## 3 Results and Discussion
### GWO-WWOA Learning Process  
As an example, the self-designed articulated manipulator shown in Figure 3 is selected and its dynamics are expressed as follows:  
\[M(\theta)\ddot{\theta}+C(\theta,\dot{\theta})\dot{\theta}+G(\theta)+d=\tau, \tag{65}\]  
where \(\theta\), \(\theta\), \(\dot{\theta}\in R^{3\times 1}\) represent the angle, angular velocity, and angular acceleration vectors, respectively, \(\tau\in R^{3\times 1}\) is the control torque vector and \(\dot{d}\in R^{3\times 1}\) is the bounded lumped uncertainty vector, which include the nonlinear dynamics such as friction and external disturbance. \(M(\theta)\in R^{3\times 3}\) is the inertia matrix, \(C(\theta,\ \dot{\theta})\in R^{3\times 3}\) is the Coriolis and the centripetal matrix, and \(G(\theta)\in R^{3\times 1}\) is the gravity vector given as  
\[M(\theta)=\left[ \begin{array}{ccc}M_{11}&M_{12}&M_{13}\\ M_{21}&M_{22}&M_{23}\\ M_{31}&M_{32}&M_{33}\end{array} \right],C(\theta,\dot{\theta})=\left[ \begin{array} []{ccc}C_{11}&C_{12}&C_{13}\\ C_{21}&C_{22}&C_{23}\\ C_{31}&C_{32}&C_{33}\end{array} \right],\] \[M_{11}=I_{z2}s^{2}\theta_{2}+I_{z3}s^{2}\theta_{23}+I_{z1}+I_{z2} c^{2}\theta_{2}+I_{z3}c^{2}\theta_{23}+m_{2}r_{2}^{2}c^{2}\theta_{2}+m_{3}(I_{2}c \theta_{2}+r_{3}c^{2}\theta_{23}),\] \[M_{12}=0,M_{13}=0,M_{21}=0,M_{22}=I_{z2}+I_{z3}+m_{3}I_{2}^{2}+m_ {1}r_{2}^{2}+m_{3}r_{3}^{2}+2m_{3}l_{2}r_{3}c\theta_{3},\] \[M_{23}=\frac{1}{12}m_{3}l_{3}^{2}+m_{3}l_{2}^{2}+\frac{1}{3}m_{3} l_{3}c\theta_{3},M_{31}=0,M_{32}=I_{z3}+m_{3}r_{3}^{2}+m_{3}l_{2}r_{3}c\theta_{3},M_{33}=L_{z3}+m_{3}r_{3}^{2},\] \[C_{11}=-\frac{1}{4}m_{2}l_{2}^{2}c\theta_{2}\theta_{2}-m_{3}[l_{2 }c\theta_{2}+\frac{1}{2}l_{3}c\theta_{23}(I_{2}s\theta_{2}+\frac{1}{2}l_{3}s \theta_{23})]+\frac{1}{2}m_{3}l_{3}s\theta_{23}(l_{2}c\theta_{2}+\frac{1}{2}l_ {3}c\theta_{23}),\] \[C_{12}=-\frac{1}{4}m_{2}l_{2}^{2}c\theta_{2}s\theta_{2}-m_{3}(l_{ 2}c\theta_{2}+\frac{1}{2}l_{3}c\theta_{23})(l_{2}s\theta_{2}+\frac{1}{2}l_{3}s \theta_{23}),\] \[C_{13}=-\frac{1}{4}m_{3}l_{3}s\theta_{23}(l_{2}c\theta_{2}+\frac{ 1}{2}l_{3}c\theta_{23}),\] \[C_{21}=\frac{1}{4}m_{2}l_{2}^{2}c\theta_{2}s\theta_{2}+m_{3}(l_{ 2}c\theta_{2}+\frac{1}{2}l_{3}c\theta_{23})(l_{2}s\theta_{2}+\frac{1}{2}l_{3} s\theta_{23}),\] \[C_{22}=-\frac{1}{2}m_{3}l_{2}l_{3}s\theta_{3},C_{23}=-m_{3}l_{2} l_{3}s\theta_{3},C_{31}=\frac{1}{2}m_{3}l_{3}s\theta_{23}(l_{2}s\theta_{2}+\frac{1}{2}l_ {3}c\theta_{23}),C_{32}=m_{3}l_{2}l_{3}s\theta_{3},\] \[C_{33}=0,G(\theta)=\left[ \begin{array}{c}0\\ -(m_{2}gr_{2}+m_{3}g_{2}l_{2})c\theta_{2}-m_{3}r_{3}c\theta_{23}\\ -m_{3}g_{3}r_{3}c\theta_{23}\end{array} \right],\]  
where \(I_{zi}\) denotes the moment of inertia of the ith joint; \(m_{i}\) denotes the mass of each link; \(l_{i}\) denotes the length of each link; \(\theta_{i}\) denotes the joint angle of each joint; \(s\theta_{i}=\sin\theta_{i}\), \(c\theta_{i}=\cos\theta_{i}\), and \(\theta_{ij}=\theta_{i}+\theta_{j}\). The specifications of the manipulator dynamics are given as \(m_{1}=1.2\) kg, \(m_{2}=0.6\) kg, \(m_{3}=0.7\) kg, \(l_{1}=0.05\) m, \(l_{2}=0.2\) m, \(l_{3}=0.2\) m. The state space model of (65) is constructed as follows:  
\[\begin{array}{l}\dot{x}_{1,1}=x_{1,2},\\ \dot{x}_{1,2}=f_{1,2}(\overline{x}_{1,2},x_{3,1},x_{3,2})+g_{1,2}(x_{1,1},x_{2,1 },x_{3,1})u_{1}+d_{1,2},\\ \dot{x}_{2,1}=x_{2,2},\\ \dot{x}_{2,2}=f_{2,2}(\overline{x}_{2,2},x_{3,1})+g_{2,3}(x_{2,1},x_{3,1}  
The test command signal in each joint for GWO-WWOA learning was used for \(0\leq t\leq 5\) s as follows: \(q_{d}=0.25\sin(1.2568t)+1.8405~{}(rad)\). Figure 4 shows the comparative ITAE values for the WOA, GWO, GWO-WOA, and WWOA techniques, where the GWO-WWOA technique has a much smaller ITAE value compared to the other techniques. Therefore, the GWO-WWOA technique was used to optimize the parameters of the fractional-order sliding mode backstepping (FSBSC) control system.