## 3 Kinetostatic Modeling and FEA
### Model of Distributed Compliant Beams  
Figure 2 illustrates the compliant beam between nodes \(j\) and \(k\), denoted as \(Beam-jk\). Node \(k\) represents the initial input position of the current compliant beam, which is subjected to the input force and displacement from the compliant beam or actuator in the previous stage at the same node. Node \(j\) represents the final input position of the compliant beam, which is also its output position. The force and displacement generated at this point will serve as the input conditions for the next stage of the compliant beam at node \(j\) or the final output position. Each node has three degrees of freedom: axial displacements \(u_{j}\), \(u_{k}\); lateral displacements \(w_{j}\), \(w_{k}\); and rotations \(\theta_{j}\), \(\theta_{k}\). Beam motion or deformation in the plane can be described using coordinate systems. \(O_{j}^{jk}-X_{j}^{ik}Y_{j}^{jk}\) and \(O_{k}^{jk}-X_{k}^{jk}Y_{k}^{jk}\) represent local coordinate systems. The _X-axis_ of both coordinate systems is aligned with the beam, with its positive direction pointing from the input to the output positions. The above-mentioned forces and displacements can also be represented in any reference coordinate frames. The positive direction of the y-coordinate is defined as the vertical direction, and the positive direction of the x-coordinate is defined  
Figure 2: Definition of coordinate frame for a single strip beam. as the horizontal direction. \(O_{j}^{t,jk}-X_{j}^{t,ik}Y_{j}^{t,jk}\) and \(O_{k}^{t,jk}-X_{k}^{t,jk}Y_{k}^{t,jk}\) are established. The statics model of general small deformation compliant beams can be uniformly described using the concept of a stiffness matrix. This matrix exhibits a form similar to [PERSON]'s law [31]. Consequently, the force-position equilibrium relationship for distributed compliant beams can be established as Equation (1). \[F_{jk}=K_{jk}U_{jk}, \tag{1}\]  
where \(F_{jk}\) represents the resultant force acting on the beam at points \(j\) and \(k\), \(U_{jk}\) represents the resultant displacement generated at points \(j\) and \(k\), and \(K_{jk}\) represents the stiffness matrix that maps displacement to force. The specific parameters of \(K_{jk}\) are calculated by considering the mechanical properties, geometry, and applied loads of the compliant beam. The stiffness matrix used in this paper is referenced from [27; 28]. Vectors \(F_{jk}\), \(U_{jk}\), and \(K_{jk}\) can be represented, respectively, by formulas Equations (2)-(4). \[F_{jk}=\left[f_{j,x}^{jk}\;f_{j,y}^{jk}\;m_{j,z}^{jk}\;f_{k,x}^{jk}\;f_{k,y}^{ jk}\;m_{k,z}^{jk}\right], \tag{2}\]  
\[U_{jk}=\left[u_{j,x}^{jk}\;u_{j,y}^{jk}\;u_{j,\theta}^{jk}\;u_{k,x}^{jk}\;u_{ k,y}^{jk}\;u_{k,\theta}^{jk}\;\right], \tag{3}\]  
\[K_{jk}=\left[ \begin{array}{ccccc}\frac{EA_{jk}}{L_{jk}}&0&0&-\frac{EA_{jk}}{ L_{jk}}&0&0\\ 0&\frac{12 EI_{jk}}{L_{jk}^{3}}&\frac{6 EI_{jk}}{L_{jk}^{3}}&0&-\frac{12 EI_{jk}}{L_{jk} }&\frac{6 EI_{jk}}{L_{jk}^{3}}\\ 0&\frac{6 EI_{jk}}{L_{jk}^{3}}&\frac{4 EI_{jk}}{L_{jk}}&0&-\frac{6 EI_{jk}}{L_{jk} }&\frac{2 EI_{jk}}{L_{jk}}\\ -\frac{EA_{jk}}{L_{jk}}&0&0&\frac{EA_{jk}}{L_{jk}}&0&0\\ 0&-\frac{12 EI_{jk}}{L_{jk}^{3}}&-\frac{6 EI_{jk}}{L_{jk}^{3}}&0&\frac{12 EI_{jk}} {L_{jk}}&-\frac{6 EI_{jk}}{L_{jk}}\\ 0&\frac{6 EI_{jk}}{L_{jk}^{3}}&\frac{2 EF_{jk}}{L_{jk}}&0&-\frac{6 EI_{jk}}{L_{jk} ^{3}}&\frac{4 EI_{jk}^{3}}{L_{jk}}\end{array} \right], \tag{4}\]  
where \(E\) represents the elastic modulus of the beam, \(I\) represents the moment of inertia of the cross-section, \(A\) represents the cross-sectional area, and \(L\) denotes the length of the beam. Since discussions on the compliance (or stiffness) matrix are meaningful only within the same coordinate system, to establish the compliance (or stiffness) matrix for the entire compliant system, it is necessary to transform the coordinates of the output end to the input end coordinate system using a rotation matrix. Based on the established global coordinate system, this rotation matrix is established as shown in Equation (5):  
\[R=\left[