# Fast Fractional-Order Terminal Sliding Mode Control for Seven-Axis Robot Manipulator
## 1 Introduction  
Sliding mode control (SMC) has been widely studied and applied to robot manipulators because of its simplicity of implementation, fast global convergence, high robustness to external variations, and insensitivity to modeling error and system parameter variations [1]. SMC includes a conventional linear sliding mode (LSM) control, which is asymptotically stable, and a terminal sliding mode (TSM) control with finite-time stability [2; 3]. Typically, it includes two steps to design an SMC: the choice of sliding surface and the control/reaching law to reach the sliding surface and maintain stay on it. TSM control (TSMC) utilizes a nonlinear sliding surface instead of a linear sliding surface in sliding surface design. It generates faster convergence, less tracking errors than LSM control (LSMC), and finite stability for reaching the equilibrium point. The control/reaching law in TSMC also contains a nonlinear term, which allows the sliding surface to be reached in finite time with fast convergence. Compared to the LSMC, TSMC shows various superior properties such as fast and finite-time convergence and few tracking errors. However, in TSMC, the first-order derivative of the nonlinear term in TSM causes a singularity problem. In recent years, many researchers have focused on solving the singularity [4; 5]. Using fractional calculus can avoid this singularity because of the hereditary and memory property, such as fractional-order TSMC (FOTSMC) [6]. FOTSMC was proposed with a terminal sliding surface using the fractional-order derivative. It can help avoid the singularity, which exists by taking the first-order derivative to the sliding surface. In addition to this, compared to conventional TSMC, FOTSMC has a faster convergence speed. The study on applying the fractional-order derivative to SMC techniques starts by applying it to the LSMC first. A fraction-order sliding mode control (FOSMC) was proposed based on LSM [7], and it exhibited better performance than the conventional LSMC (integer-order derivative-based). The reason has been roughly analyzed and discussed. In [8], researchers analyzed and found that the solution of a fractional-order (larger than zero and less or equal to one) differential equation could be presented as a Mittag-Leffler function. This mathematical solution explains why FOSMC exhibited a higher convergence speed than LSMC (see Appendix A). However, FOSMC is not globally faster than LSMC. It has slow convergence near the terminal where small values are, which is discussed in detail in Appendix A.  
FOSMC can compensate for the slow convergence drawback near the equilibrium of FOSMC because the terminal sliding surface has a faster convergence speed than the linear sliding surface, especially in the field of small values. Compared to TSMC and FOSMC, FOSMC is a complementary method, the terminal sliding surface helps the slow convergence in the interval of small values, and the fractional-order derivative accelerates the convergence in other intervals. Even though many researchers have demonstrated the performance of FOSMC through real applications and simulations [9; 10; 11; 12], the specific reason why FOSMC performs better than FOSMC has not been introduced clearly, which is discussed in Appendix A. Additionally, all previous works on FOSMC only focused on applying the fractional-order derivative on the sliding surface design, without paying attention to the design of the control/reaching law for reaching the sliding surface, e.g., FOSMC in [9] only utilized the fractional calculus in designing the sliding surface. It applied a control law, obtained by taking the first-order derivative to the sliding surface with respect to time, to reach the sliding surface and maintain stay on it. The control input of FOSMC was merely derived simply by the desired dynamics of the sliding surface.