# Modeling and Analysis of Dynamics of Rigid-Flexible Coupled Parallel Robots
## 4 Simulation-Based Vibration Analysis  
Since the multi-energy domain flexible parallel robotic system is a complex system coupled with a control and mechanical subsystem, both the control and mechanical subsystem may cause system vibration. According to these two vibration sources, the vibration analysis of different vibration forms is discussed in the frequency domain. The servomotor in the control subsystem provides the driving force to drive the mechanical subsystem and receives the actual position/velocity feedback from the mechanical subsystem. Therefore, for the vibration analysis of the control subsystem, the coupling effect of the mechanical subsystem should also be considered. The velocity loop is an important part of the servo motor control system, and its control performance is an important index for evaluating the overall system performance. Without considering the mechanical subsystem, the frequency domain toolbox of 20-sim software can be used to get the closed-loop of different speed loop controller parameters. Based on the global bond graph, the corresponding state equations for this system are derived:  
\[\begin{cases}\dot{\mathbf{p}}_{Ni}=\mathbf{A}d_{H_{Ni}^{i}}^{T}\mathbf{e}_{Ei}+\mathbf{W}_{g}^{ Ni}-\mathbf{A}d_{H_{Ni}^{i}}^{T}\mathbf{e}_{Hi}-\mathbf{A}d_{H_{Ni}^{i}}^{T}\mathbf{e}_{Hi}\\ \dot{\mathbf{p}}_{Uli}=\mathbf{A}d_{H_{lli}^{i}}^{T}\mathbf{e}_{Hi}+\mathbf{W}_{g}^{Li}-\mathbf{A}d_{ H_{lli}^{i}}^{T}\mathbf{e}_{Ki}\\ \dot{\mathbf{p}}_{U^{i}}=\mathbf{A}d_{H_{U^{i}}^{i}}^{T}\mathbf{e}_{Hi}+\mathbf{W}_{g}^{U^{i}i}- \mathbf{A}d_{H_{U^{i}}^{i}}^{T}\mathbf{e}_{Ki}\\ \dot{\mathbf{p}}_{p}=\sum\limits_{i=1}^{3}\mathbf{A}d_{H_{p}^{i}}^{T}\mathbf{e}_{Ki}+\sum \limits_{i=1}^{3}\mathbf{A}d_{H_{p}^{i}}^{T}\mathbf{e}_{Ki}+\mathbf{W}_{g}^{p}+\mathbf{A}d_{H_{ p}^{i}}^{T}\mathbf{e}_{o}\\ \dot{\mathbf{p}}_{Ml1}=\mathbf{e}_{Hi}\mathbf{M}\mathbf{T}\mathbf{e}_{i1,xHi}+\mathbf{e}_{Ki}\mathbf{M}\mathbf{ T}\mathbf{e}_{i1,xKi}-\frac{\mathbf{q}_{c11}}{\mathbf{c}_{i1}}\\ \dot{\mathbf{p}}_{Ml2}=\mathbf{e}_{Hi^{\prime}}\mathbf{M}\mathbf{T}\mathbf{e}_{i2,xHi^{\prime}}+ \mathbf{e}_{Ki^{\prime}}\mathbf{M}\mathbf{T}\mathbf{e}_{i2,xKi^{\prime}}-\frac{\mathbf{q}_{c2i}}{ \mathbf{c}_{i2}}\\ \dot{\mathbf{q}}_{c11}=\frac{\mathbf{p}_{Ml1}}{\mathbf{I}_{Ml1}}\\ \dot{\mathbf{q}}_{c2i}=\frac{\mathbf{p}_{Ml2}}{\mathbf{I}_{Ml2}}\\ \dot{\mathbf{q}}_{cEi}=\dot{\mathbf{\theta}}_{i}\mathbf{M}\mathbf{T}\mathbf{F}=\frac{\mathbf{p}_{Ni}}{ \mathbf{I}_{Ni}}\mathbf{A}d_{H_{Ni}^{i}}\\ \dot{\mathbf{q}}_{cHi}=\frac{\mathbf{p}_{Ni}}{\mathbf{I}_{Ni}}\mathbf{A}d_{H_{Ni}^{i}}-\frac{ \mathbf{p}_{Mi}}{\mathbf{I}_{Uli}}\mathbf{A}d_{H_{lli}^{i}}-\frac{\mathbf{p}_{Mi1}}{\mathbf{I}_{Mi1} }\mathbf{M}\mathbf{T}\mathbf{F}_{c11,xHi}\\ \dot{\mathbf{q}}_{cKi}=\frac{\mathbf{p}_{Uli}}{\mathbf{I}_{Uli}}\mathbf{A}d_{H_{lli}^{i}}-\frac{ \mathbf{p}_{P}}{\mathbf{I}_{P}}\mathbf{A}d_{H_{p}^{i}}-\frac{\mathbf{p}_{Mi1}}{\mathbf{I}_{Mi1}} \mathbf{M}\mathbf{T}\mathbf{F}_{c11,xKi}\\ \dot{\mathbf{q}}_{cH^{\prime}i}=\frac{\mathbf{p}_{Ni}}{\mathbf{I}_{Ni}}\mathbf{A}d_{H_{lli}^{i}} -\frac{\mathbf{p}_{Ui}}{\mathbf{I}_{Ui}}\mathbf{A}d_{H_{U^{i}}^{i}}-\frac{\mathbf{p}_{Mi2}}{ \mathbf{I}_{Mi2}}\mathbf{M}\mathbf{T}\mathbf{F}_{c2,xHi}\\ \dot{\mathbf{q}}_{cKi^{\prime}i}=\frac{\mathbf{p}_{Ui^{\prime}}}{\mathbf{I}_{Ui^{\prime}}} \mathbf{A}d_{H_{U^{i}}^{i}}-\frac{\mathbf{p}_{P}}{\mathbf{I}_{P}}\mathbf{A}d_{H_{p}^{i}}- \frac{\mathbf{p}_{Mi2}}{\mathbf{I}_{Mi2}}\mathbf{M}\mathbf{T}\mathbf{F}_{c2,xKi^{\prime}i}\\ \Frequency response is shown in Figure 14. The data in parentheses in the Figure are the scaling factor \(k_{pv}\) and the integration time constant \(k_{iv}\). From Figure 14, it can be seen that the higher the controller parameters, the greater the bandwidth of the control subsystem and the faster the response of the system. However, the increase in bandwidth also increases the shear frequency of the system, the phase stability margin of the system decreases, and in the middle-frequency band, the phase lag is less than the lower bandwidth at the same frequency.