# Analysis of Bifurcation Vibrations of an Industrial Robot Arm System with Joints Compliance
## 2 Mathematical Model of the Robot
Taking into account the static deformations and ignoring \(\alpha_{1 st}^{2}\) and \(\alpha_{2 st}^{2}\) as small values of higher order than \(\alpha_{1 st}\) and \(\alpha_{2 st}\), the potential energy was obtained in the following form:  
\[V=\frac{1}{2}k_{1}\alpha_{1}^{2}+\frac{1}{2}k_{2}\alpha_{2}^{2}+m_{1}gl_{c1} \sin\beta_{1}+m_{2}gl_{1}\sin\beta_{1}+m_{2}gl_{c2}\sin(\beta_{1}+\beta_{2}). \tag{10}\]  
The kinetic energy of the system is the sum of the kinetic energy of link 1 in rotation and link 2 in plane motion:  
\[E=\frac{1}{2}I_{A}^{(1)}\dot{q}_{1}^{2}+\frac{1}{2}m_{2}v_{S2}^{2}+\frac{1}{2 }I_{S2}^{(2)}\left(\dot{q}_{1}+\dot{q}_{2}\right)^{2}, \tag{11}\]  
Figure 2: A two-link planar manipulator with flexible joints. where \(v_{S2}\) is the velocity of the centre of mass of link 2. Its value was defined as \(v_{S2}=\sqrt{\dot{x}_{S2}^{2}+\dot{y}_{S2}^{2}}\), where the velocity components are derivatives of the coordinates of point S\({}_{2}\) in the xy coordinate system, given by the following system of equations:  
\[ \begin{cases}x_{S2}=l_{1}\cos(q_{1})+l_{c2}\cos(q_{1}+q_{2})\\ y_{S2}=l_{1}\sin(q_{1})+l_{c2}\sin(q_{1}+q_{2})\end{cases} . \tag{12}\]  
taking into account Equations (1) and (2) in the kinematics Equation (12), an equation determining the kinetic energy of the system was obtained in the following form:  
\[ \begin{split} E=\frac{1}{2}\Big{[}I_{A}^{(1)}+I_{S2}^{(2)}+m_{2} l_{1}^{2}+m_{2}l_{c2}^{2}+m_{2}l_{1}l_{c2}\cos(\beta_{2}+\alpha_{2})\Big{]} \dot{\alpha}_{1}^{2}+\big{[}m_{2}l_{c2}^{2}+\\ m_{2}l_{1}l_{c2}\cos(\beta_{2}+\alpha_{2})+l_{S2}^{(2)}\Big{]}\dot{ \alpha}_{1}\dot{\alpha}_{2}+\frac{1}{2}\Big{(}m_{2}l_{c2}^{2}+l_{S2}^{(2)} \Big{)}\dot{\alpha}_{2}^{2}.\end{split}  \tag{13}\]  
the generalised forces were determined using the principle of virtual work. The virtual work of nonpotential forces \(dW\) is equal to virtual work of generalized force \(Q_{i}d\alpha_{i}\), so  
\[dW=F_{i}d\alpha_{i}-\frac{\partial R}{\partial\dot{\alpha}_{i}}d\alpha_{i}=Q_{ i}d\alpha_{i}. \tag{14}\]  
where \(F_{i}\)--external forces for link \(i\); \(R\)--energy dissipation function; \(i=1,2\). The energy dissipation function is given in the following form:  
\[R=\frac{1}{2}d_{1}\dot{\alpha}_{1}^{2}+\frac{1}{2}d_{2}\dot{\alpha}_{2}^{2}. \tag{15}\]  
the external force is as follows:  
\[F_{i}=u_{i}+\tau_{ext\ i}. \tag{16}\]  
where \(u_{i}\)--control input for link \(i\); \(\tau_{ext\ i}\)--the torque of any forces acting on the end-effector of the robot. Taking into account Equations (14)-(16), the generalized force \(Q_{i}\) is written as follows  
\[Q_{i}=u_{i}+\tau_{ext\ i}-d_{i}\dot{\alpha}_{i}^{2}. \tag{17}\]  
using Equation (4), the equations of motion of the system were obtained in the general form of Equation (3), where the vectors and matrices have the following form:  
\[\boldsymbol{\alpha}= \begin{bmatrix}\alpha_{1}\\ \alpha_{2}\end{bmatrix} , \tag{18}\]  
\[\boldsymbol{M}(\boldsymbol{\alpha},\boldsymbol{\beta})= \begin{bmatrix}a_{1}+2 a_{2}\cos(\beta_{2}+\alpha_{2})&a_{3}+a_{2}\cos(\beta_{2}+\alpha_{2})\\ a_{3}+a_{2}\cos(\beta_{2}+\alpha_{2})&a_{3}\end{bmatrix} , \tag{19}\]  
\[\boldsymbol{C}(\boldsymbol{\alpha},\dot{\boldsymbol{\alpha}},\boldsymbol{ \beta})= \begin{bmatrix}-a_{2}\sin(\beta_{2}+\alpha_{2})\dot{\alpha}_{2}&-a_{2 }\sin(\beta_{2}+\alpha_{2})\big{(}\dot{\alpha}_{1}+\dot{\alpha}_{2})\\ a_{2}\sin(\beta_{2}+\alpha_{2})\dot{\alpha}_{1}&0\end{bmatrix} , \tag{20}\]  
\[\boldsymbol{D}= \begin{bmatrix}d_{1}&0\\ 0&d_{2}\end{bmatrix} , \tag{21}\]  
\[\boldsymbol{K}= \begin{bmatrix}k_{1}&0\\ 0&k_{2}\end{bmatrix} , \tag{22}\]  
\[\boldsymbol{u}= \begin{bmatrix}u_{1}\\ u_{2}\end{bmatrix} , \tag{23}\]  
\[\boldsymbol{\tau}_{ext}= \begin{bmatrix}\tau_{ext\ 1}\\ \tau_{ext\ 2}\end{bmatrix} =I_{a}^{T}(\boldsymbol{\alpha},\boldsymbol{\beta}) \boldsymbol{F}_{ext}. \tag{24}\]the \(a_{i}\) parameters result from the following coefficients:  
\[