## 2 Dynamic Model  
The system studied in this paper is a flexible single-link manipulator at the end of the arm in a plane. A schematic diagram of the system is shown in Figure 1. In the natural state of the flexible manipulator, with the central axis of the flexible manipulator as the axis and the rotation axis of the motor and flexible manipulator as the origin, a world coordinate system is established. After the deformation, the direction of tangent line of the central axis of the flexible manipulator is the y axis, and a fixed coordinate system is established with the origin of world coordinate system. Some basic parameters and units of the flexible manipulator defined below are shown in Table 1:  
According to [PERSON]'s principle [17; 18; 19; 20], the relationship between the kinetic energy of the system and the potential energy, as well as the non-conservative force, is expressed during the time interval \(t_{1}\) to \(t_{2}\):  
\[\int_{t_{1}}^{t_{2}}\left(\varepsilon T-\varepsilon U-\varepsilon W_{nc}\right) dt=0 \tag{1}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Physical Meaning** & **Unit** \\ \hline \(L\) & Length of the flexible manipulator in natural state & m \\ \(EI\) & The bending stiffness & N\(\cdot\)m\({}^{2}\) \\ \(m\) & Load mass at the end of the flexible manipulator & kg \\ \(I\) & Moment of inertia of the flexible manipulator & kg\(\cdot\)m\({}^{2}\) \\ \(\tau(t)\) & Input torque of the initial end motor & N\(\cdot\)m \\ \(M(t)\) & End load motor input control torque & N\(\cdot\)m \\ \(\theta(t)\) & Rotation angle of joint of the flexible manipulator & rad \\ \(\rho\) & Mass per unit length of the flexible manipulator & kg/m \\ \(y(x,t)\) & Elastic deformation of the flexible manipulator at point \(x\) & m \\ \hline \hline \end{tabular}
\end{table}
Table 1: The basic parameters and units of the flexible manipulator.  
Figure 1: Structure diagram of the flexible manipulator.  
Here, \(T\) is the kinetic energy of the flexible manipulator, \(U\) is the potential energy of the flexible manipulator, \(W_{nc}\) is the power by non-conservative force of the flexible manipulator, and \(\varepsilon(A)\) means the variation of \((A)\). The kinetic energy of the system is expressed as the following equation:  
\[T=\frac{1}{2}I\dot{\theta}^{2}(t)+\frac{1}{2}\int_{0}^{L}\rho\dot{u}^{2}(y,t)dy +\frac{1}{2}m\dot{u}^{2}(L,t) \tag{2}\]  
where \(u(x,t)\) is the offset of the flexible manipulator at any time \(t\). The potential energy of the system is expressed as following equation:  
\[U=\frac{1}{2}\int_{0}^{L}EIx^{\prime 2}(y,t)dy \tag{3}\]  
The power by non-conservative force of the flexible manipulator is expressed as the following equation:  
\[W_{nc}=\tau(t)\theta(t)+F(t)u(L,t) \tag{4}\]  
In the case of initial position 0, the boundary conditions are \(u(0,t)=0\), \(u^{\prime}(0,t)=\theta\). Bringing Equations (2)-(4) into Equation (1), the following equation can be obtained:  
\[\int_{t_{1}}^{t_{2}}(\varepsilon T-\varepsilon U+\varepsilon W_{nc})dt=-\int_ {t_{1}}^{t_{2}}\int_{0}^{L}A\varepsilon u(x,t)dydt-\int_{t_{1}}^{t_{2}}B \varepsilon u^{\prime}(0,t)dt-\int_{t_{1}}^{t_{2}}C\varepsilon u(L,t)dt-\int _{t_{1}}^{t_{2}}D\varepsilon u^{\prime}(L,t)dt=0 \tag{5}\]  
Parameters of \(A\), \(B\), \(C\), and \(D\) are respectively expressed as: \(A=\rho\dot{u}(x,t)+EIu^{(4)}(L,t)\), \(B=I\ddot{u}^{\prime}(0,t)-E\dot{I}u^{\prime\prime}(0,t)-\tau(t)\), \(C=m\ddot{u}(L,t)-EIu^{(3)}(L,t)-M(t)\), \(D=EIu^{\prime\prime}(L,t)\). As \(\varepsilon u(x,t)\), \(\varepsilon u^{\prime}(0,t)\), \(\varepsilon u(L,t)\) and \(\varepsilon u^{\prime}(L,t)\) are independent variables, all of these equations are linearly independent, \(A=B=C=D=0\). Thus, these partial differential equations (PDEs) [21; 22] of the flexible manipulator are established as follows:  
\[\rho\ddot{u}(x,t)=-EIu^{(4)}(x,t) \tag{6}\]  
\[\tau(t)=I\ddot{u}(0,t)-EIu^{\prime\prime}(0,t) \tag{7}\]  
\[M(t)=m\ddot{u}(L,t)-EIu^{(3)}(L,t) \tag{8}\]  
\[u^{\prime\prime}(L,t)=0 \tag{9}\]