## 3 Kinetostatic Model
### CBCM of Three-Stage CBBM  
As shown in Figure 7, the local coordinate system \(O_{1}X_{1}Y_{1}\) is defined at the fixed end, \(O_{1}\), of the first flexible beam, and the local coordinate system \(O_{2}X_{2}Y_{2}\) is defined at the fixed end of the second flexible beam. \(L_{1}\) and \(L_{2}\) represent the lengths of the two flexible  
Figure 6: Element _ij_ at its deflection position.  
beams along directions \(X_{1}\) and \(X_{2}\) in the local coordinate systems \(O_{1}X_{1}Y_{1}\) and \(O_{2}X_{2}Y_{2}\), respectively. As the deformation of the rigid beam is much smaller than that of the flexible beam, length \(L_{3}\) of the rigid beam can be expressed as the length of the line connecting points A and B. \(\tau_{1}\) and \(\tau_{2}\) represent the angles between local coordinate systems \(O_{1}X_{1}Y_{1}\) and \(O_{2}X_{2}Y_{2}\) and natural coordinate system \(OXY\), respectively, and \(\tau_{3}\) represents the angle between the line connecting points A and B and the \(X\) axis in the natural coordinate system. Owing to the geometric nonlinear deformation of the internal flexible beam of the bistable mechanism during motion, modeling is complicated. We adopt the CBCM to describe the flexible bistable mechanism. The CBCM discretizes the initial circular beam into \(N\) elements, which are separately modeled using the BCM. Through discretization, the CBCM eliminates the limitation of the BCM for describing curved beams.  
Figure 8 shows the free body diagram of the bistable limbs of the proposed CBBM with load balance to the intermediate rigid segment.  
When the displacement of the intermediate shuttle is \(d\), the deflection geometry of flexible circular beam \(i\) (\(i=1,2\)) is provided by [28]:  
\[\begin{array}{l}\alpha_{i}=\sum\limits_{j=1}^{N}\alpha_{ij}\quad\lambda_{i} =\sum\limits_{j=1}^{N}\lambda_{ij}\quad\delta_{i}=\sum\limits_{j=1}^{N} \delta_{ij}\\ F_{i}=\sum\limits_{j=1}^{N}F_{ij}\quad P_{i}=\sum\limits_{j=1}^{N}P_{ij}\\ \\ \alpha_{1}=\alpha_{2}=\alpha_{3}\end{array} \tag{10}\]  
Figure 8: Free body diagram of circular-beam bistable limb in proposed mechanism.  
Figure 7: Geometric parameters of circular-beam bistable limb.  
In addition,  
\[[P_{1}\cos\tau_{1}-F_{1}\sin\tau_{1}]+[P_{2}\cos(\pi+\tau_{2})-F_{2}\sin(\pi+\tau_ {2})]=0 \tag{12}\]  
\[[P_{1}\sin\tau_{1}+F_{1}\cos\tau_{1}]+[P_{2}\sin(\pi+\tau_{2})+F_{2}\cos(\pi+ \tau_{2})]=0 \tag{13}\]  
\[M_{1}+M_{2}-F_{1}L_{3}\cos(\tau_{3}-\tau_{1}+\alpha_{3})+P_{1}L_{3}\sin(\tau_{ 3}-\tau_{1}+\alpha_{3})=0 \tag{14}\]  
The geometric relationship when the shuttle displacement is \(d\) is expressed as  
\[\left[\begin{array}{c}L_{X}\\ L_{Y}-d\end{array}\right] = \left[\begin{array}{cc}\cos\tau_{1}&-\sin\tau_{1}\\ \sin\tau_{1}&\cos\tau_{1}\end{array}\right]\left[\begin{array}{c}L_{1}+ \lambda_{1}\\ \delta_{1}\end{array}\right] \tag{15}\] \[+ L_{3}\left[\begin{array}{c}\cos(\alpha_{3}+\tau_{3})\\ \sin(\alpha_{3}+\tau_{3})\end{array}\right]\] \[- \left[\begin{array}{cc}-\cos\tau_{2}&\sin\tau_{2}\\ -\sin\tau_{2}&-\cos\tau_{2}\end{array}\right]\left[\begin{array}{c}L_{2}+ \lambda_{2}\\ \delta_{2}\end{array}\right]\]  
The load balance for the second flexible beam column can be obtained as follows:  
\[F_{i}=P_{2}\sin(\pi+\tau_{2})+F_{2}\cos(\pi+\tau_{2}) \tag{16}\]  
\[F_{t}=N_{t}F_{i} \tag{17}\]  
where \(F_{t}\) is the input force required to displace the mechanism and \(N_{t}\) is the number of fixed limbs bistabiles. The CBBM force-deformation relationship provided by the CBCM can be obtained from Equations (9)-(17). The relationship between \(F_{t}\) and \(d\) was solved by using the 'fsolve' function in MathWorks MATLAB.