# Feasibility Study of a Gripper with Thermally Controlled Stiffness of Compliant Jaws
## 2 Theoretical Basis  
The compound parallelogram mechanism (Figure 1) is a well-behaved translational joint with a high compliance in the direction of the degree of freedom (primary motion) and very large stiffnesses in the directions of the degree of constraints. It is also a symmetrical design without undesired parasitic motions inherently accompanying the primary motion. The primary motion of the compound parallelogram mechanism (i.e., the jaw's gripping motion) is controlled by the equation below [8]:  
\[y_{\mathrm{s}}=\frac{f_{\mathrm{y}}}{48+2.4(\frac{12\
u_{\mathrm{s}}^{2}+2p_{ \mathrm{in}}/d}{1/d+y_{\mathrm{s}}^{2}/700})} \tag{1}\]  
In Equation (1), all symbols are dimensionless and represented by corresponding lower-case symbols as defined in [8], which means that all geometrical parameters are divided by the length (L) of identical beams; forces by \(E(UT^{3}/12)/L^{2}\). \(T\) and \(U\) are the beams' in-plane and out-of-plane thicknesses, respectively; \(E\) is the Young's modulus of the beam material. The symbol \(y_{\mathrm{s}}\) is the relative motion between the actuation displacement and the jaw's actual displacement, which is equal to the deformation displacement of the object; The symbol \(f_{\mathrm{y}}\) is the reaction force from the object when gripping occurs, which is equal to the gripping force. \(d\) is equal to \(12/(T/L)^{2}\). \(p_{\mathrm{in}}\) is the initial internal  
Figure 1: Schematic design of the proposed monolithic symmetrical gripper.  
axial force applied on each beam, which should be much larger than \(-10\) in order to avoid first-order buckling, and will be induced by the thermal control in this paper. Here, a positive value of \(p_{\text{in}}\) represents the tensional axial force, and a negative one stands for the compressive axial force. It can be observed that when the magnitude of the negative \(p_{\text{in}}\) increases, the force/stiffness at the same displacement (\(y_{\text{s}}\)) reduces. The geometrical parameters also affect the stiffness of the jaw, which is not discussed in this paper, but can be seen in [8].  
The relation between the initial internal axial force and the thermal change is explained as follows. Considering a slender beam (Figure 1) without any constraints on either of its ends, we can have the following expression to consider the change of beam length that corresponds to a change in temperature [8]:  
\[\Delta L=\alpha\int_{0}^{L}\Delta T(L)\text{d}L \tag{2}\]  
where \(\Delta L\) is the final length change when the final steady-state temperature change is \(\Delta T\) (varying along the beam). The symbol \(\alpha\) is a simplified constant value; i.e., mean coefficient of thermal expansion.  
Due to the fact that both ends of the beam in the compound parallelogram mechanism are fixedly constrained, the length change by temperature increase is actually zero, which in return induces the initial internal compressive axial force. The magnitude of initial internal axial force can be calculated in terms of the temperature change by using the result in Equation (2) as below [8]:  
\[\begin{array}{l}p_{\text{in}}/d=\Delta L/L\\ \Rightarrow p_{\text{in}}=d\alpha\int_{0}^{L}\Delta T(L)\text{d}L/L\end{array} \tag{3}\]  
If temperature is uniform along the beam for simplification, Equation (3) reduces to \(p_{\text{in}}=d\alpha\Delta T\), suggesting that the internal axial force magnitude is proportional to the temperature increase.