# Analysis of Bifurcation Vibrations of an Industrial Robot Arm System with Joints Compliance
## 2 Mathematical Model of the Robot
Equation (3) defines the general structure of the dynamics equations describing the vibrations of the links relative to the configuration defined by the \(\mathbf{\beta}\) vector. The equation of motion of the robot arm in the form (3) for the selected robot can be obtained using various mathematical formalisms. The remainder of this section presents an approach based on the Lagrange formalism for a fragment of an industrial robot arm with two degrees of freedom. Detailed derivations of the equations of motion are presented in [30], where a mathematical model is presented without taking into account viscous damping and an excitation vector. For this reason, this article is limited to presenting the most important information about the mathematical model of the robot arm with two degrees of freedom. The manipulator diagram is shown in Figure 2. The model takes into account the compliance of the joints at points A and B. The stiffness of the joints at points A and B are represented by the coefficients \(k_{1}\) and \(k_{2}\), respectively. The viscous damping coefficients in the joints are \(d_{1}\) and \(d_{2}\). The lengths of links 1 and 2 are \(l_{1}\) and \(l_{2}\), respectively. The masses of the links are \(m_{1}\) and \(m_{2}\), respectively, the moment of inertia of link 1 relative to point A is \(I_{A}^{(1)}\) and the moment of inertia of link 2 relative to point S\({}_{2}\) is \(I_{S2}^{(2)}\). Points S\({}_{1}\) and S\({}_{2}\) are the centres of mass of the links, defined relative to joints A and B by the values \(l_{c1}\) and \(l_{c2}\), respectively. The moments that cause the arm to vibrate are \(u_{1}\) and \(u_{1}\), and \(F_{ext}\) is an external force vector. The equations of motion of the manipulator were formulated using the Lagrange formalism, based on the following system of equations:  
\[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{\alpha}_{i}}\right)-\frac{ \partial L}{\partial\alpha_{i}}=Q_{i}, \tag{4}\]  
where \(L=E-V\) is the Lagrange function, which is the difference between kinetic and potential energy and \(Q_{i}\) is generalised force. The potential energy of the system is the sum of the potential energy of the masses of the links \(V_{g1}\) and \(V_{g2}\) and the potential energy of the elastic joints \(V_{k1}\) and \(V_{k2}\):  
\[V=V_{k1}+V_{k2}+V_{g1}+V_{g2}, \tag{5}\]  
where  
\[V_{k1}=\frac{1}{2}k_{1}(\alpha_{1}+\alpha_{1 st})^{2}, \tag{6}\]\[V_{k2}=\frac{1}{2}k_{2}(\alpha_{2}+\alpha_{2 st})^{2}, \tag{7}\]  
\[V_{g1}=m_{1}gl_{c1}\sin(\beta_{1}+\alpha_{1}), \tag{8}\]  
\[V_{g2}=m_{2}g[l_{1}\sin(\beta_{1}+\alpha_{1})+l_{c2}\sin(\beta_{1}+\beta_{2}+ \alpha_{1}+\alpha_{2})], \tag{9}\]  
where \(\alpha_{1 st}\) and \(\alpha_{2 st}\) represent static deformations of joints under the influence of moments resulting from the gravitational forces of the joints. Expansions of the type \(\sin(a+b)=\sin a\cos b+\cos a\sin b\) were applied to the sines of the sums of the angles appearing in Equations (8) and (9). Maintaining the assumption that the angular coordinates \(\alpha_{1}\) and \(\alpha_{2}\) describing vibrations have small values, it was assumed that \(\sin\alpha_{1}=\alpha_{1}\), \(\cos\alpha_{1}=1\), \(\sin(\alpha_{1}+\alpha_{2})=\alpha_{1}+\alpha_{2}\) and \(\cos(\alpha_{1}+\alpha_{2})=1\). Then, using the Dirichlet criterion, the static deformations of the joints that occur in the static equilibrium position in conditions where \(\alpha_{1}=\alpha_{2}=0\) were determined.