# Design and Analysis of a Novel Variable Stiffness Joint Based on Leaf Springs
## 2 Principle of the VSJ
### Stiffness Model  
Referring to Figure 1, in the small deformation hypothesis, the deformation of the leaf spring is approximately defined by the geometric relationship as  
\[\omega=l_{2}\tan\varphi \tag{1}\]  
where \(l_{2}\) is the distance from the slider to the output shaft, and \(\varphi\) is the deflection angle of the VSJ input shaft relative to its output shaft.  
The effective length of the leaf spring can be changed by moving the slider along the leaf spring. According to [PERSON]'s law and the small deformation cantilever beam model, the deflection of the leaf spring at the contact point with the slider, as well as the deformation angle \(\theta\) of the leaf spring at the free end relative to the initial position, are defined as  
\[\begin{cases}\omega=Fl_{1}^{3}/(3 EI)\\ \theta=Fl_{1}^{2}/(2 EI)\end{cases} \tag{2}\]  
where \(F\) is the force applied to the leaf spring from the slider, \(l_{1}\) is the distance from the slider to the leaf spring holder, and \(E\) is the Young's modulus of the leaf spring.  
It is worth mentioning that, although large deflection of the leaf spring can help improve the reverse drive and passive elastic energy performance of the joint actuator, the large deflection theory of a cantilever beam is complex, and the beam spring is prone to fatigue damage under large deformation, which can lead to an increase in the hysteresis of elastic deformation. This could also even lead to a large transmission backlash in the actuator, which is not conducive for compliant operation. Therefore, based on experience and design parameters, in order to meet the assumption of small spring deformation, avoid model nonlinear deviation during large deformation, and reduce the hysteresis effect of spring deformation, \(\theta\) is limited to a conservative range where \(\theta<10^{o}\), which can easily be ensured via mechanical limiting in the prototype.  
By combining Equations (1) and (2), the concentration force on the free end can be derived by the following equation:  
\[F=\frac{3 EI_{2}\tan\varphi}{l_{1}^{3}} \tag{3}\]  
where \(I\) is the moment of inertia of the leaf spring section, defined as  
\[I=\frac{bh^{3}}{12} \tag{4}\]  
where \(b\) and \(h\) are the width and thickness of the leaf spring, respectively.  
By combining Equations (3) and (4), the concentration force on the free end can be rewritten as follows:  
\[F=\frac{Ebl^{3}l_{2}\tan\varphi}{4l_{1}^{3}} \tag{5}\]  
The designed VSJ is a rotational joint, composed of \(N\) parallel leaf springs. The elastic torque \(\tau\) acting on the joint can be calculated as follows:  
\[\tau=NFl_{2}\cos\varphi \tag{6}\]  
By combining Equations (5) and (6), the elastic torque can be defined as  
\[\tau=\frac{NEbh^{3}l_{2}^{2}\sin\varphi}{4l_{1}^{3}} \tag{7}\]The stiffness of the VSJ is given as  
\[K=\frac{d\tau}{d\varphi} \tag{8}\]  
Therefore, by combining Equations (7) and (8), the total stiffness of the VSJ can be approximated as  
\[K=\frac{NEbh3l_{2}^{2}\cos\varphi}{4l_{1}^{3}} \tag{9}\]  
When the VSJ is subjected to an external load, the elastic deflection between the input and output shafts will be equivalent to the elastic potential energy. This energy, which will be expected to be reused in some cyclic motion cases, is defined as  
\[V=\int_{0}^{\theta}\tau d\varphi=\frac{Ebh3l_{2}^{2}}{4l_{1}^{3}}(1-\cos\varphi) \tag{10}\]  
When the elastic deflection angle is not zero, stiffness adjustment requires the stiffness motor to overcome resistant torque \(\tau_{r}\), which is defined as follows:  
\[\tau_{r}=\frac{dV}{dl_{1}} \tag{11}\]  
\(l_{2}=r_{2}+l_{1}\), \(r_{2}\) is the distance from the leaf spring holder to the center axis of the output shaft. By combining Equations (10) and (11), \(\tau_{r}\) is calculated as follows:  
\[\tau_{r}=\frac{Ebh^{3}(1-\cos\varphi)}{4}\frac{l_{1}^{2}+4r_{2}l_{1}+3r_{2}^{2 }}{-l_{1}^{4}} \tag{12}\]