## 2 System Model
### Dynamic Modeling of the Coupled System  
Since the system moves on the horizontal plane, the gravity is negligible. Through employing the Lagrange method, the dynamic equation of the system should satisfy the following form:  
\[\frac{d}{dt}\left(\frac{\partial L}{\partial q_{i}}\right)-\frac{\partial L}{ \partial q_{i}}+\frac{\partial D}{\partial q_{i}}=Q_{i},\ i=0,1,2,\ldots \tag{3}\]where \(L\) represents the Lagrange multiplier, \(q_{i}\) denotes generalized coordinates, \(D\) is the Rayleigh dissipated energy, and \(Q_{i}\) indicates the generalized force acting on the generalized coordinate \(q_{i}\). The Lagrange multiplier \(L\) can be expressed as:  
\[L=T-V \tag{4}\]  
where \(T\) and \(V\) represent the kinetic and potential energy of the system, separately. Consider a coupling system of dual joints and a flexible beam in a horizontal plane, which can be found in Figure 2. Figure 2 illustrates the structural deformation diagram of the flexible plate, where the two-dimensional plate is simplified into a one-dimensional beam model. In fact, not only is the torsional rigid body motion \(\theta\) shown, but also the bending deformation of the plate based on the torsional rigid body motion \(\theta\) is demonstrated. The body-fixed coordinate \((x,y)\) is introduced, aiming to efficiently depict the local deflection of the beam; the origin is at the joint. Then, in the inertial frame, the position \((x,y)\) of a point on the deformed centerline of the beam can be expressed as:  
\[\left\{ \begin{array}{l}X=x\cos\theta-y(x,t)\sin\theta\\ Y=x\sin\theta+y(x,t)\cos\theta\end{array} \right. \tag{5}\]  
where \(x\) indicates the location of the undeformed point on the flexible beam and \(y(x,t)\) represents the transverse elastic displacements with \(\theta\) denoting the angular motions of the joint. According to Equation (5), we obtain  
\[\left\{ \begin{array}{l}\dot{X}=(\dot{x}\cos\theta-x\dot{\theta}\cos\theta)-( \dot{y}\sin\theta+y\dot{\theta}\cos\theta)\\ \dot{Y}=(\dot{x}\sin\theta+x\dot{\theta}\cos\theta)+(\dot{y}\cos\theta-y\dot{ \theta}\sin\theta)\end{array} \right. \tag{6}\]  
Then, the absolute velocity \(v\) at any point \((x,y)\) on the beam can be formulated as:  
\[v(x,y)=\left(\dot{X}^{2}+\dot{Y}^{2}\right)^{\frac{1}{2}}=\left[\left(x^{2}+y ^{2}\right)\dot{\theta}^{2}+2\left(x\dot{y}-\dot{x}y\right)\dot{\theta}+\left( \dot{x}^{2}+\dot{y}^{2}\right)\right]^{\frac{1}{2}} \tag{7}\]  
For \(\dot{x}=0\), the absolute velocity \(v\) can be calculated as follows:  
\[v(x,y)=\left(\dot{X}^{2}+\dot{Y}^{2}\right)^{\frac{1}{2}}=\left[\left(x^{2}+y ^{2}\right)\dot{\theta}^{2}+2x\dot{y}\dot{\theta}+\dot{y}^{2}\right]^{\frac{1 }{2}} \tag{8}\]  
Figure 2: The simplified dynamic model for the coupling system of dual joints and single flexible plate. Ignoring the longitudinal deformation of the beam, the kinetic energy of the system is  
\[ \begin{split} T&=2\cdot\frac{1}{2}J\dot{\theta}^{2}+ \frac{1}{2}J_{0}^{l}\rho Av(x,y)^{2}dx\\ &=J\dot{\theta}^{2}+\frac{1}{2}\int_{0}^{l}\rho A\left[(x^{2}+y^{2 })\dot{\theta}^{2}+2x\dot{y}\dot{\theta}+\dot{y}^{2}\right]dx\end{split}  \tag{9}\]  
where \(\rho\) refers to the density of the beam, \(A\) represents the cross-sectional area of the beam, \(J\) indicates the moment of inertia of one joint, and \(l\) denotes the length of flexible plate, as shown in Figure 1.  
The potential energy of the system is  
\[V=\frac{1}{2}\int_{0}^{l}EI\left[\frac{\partial^{2}y(x,t)}{\partial x^{2}} \right]^{2}dx+2\cdot\frac{1}{2}k\theta^{2} \tag{10}\]  
where \(EI\) indicates the bending stiffness of the beam and \(k\) represents the torsional stiffness of one joint, as shown in Equation (1). Due to the damping characteristics of the joint mechanism, the dissipated energy of dual joints becomes  
\[D=2\cdot\frac{1}{2}c\dot{\theta}^{2} \tag{11}\]  
where \(c\) is the damping coefficient of one joint, as shown in Equation (2). Here, in our theoretical model, only the electromagnetic damping of dual joints is considered, while the structural damping of the flexible plate is not considered.