# Modeling and Analysis of Dynamics of Rigid-Flexible Coupled Parallel Robots
## 2 Modal Vector Bond Graph
### Hybrid Bond Graph Rigid-Flexible Coupled Rod System  
As shown in Figure 4, it is a rod system composed of two connecting rods connected by a revolute joint, assuming that the connecting rod \(i\) is a rigid body, and the connecting rod \(k\) is an elongated rod of equal cross-section with a length of \(L\), which is regarded as a flexible rod. The center of the revolute joint is \(B\), the angle of the flexible rod is \(\theta_{j}\), \(O\)-\(xy\) is the inertial coordinate system \(\Psi_{0}\), the center of mass of the flexible rod is \(k\), and \(Ock\)-\(XckYck\) is the link coordinate system \(\Psi_{ck}\). The flexible rod modal vector bond graph unit module model, combined with the rigid body and joint screw bond graph model, is used to construct the rigid-flexible coupled rod system bond graph model, as shown in Figure 5.  
In the integral causality, the \(p\) variable on the energy element \(I\)-element and the \(q\) variable on the \(C\)-element are chosen as the state variables of the system, and the equation of the state of the system is obtained by deriving according to the rules. Figure 4: Rigid–flexible coupling linkage system. Figure 5: Bond graph model of rigid–flexible coupled linkage system. \[ \begin{cases}\dot{p}_{2}=e_{1}-e_{3}-e_{7}MTF_{\text{\textcircled{1}}}\\ \dot{p}_{14}=e_{7}MTF_{\text{\textcircled{2}}}-\frac{q_{14}}{c_{13}}\\ \dot{p}_{17}=e_{7}MTF_{\textcircled{3}}-e_{18}-e_{19}\\ \dot{q}_{13}=\frac{p_{14}}{14}\\ \dot{q}_{9}=\frac{p_{2}}{l_{2}}MTF_{\textcircled{1}}-\frac{p_{14}}{l_{14}}MTF_{ \textcircled{2}}-\frac{p_{17}}{l_{17}}MTF_{\textcircled{3}}\end{cases}  \tag{9}\]  
where one of the outputs of the system is the joint binding,  
\[ \begin{array}{ll}e_{7}&=e_{8}+e_{9}=R_{8}f_{8}+\frac{q_{9}}{C_{9}}\\ &=R_{8}\left(\frac{p_{2}}{l_{2}}MTF_{\textcircled{1}}-\frac{p_{14}}{l_{14}}MTF_ {\textcircled{2}}-\frac{p_{12}}{l_{17}}MTF_{\textcircled{3}}\right)+\frac{q_{9} }{C_{9}}\end{array}  \tag{10}\]  
Describing the system state Equation (10) using the matrix form, the  
\[\dot{\widetilde{X}}=A\widetilde{\dot{X}}+\mathbf{U} \tag{11}\]  
Among them, \(\widetilde{\dot{X}}= \begin{bmatrix}\mathbf{\mathcal{I}}_{i}\mathbf{T}_{i}^{i}&\mathbf{K}_{ \text{br}}\dot{\mathbf{\eta}}_{kn}&\mathbf{\mathcal{I}}_{k}\dot{\mathbf{\eta}}_{k0}&\mathbf{ \eta}_{kn}&\mathbf{q}_{9}\end{bmatrix} ^{\text{T}}\);  
\[\mathbf{A}= \begin{bmatrix}-\frac{R_{8}}{\mathbf{\mathcal{I}}_{i}}\left(\mathbf{A}\mathbf{d}_{H _{i}^{j}}\right)^{2}&\frac{R_{8}}{\mathbf{M}}R_{j}^{\text{circled{3}}}\widetilde{ \dot{Y}}(x_{B})\mathbf{A}\mathbf{d}_{H_{i}^{j}}&-\frac{R_{8}}{\mathbf{\mathcal{I}}_{k}}\mathbf{ A}\mathbf{d}_{H_{k}^{j}}\mathbf{A}\mathbf{d}_{H_{i}^{j}}&0&-\frac{1}{C_{9}}\mathbf{A}\mathbf{d}_{H_{i}^{j}} \\ \frac{R_{8}}{\mathbf{\mathcal{I}}_{i}}\mathbf{A}\mathbf{d}_{H_{i}^{j}}\mathbf{R}_{\text{Y}}^{ \text{circled{3}}}\widetilde{\dot{Y}}(x_{B})&-\frac{R_{8}}{\mathbf{M}}\left[R_{ \text{Y}}^{\text{circled{3}}}\widetilde{\dot{Y}}(x_{B})\right]^{2}&\frac{R_{8}} {\mathbf{\mathcal{I}}_{k}}\mathbf{A}\mathbf{d}_{H_{k}^{j}}\mathbf{R}_{\text{Y}}^{\text{circled{3 }}}\widetilde{\dot{Y}}(x_{B})&-K&\frac{1}{C_{9}}\mathbf{R}_{\text{Y}}^{\text{circled{3 }}}\widetilde{\dot{Y}}(x_{B})\\ \frac{R_{8}}{\mathbf{\mathcal{I}}_{i}}\mathbf{A}\mathbf{d}_{H_{k}^{j}}\mathbf{A}\mathbf{d}_{H_{k}^{ j}}&-\frac{R_{8}}{\mathbf{M}}\mathbf{R}_{\text{Y}}^{\text{circled{3}}}\widetilde{\dot{Y}}(x_{B}) \mathbf{A}\mathbf{d}_{H_{k}^{j}}&-\frac{R_{8}}{\mathbf{\mathcal{I}}_{k}}\left(\mathbf{A}\mathbf{d}_ {H_{k}^{j}}\right)^{2}&0&\frac{1}{C_{9}}\mathbf{A}\mathbf{d}_{H_{k}^{j}}\\ 0&\frac{1}{\mathbf{\mathcal{I}}_{i}}\mathbf{A}\mathbf{d}_{H_{k}^{j}}&-\frac{R_{8}}{\mathbf{M}} \mathbf{R}_{\text{Y}}^{\text{circled{3}}}\widetilde{\dot{Y}}(x_{B})&-\frac{R_{8}}{ \mathbf{\mathcal{I}}_{k}}\mathbf{A}\mathbf{d}_{H_{k}^{j}}&0&0\end{bmatrix}