# RBF Neural Network Based Backstepping Control for an Electrohydraulic Elastic Manipulator
## 2 Electrohydraulic Elastic Manipulator Dynamics
### Mechanical Dynamics  
The two-link manipulator of the EEM is shown in Figure 2 with two different coordinate systems for the two degrees of freedom (DOF) of the manipulator. The parameters are also shown. \(l_{0}\) presents the position of the 1 st link along with the \(z\)-axis. \(l_{i}(I=1,2)\) are the lengths of the links, and \(\theta_{i}(i=1,2)\) are the angle of the \(i\)th from the coordinate system. The EEM dynamics without considering the adjustable stiffness mechanism are expressed based on the dynamics of a 2-DOF manipulator [27]. The dynamic equations are as follows:  
\[M(\theta)\ddot{\theta}+C\big{(}\theta,\dot{\theta}\big{)}+G(\theta)=\tau+\tau _{f} \tag{7}\]  
where \(\dot{\theta},\) and \(\ddot{\theta}\in R^{2\times 1}\) are the joint velocity and acceleration vectors, respectively; \(M(\theta)\in R^{2\times 2}\) is the inertia matrix; \(C\big{(}\theta,\dot{\theta}\big{)}\in R^{2\times 2}\) presents the matrix of centripetal and Coriolis forces; \(G(\theta)\in R^{2\times 1}\) is the gravity vector; \(\tau_{f}\) is the friction vector; and \(\tau\) is the torque vectors. Detailed descriptions are given in Appendix A. Figure 3 is a simple form of the EEM which is used to analyze the forces acting on the two-link manipulator: the force of hydraulic cylinder and the spring forces of the ASBM system. Three black lines represent the cylinder, the first, and the second link. From this figure, the excited torque produced by the hydraulic actuator can be obtained as  
\[\tau_{c}=l_{1}F_{c}\sin\gamma \tag{8}\]  
where \(\gamma\) is the angle between the first link and the cylinder. During the system operation, the angle \(\gamma\) is around 90 degrees. So Equation (8) can be rewritten as follows:  
\[\tau_{c}=l_{1}F_{c} \tag{9}\]  
Figure 2: The architecture of the two-link robot manipulator in the electrohydraulic elastic manipulator (EEM). Additionally, the relationship between the cylinder displacement \(x_{c}\) and \(\theta_{1}\) is expressed as:  
\[\cos(\theta_{0}+\theta_{1})=\frac{l_{1}^{2}+l_{0}^{2}+d_{0}^{2}-{d_{3}}^{2}}{2l_ {1}\sqrt{\left(l_{0}^{2}+d_{0}^{2}\right)}} \tag{10}\]  
where \(d_{3}=d_{30}+x_{c}\), and \(d_{30}\) is the length of the cylinder when \(x_{c}=0\). The angular motion of the second link is caused by the spring forces. The ASBM system dynamics are analyzed using the distance between the second rotary shaft and the spring position of \(a_{3}\), the spring axes, and the geometric relationships of the VSA structure as shown in Figure 3.  
The forces produced by the two springs are represented as follows:  
\[ \begin{array}{l}F_{1}=\Delta x_{1}K_{1}\\ F_{2}=\Delta x_{2}K_{2}\end{array}  \tag{11}\]  
where \(\Delta x_{i}(i=1,2)\) are the variations of the springs, and \(K_{i}(i=1,2)\) are the stiffnesses of the springs. These forces excite the coupling torques in the two links. The torques are expressed as follows:  
\[ \begin{array}{l}\tau_{12}=(F_{1}\sin\beta_{1}-F_{2}\sin\beta_{2})a_{3}\\ \tau_{21}=-(F_{1}\sin\beta_{1}-F_{2}\sin\beta_{2})(l_{1}+a_{3})\end{array}  \tag{12}\]  
where \(\tau_{12}\) and \(\tau_{21}\) present the coupling torques, which impact the two links and are caused by the springs in the adjustable stiffness mechanism, and \(\beta_{i}\) is an angle between the ball screws and the \(i\)th spring. The considered hydraulic friction torque in this model is a combination of the Coulomb friction and the viscous friction [28] as  
\[\tau_{fc}=l_{1}F_{f}=l_{1}\Big{(}F_{Coul}\text{sign}\Big{(}\dot{x}_{c}\Big{)}+ F_{v}\dot{x}_{c}\Big{)} \tag{13}\]  
where \(F_{Coul}\) derives a Coulomb coefficient, and \(F_{v}\) depicts a viscous coefficient.