# Adaptive Fuzzy Command Filtered Tracking Control for Flexible Robotic Arm with Input Dead-Zone
## 2 Preliminaries  
### System  
The flexible robotic arm is a non-rigid single-link system with a DC motor, as shown in Figure 1. The system's dynamic equations can be derived using the Euler-Lagrange equations, yielding the mechanical arm system's equations of motion. \[\left\{ \begin{array}{l}J_{1}\ddot{q}_{1}+F_{1}\dot{q}_{1}+K(q_{1}-\frac{q_{2 }}{N})+mgdcosq_{1}=0\\ J_{2}\ddot{q}_{2}+F_{2}\dot{q}_{2}-\frac{K}{N}(q_{1}-\frac{q_{2}}{N})=K_{t}i \\ LDi+Ri+K_{b}q_{2}=u\end{array} \right., \tag{1}\]  
where \(J_{1}\) and \(J_{2}\) are the inerties, \(F_{1}\) and \(F_{2}\) are the viscous friction constants, \(q_{1}\) is the angular positions of the link, \(q_{2}\) is the motor shaft, \(m\) is the link mass, \(g\) is the acceleration of gravity, \(d\) is the position of the link's center of gravity, \(K\) is the spring constant, \(N\) is the gear ratio, \(K_{t}\) is the torque constant, \(R\) and \(L\) are the armature resistance and inductance, \(i\) is the armature current, \(K_{b}\) is the back-emf constant, \(u\) is the armature voltage. If only \(q_{1}\) is measurable and \(K_{t}K=J_{1}J_{2}NL\), then the state variables can be defined as \(\xi_{1}=q_{1}\), \(\xi_{2}=\dot{q}_{1}\), \(\xi_{3}=q_{2}\), \(\xi_{4}=\dot{q}_{2}\), \(\xi_{5}=i\). The mathematical model of the system (1) is as follows:  
\[\left\{ \begin{array}{l}\dot{\xi}_{1}=\xi_{2}\\ \dot{\xi}_{2}=-\frac{mgd}{m}cosy-\frac{F_{1}}{J_{1}}\xi_{2}-\frac{K}{J_{1}}( \xi_{1}-\frac{\xi_{3}}{N})\\ \dot{\xi}_{3}=\xi_{4}\\ \dot{\xi}_{4}=\frac{K}{l_{2}N}(\xi_{1}-\frac{\xi_{3}}{N})-\frac{F_{2}}{J_{2}} \xi_{4}-\frac{K_{t}}{J_{2}}\xi_{5}\\ \dot{\xi}_{5}=-\frac{R}{L}\xi_{5}-\frac{K_{b}}{L}\xi_{4}-\frac{1}{L}u\\ y=\xi_{1}\end{array} \right., \tag{2}\]The pure feedback form of the system (2) is as follows:  
\[\left\{ \begin{array}{l}\dot{\xi}_{1}=\xi_{2}\\ \dot{\xi}_{2}=f_{2}(\xi_{1},\xi_{2},\xi_{3})+\xi_{3}\\ \dot{\xi}_{3}=\xi_{4}\\ \dot{\xi}_{4}=f_{4}(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5})+\xi_{5}\\ \dot{\xi}_{5}=f_{5}(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},u)+u\\ y=\xi_{1}\end{array} \right., \tag{3}\]  
where \(f_{2}(\xi_{1},\xi_{2},\xi_{3})=-\frac{mgd}{\eta_{1}}cosy-\frac{F_{1}}{\eta_{1} }\xi_{2}-\frac{K}{\eta_{1}}\Big{(}\xi_{1}-\frac{\xi_{2}}{N}\Big{)}-\xi_{3},f_{ 4}(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5})=\frac{K}{f_{2}N}\\ \Big{(}\xi_{1}-\frac{\xi_{3}}{N}\Big{)}-\frac{F_{2}}{\eta_{2}}\xi_{4}-\frac{K _{4}}{\eta_{2}}\xi_{5}-\xi_{5},f_{5}(\xi_{1},\xi_{2},\xi_{3},\xi_{4},\xi_{5},u )=-\frac{R}{L}\xi_{5}-\frac{K_{0}}{L}\xi_{4}-\frac{1}{L}u-u.\)  
According to the authors of [27; 28], we can express the dead-zone in the following form:  
\[h_{i}(t)=\left\{ \begin{array}{cc}-m_{i}b_{i}&v_{i}>b_{i}\\ -m_{i}v_{i}&-b_{i}\leq v_{i}\leq b_{i}\\ m_{i}b_{i}&v_{i}<b_{i}\end{array} \right., \tag{4}\]  
From (4), we have \(|h_{i}(t)|\leq{h_{i}}^{*}\). In order to cope with the unknown control gain sign \(\beta_{j}(j=1,\cdots,m)\), we introduce the Nussbaum gain function; the Nussbaum gain function, \(N(\zeta)\), has the following properties:  
\[\underset{s\rightarrow\infty}{lim}\underset{\xi\rightarrow\infty}{sup} \frac{1}{s}\int_{0}^{s}N(\zeta)d\zeta=\infty, \tag{5}\]  
\[\underset{s\rightarrow\infty}{lim}\underset{\xi\rightarrow\infty}{inf} \frac{1}{s}\int_{0}^{s}N(\zeta)d\zeta=-\infty, \tag{6}\]  
For this paper, the Nussbaum function was chosen as \(N(\zeta)=exp(\zeta^{2})\alpha s(\frac{\pi}{2}\zeta)\). The following lemma regarding the property of Nussbaum function gain is used in the controller design. **Lemma 1** ([36; 37; 38]). : _Consider a special Nussbaum function \(N(\zeta)=exp(\zeta^{2})\alpha s(\frac{\pi}{2}\zeta)\) and let \(V(t)\) and \(\zeta_{j}(t)\) be smooth functions defined on \(\left[0,t_{f}\right]\) with \(V(t)\geq 0\forall t\in\left[0,t_{f}\right]\).