## 2 Design Methodology
### Dynamic Analysis  
Using the PRBM [1], the flexure hinge is usually approximated as a traditional rigid rotary joint and a torsional spring to simplify and convert the CM into an equivalent rigid-body model. However, for more general investigation, this paper treats each flexure  
Figure 4: Compliant matrix analysis models: (**a**) compliant configuration, and (**b**) the equivalent dynamic model. hinge with 3 DOFs exhibiting axial, bending, and torsion deformation, to build the natural frequency mathematical model and the analytical displacement control model. This dynamic model derivation aims to facilitate a quick evaluation of the initial design phase, and assists in a preliminary prediction for the possibility to control it in the later stage. In a plane strain model, a flexure hinge exhibits three degree-of-freedoms, which include two translational movements \(\left(u_{x},\ u_{y}\right)\) and a rotational movement \(\left(\theta_{z}\right)\). The semi-circular elastic hinge in Figure 5a has been modelled as a 3-DOF joint in Figure 5b. The corresponding spring constants \(k_{x},\ k_{y}\), and \(k_{z}\) represent the three displacement possibilities. In the mathematical model of the mechanism in Figure 6, the notations \(\left(m_{i},\ L_{i}\right)\), \(\left(i=1,\ 2\ldots 4\right)\), and \(\theta_{j}\)\(\left(j=1,\ 2\right)\) are the mass, length, and rotaion angle of the corresponding rigid links, respectively. The dynamic parameters of the model, including translational stiffness \(\left(K_{ix},\ K_{iy}\right)\) and torsional stiffness \(\left(K_{iz}\right)\) of the flexure hinges, and the second moment of area of the rigid links \(\left(I_{i}\right)\), are presented and calculated using the Equations (23)-(26) [33; 35; 38]. \[K_{iz}=\frac{1}{C_{\theta}} \tag{23a}\] \[C_{\theta}=\frac{\Delta\theta_{z}}{M_{z}}=\frac{24R}{EhT^{3}(2R+T)(4R+T)^{3}} \Gamma_{\theta}\] (23b) \[\Gamma_{\theta}=T(4R+T)\Big{(}6R^{2}+4 RT+T^{2}\Big{)}+6R(2R+T)^{2}\sqrt{T(4R+T )}\text{arctan}\sqrt{1+\frac{4R}{T}}\] (23c) \[K_{ix}=\frac{1}{C_{x}}\] (24a) \[C_{x}=\frac{\Delta x}{F_{x}}=\frac{1}{Eh}\Bigg{[}\frac{2(2R+T)}{ \sqrt{T(4R+T)}}\text{arctan}\sqrt{1+\frac{4R}{T}}-\frac{\pi}{2}\Bigg{]}\] (24b) \[K_{iy}=\frac{1}{C_{y}}\] (25a) \[C_{y}=\frac{\Delta y}{F_{y}}=\frac{3}{4 EhT(2R+T)}(\Psi_{1}+\Psi_ {2}+\Psi_{3}+\Psi_{4})\] (25b) \[\Psi_{1}=2R(2+\pi)+\pi T\] (25c) \[\Psi_{2}=\frac{8R^{3}\big{(}44R^{2}+28 RT+5T^{2}\big{)}}{(4R+T)^{2}T^{2}} \tag{25d}\]  
\[K_{iz}=\frac{1}{C_{\theta}} \tag{23a}\]0 Coordinates of a semicircular elastic hinge (**a**) and its equivalent PRB model (**b**). \[K_{iz}=\frac{1}{C_{\theta}} \tag{23a}\]1 \[K_{iz}=\frac{1}{C_{\theta}} \tag{23a}\]2 (25f) \[K_{iz}=\frac{1}{C_{\theta}} \tag{23a}\]3  
where, \(E\) is the Young's modulus of the material; \(T_{i}\), \(R_{i}\), and \(h\) are the minimum thickness, radius, and height of the elastic hinges, respectively. When the mechanism in Figure 6 is operating, the total dynamic energy is calculated as:  
\[K_{iz}=\frac{1}{C_{\theta}} \tag{23a}\]4  
Cascading the above general formula for each link, Equation (28) can be easily obtained:  
\[ \begin{array}{c}T_{0}=\frac{1}{2}m_{1}\big{(}y_{in}\big{)}^{2}+m_{2}\big{(} y_{2x}\big{)}^{2}+I_{2x}\dot{\theta}_{1}^{2}+2m_{3}\big{(}\dot{y}_{3x}\big{)}^{2}+2I_{3x} \dot{\theta}_{2}^{2}+\frac{1}{2}m_{4}\big{(}\dot{y}_{5}\big{)}^{2}\\ =\frac{1}{2}m_{1}\big{(}\dot{y}_{in}\big{)}^{2}+m_{2}\big{(}\frac{\dot{y}_{2}+ \dot{y}_{3}}{2}\big{)}^{2}+I_{2x}\big{(}\frac{\dot{y}_{3}-\dot{y}_{2}}{L_{1}+ L_{2}}\big{)}^{2}+2m_{3}\Big{(}\frac{\dot{y}_{4}+\dot{y}_{5}}{2}\big{)}^{2}+2I_{3x} \Big{(}\frac{\dot{y}_{5}-\dot{y}_{4}}{L_{3}+L_{4}}\Big{)}^{2}+\frac{1}{2}m_{4} \big{(}\dot{y}_{5}\big{)}^{2}\end{array}  \tag{28}\]  
The total strain energy is also calculated using Equation (29):  
\[