# Design and Analysis of a Novel Variable Stiffness Joint Based on Leaf Springs
## 2 Principle of the VSJ
### Performance Analysis of Stiffness Adjustment Mechanism
According to Equation (16), when the stiffness adjustment motor regulates the rotation angle \(\theta\) of the stiffness adjustment disk, the effective length \(l\) of the leaf spring approximately changes in a linear manner, as shown in Figure 5. This will help reduce the complexity of the stiffness adjustment process and improve the response speed of this adjustment. At the same time, according to Equation (9), as the effective length of the leaf spring increases, the joint stiffness demonstrates a nonlinear trend. It is evident that, when the effective length of the leaf spring is approximately zero, the joint stiffness approaches infinity. The joint stiffness changes dramatically when the effective length of the leaf spring is within the range of 0-6 mm. Although a high stiffness adjustment speed was demonstrated, from the application point of view, the stiffness adjustment resolution decreased sharply within this range, whereas its adjustment was almost meaningless. Therefore, the effective adjustment range of the leaf spring is set to be 6-25 mm, while the limit rotation angle of the corresponding stiffness adjustment disk is set to 47\({}^{\circ}\). As shown in Figure 6, the output stiffness of the VSJ decreases nonlinearly as the effective length of the leaf spring increases. The theoretical stiffness range is about 9-inf. Nm/rad. The shaded area surrounded by pink lines represents an area with relatively slow stiffness changes, and the stiffness adjustment accuracy in this area is relatively high. Figure 4: Relationship between the joint stiffness and elastic deflection angle. Figure 5: Relationship between the effective length of the leaf spring and the rotation angle of the stiffness-varying disk. The relationship between the elastic torque and the effective arm length of the leaf spring can be obtained via Equation (7). Figure 7 illustrates this, along with the elastic deflection angle of the joint. Unlike the output stiffness of the joint, the elastic deflection angle of the joint has a significant influence on the elastic torque, especially when the effective length of the leaf spring is relatively small, which is more prominent. In Figure 7, the \(x\)-axis coordinate starts from 6 mm, because when \(l\) approaches infinity, the stiffness value also approaches infinity, while the elastic torque of the VSJ also theoretically tends to infinity. However, this situation is not a realistic scenario, because the stiffness of the VSJ depends on the structural strength, and there is a high risk of damage due to overload. The elastic mechanism of VSJ can store energy in periodic motions, and then repeatedly reuse part of this energy in order to achieve energy-efficient motion control and shock absorption. This feature has great potential in applications of periodic motion control and in physical interaction shock absorption. The elastic potential energy of the VSJ joint under different deflection amounts can be obtained using Equation (10). Its characteristics are shown in Figure 8. It is easy to conclude that the joint potential energy increases as the effective length of the leaf spring decreases, whereas a positive correlation with the change of the joint deflection angle is also clear. When the elastic deflection angle of the joint is close to zero, the joint potential energy is minimized. Figure 6: Relationship between joint stiffness and the effective length of the leaf spring. Figure 7: Theoretical elastic torque characteristics. According to Equation (12), under the coupling effect of the effective length and elastic deflection angle of the leaf spring, to overcome the elastic potential energy under different elastic deflection states is necessary, particularly when the leaf spring has different lengths during stiffness adjustments.