# Design and Analysis of a Novel Variable Stiffness Joint Based on Leaf Springs
## 2 Principle of the VSJ
### Performance Analysis of Stiffness Adjustment Mechanism  
Based on Equations (7)-(12), the performance of the VSJ is related to the number, width, and thickness of the leaf springs, the length of the joint rotation moment arm, the effective force arm length of the leaf springs, as well as the joint deflection angle. Once the parameters, such as \(N\), \(b\), \(h\), and \(r_{2}\), are completely determined, the stiffness adjustment performance of the joint is influenced by the effective arm length of the leaf spring and the joint rotation angle. The theoretical simulation analysis of the variable \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Definition** \\ \hline \(O\) & the center position of the stiffness regulation disc \\ \(O_{1}\) & the center position of the arcuate slots \\ \((x_{01},y_{01})\) & the abscissa and ordinate of the point of \(O_{1}\) \\ \(A,S,C\) & the position of the slider \\ \(L_{ASC}^{0},L_{ASC}^{-1}L_{ASC}^{\max}\) & the arc length of the slider \\ \((0,r_{2})\) & the coordinates of A \\ \((l_{2}\sin\alpha_{1},l_{2}\cos\alpha)\) & the coordinates of S \\ \((r_{1}\sin\alpha_{1},r_{1}cos\alpha_{1})\) & the coordinates of C \\ \(r_{1}\) & the distance between the origin \(O\) and the point C \\ \(l_{2}\) & the distance between the origin \(O\) and the point S \\ \(r_{2}\) & the distance between the origin \(O\) and the point A \\ \(\alpha\) & the angle between the line \(\overline{OA}\) and the line \(\overline{OS}\) \\ \(\alpha_{1}\) & the angle between the line \(\overline{OA}\) and the line \(\overline{OC}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbol definition. Figure 2: Schematic diagram of the effective length adjustment mechanism for leaf spring. stiffness characteristics of the joint was performed using Matlab software (R2019a) in order to understand their mutual changes, thus providing a basis for subsequent VSJ structure designs and optimization. The design parameters of the VSJ required for simulation can be found in Table 2.  
According to Equation (9), when the output part of the joint is subjected to a load, it will undergo elastic deflection. The joint stiffness is collaboratively constrained by the effective length of the leaf spring and the deflection angle. The stiffness characteristics of the joint are illustrated in Figure 3, where one can see that the longer the effective arm length of the leaf spring, the lower the output stiffness. Meanwhile, the influence of the joint deflection angle on the output stiffness is relatively small. Figure 4 shows the relationship between the output stiffness and the deflection angle under the different effective arm lengths of the leaf spring. It is demonstrated that the output stiffness hardly changes with the change of joint deflection angle, while it can be approximately considered that the output stiffness of the joint is solely determined by the arm length of the spring leaf. This represents a weak coupling between the load and the VSJ stiffness, which is conducive to motion control. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline Diameter & 140 & mm \\ Height & 114 & mm \\ Weight & 4 & kg \\ Max. elastic torque with \(N=1\) & 14.6 & Nm \\ Stiffness adjustment range & \(N\times(9\)-inf.) & Nm/rad \\ Output rotation range & 0–360 & \(\circ\) \\ Deflection angle range & \(\pm 3.03\) & \(\circ\) \\ The leaf spring number \(N\) & 1 & / \\ The leaf spring width \(b\) & 25 & mm \\ The leaf spring thickness \(h\) & 1 & mm \\ Young’s modulus of the spring & 206 & GPa \\ material \(E\) & 206 & GPa \\ The distance of \(\overline{OC}\)\(r_{1}\) & 54 & mm \\ The distance of \(\overline{OA}\)\(r_{2}\) & 25 & mm \\ The angle between \(\overline{OC}\) and \(\overline{OA}\)\(\alpha_{1}\) & 47 & \(\circ\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the VSJ and the leaf spring for simulation. Figure 3: Theoretical stiffness characteristics.