# Fast Fractional-Order Terminal Sliding Mode Control for Seven-Axis Robot Manipulator
## 4 Simulation and Experiment  
### Simulation  
The designed controller, FFOTSMC with SPO (FFOTSMCSPO), was applied to a simple second-order SISO system simulated by MATLAB Simulink. It compared with four different controllers such as TSMC with SPO (TSMCSPO), LSMC with SPO (SMCSPO), FOSMC with SPO (FOSMCSPO) and FOSMC with SPO (FOTSMCSPO). The advantages of using the fractional-order derivative and the terminal sliding surface were discussed via the simulation results. The dynamics of the second-order SISO system corresponding to the model (1) were defined as follows. \[m\ddot{x}+c\dot{x}=\tau+d \tag{36}\]  
where \(m=6.7\ \mathrm{kg}\cdot\mathrm{m}^{2}\), \(c=2400\ \mathrm{kg}\cdot\mathrm{m}^{2}\mathrm{s}^{-1}\) and \(d=10\sin(10t)\ \mathrm{N}\cdot\mathrm{m}\). The given dynamics were used to assume a hydraulic servo system, whose frictional coefficient term \(c\) has more impact than the moment of inertia \(m\). The control inputs of all controllers were calculated using (27) with different parameters, as shown in Table 1. The simulation simulated as a regulator (\(\mathbf{x_{d}=0}\)) with the initial state \(\mathbf{x_{0}=[0,0,0]^{T}}\). Where \(\beta\) determines whether the fractional-order derivative is used or not in the control law (27); \(\alpha\) represents the fractional/integer-order of the derivative in the sliding surface design; when \(\beta=1\)\(\alpha=1\) it means the original first-order derivative. When selecting \(\beta\in[0.5,1)\) and \(\alpha\in[0.5,1)\), it means the fractional derivative is used. Selecting \(\alpha\) and \(\beta\) closer to one, the fractional derivative is more similar to the first-order derivative. As the discussion in Appendix A shows, a smaller fractional-order (close to 0.5) results in a faster convergence speed in the beginning, leading to a short rising time in transient response; however, it causes a slow convergence in the terminal interval, resulting in a considerable settling time. \(r_{1}\) is selected less than one when using the sliding surface of TSM; \(r_{2}\) corresponding to the reaching law is associated with the fast control law in TSMC [3]. \(r_{1}\) and \(r_{2}\) can be viewed as terms to choose whether to approach linear control or discontinuous control between a linear control (\(r_{1}=1\) or \(r_{2}=1\)) and a discontinuous one (\(r_{1}=0\) or \(r_{2}=0\)). The discontinuous control input can generate faster convergence; however, the more discontinuous it is, the more chattering there is. The other parameters in (27) were selected as \(\lambda_{1}=\eta_{1}=10,\ \mathrm{k_{1}}\)\(=20\ \mathrm{k_{2}}=0.35\) and \(\mathrm{k_{3}}=0.01\) in all five controllers. Large \(\eta_{1}\) can help a more accurate estimating of SPO; however, it is limited by the sampling time [13]. The sliding surface slope \(\lambda_{1}\) relates to the sliding manifold convergence speed, and it is also limited by the control input of the hardware limitation (maximum torque or voltage) in real applications. The reaching law gains (\(\mathrm{k_{1}}\), \(\mathrm{k_{2}}\) and \(\mathrm{k_{3}}\)) are associated with the reaching speed from \(s(t=0)\
eq 0\) to \(s=0\), and larger gains can generate a faster speed; however, chattering is also increased. The simulation results for the SISO system are shown in Figure 3. Compared to SMCSPO, FOSMCSPO, and TSMCPO, FFOTSMCSPO exhibited the fewest errors, which means that both the fractional-order sliding surface and the TSM can generate a faster convergence. This was demonstrated by the result of the FOTSMCSPO, which applies the fractional-order derivative and terminal type in the sliding surface design. Compared to FOTSMCSPO, the proposed controller with a fast control law, FFOTSMCSPO exhibited a better tracking performance. Although an exact reaching time of