fractional order nonholonomic chained system and proposed a fractional sliding order controller [37]. Based on the fractional model of a flexible underactuated manipulator, [PERSON] et al. presented a fractional PI\({}^{\lambda}\) sliding mode controller [38]. To summarize, most literature has focused on the application of fractional-order sliding mode controllers in fractional-order underactuated systems, and only very few papers have considered applying fractional sliding mode control to integer-order underactuated systems [39]. Existing papers show that the fractional sliding surfaces have better control effects than integer ones. The fractional sliding mode control has faster response and convergence speed in the initial stage due to the fractional operator. This means that the fractional sliding mode controller will yield a high output, which can more easily lead to input saturation. Moreover, in spherical robots, the output of the actuator remains constant, and when the robot system is under input saturation, it can yield a wave motion in the dynamic response process. Though many methods regarding input saturation have been developed [40; 41; 42; 43], the wave motion in the dynamic response process cannot be solved in a satisfying way, since the method cannot completely avoid input saturation. For this reason, it is not enough to improve the spherical robot speed control under input saturation by simply designing a sliding surface with faster response and convergence speed. The opposite direction at the start-up stage and wave motion in the dynamic response process also need to be taken into consideration. Focusing on this topic, the novelty of this paper can be summarized as follows:  
1. A fractional PI\({}^{\lambda}\)D\({}^{\mu}\) sliding control method based hierarchical sliding control and fractional calculus is proposed to improve the control performance;
2. A novel fractional PI\({}^{\lambda}\)D\({}^{\mu}\) sliding controller with an auxiliary system is proposed to deal with input saturation;Smooth dynamic response is achieved by adding a filter, which can decrease the output of the controller in the initial stage and make full use of the fractional sliding surface. In this paper, \(R^{+}\) and \(Z\) are the set of positive real numbers and integers, respectively, \(R^{n}\) and \(R^{n\omega n}\) are the \(n\)-dimensional Euclidean space, and the matrices of order \(n\). \(\|.\|_{\gamma}\) is the norm. For a real number \(x\), \([x]=\min\{n\in Z|x\leq n\}\) is the ceiling function, and \(\Gamma(x)=\int_{0}^{\infty}t^{x-1}e^{-t}\mathrm{d}t\) is the Gamma function. \(\mathbf{A}^{\mathrm{T}}\) is the transpose of matrix \(\mathbf{A}\), \(\mathrm{eig}(\mathbf{A})\) is the matrix eigenvalues.