# Analysis of Bifurcation Vibrations of an Industrial Robot Arm System with Joints Compliance
## 2 Mathematical Model of the Robot  
This section deals with the mathematical model of the robot that describes low-frequency vibrations. These vibrations are due to the flexibility of the joints and consist of the rotational oscillations of the links relative to the joints. In order to take them into account, it can be assumed that each configuration coordinate of the arm \(q_{i}\) can be expressed as the sum of two angular values:  
\[q_{i}=\beta_{i}+\alpha_{i}, \tag{1}\]  
where \(\beta_{i}\) is a variable describing the position of the \(i\)-th link of the manipulator, relative to which the oscillations of this element, expressed by the variable \(\alpha_{i}\), take place (Figure 1). In the ideal case, when there are no vibrations, \(\beta_{i}\) is identical to the configuration coordinate \(q_{i}\). If there is compliance in the joints, the angle \(\alpha_{i}\) is a rapidly changing value related to vibration phenomena. On the other hand, \(\beta_{i}\) is a slowly changing value whose change results from the execution of the motion path, or it can be a constant value if the link is in a fixed position. Since the present research is concerned with vibration phenomena and not with the overall movement of the arm, a simplifying assumption was made in further considerations. Figure 1: Configuration coordinates of the n-link robot arm, taking into account the displacements due to the flexibility of the joints: the links marked with a solid line indicate the links at the equilibrium position given by \(\beta_{i}\) angle, the links marked with a dashed line indicate the positions of the links deflected from the equilibrium, \(x_{i}\) is the axis associated with the \(i\)-th link, \(||x_{i}\) denotes the axis parallel to \(x_{i}\), \(\alpha_{i}\) is the angle of deviation of the link \(i\) from equilibrium position and this angle is defined relative to the \(x_{i}\) axis. The vibrational movements of the elements described by the \(\alpha_{i}\) variables were considered in relation to the positions of the elements described by the \(\beta_{i}\) coordinates, which were treated as constants. In other words, the vibrations of the kinematic chain relative to the assumed position defined by the \(\beta_{i}\) coordinates are the subject of the analysis. This approach is justified by the fact that, during the movement, changes in \(\beta_{i}\) are many times slower than changes in \(\alpha_{i}\). Therefore, based on Equation (1), the angular velocities of the elements are expressed as follows:  
\[\dot{q}_{i}=\dot{\alpha}_{i} \tag{2}\]  
The equation describing the vibrations of the robot in the selected arm position, in the case without constraints, can be written in the following form:  
\[\mathbf{M}(\mathbf{\alpha},\mathbf{\beta})\ddot{\mathbf{\alpha}}+\mathbf{C}\big{(}\mathbf{\alpha},\dot {\mathbf{\alpha}},\mathbf{\beta}\big{)}\dot{\mathbf{\alpha}}+\mathbf{D}\dot{\mathbf{\alpha}}+\mathbf{K }\mathbf{\alpha}=\mathbf{u}+\mathbf{\tau}_{ext}, \tag{3}\]  
where \(\mathbf{\alpha}\in\mathbf{R}^{n}\)--a vector of coordinates representing the vibrations of the links of the robot, such that \(\mathbf{\alpha}= \begin{bmatrix}\alpha_{1}&\dots&\alpha_{i}&\dots&\alpha_{n}\end{bmatrix} ^ {T}\); \(\mathbf{\beta}=\mathbf{const}\). \(\in\mathbf{R}^{n}\)--a vector of coordinates representing the position of the robot arm, such that \(\mathbf{\beta}= \begin{bmatrix}\beta_{1}&\dots&\beta_{i}&\dots&\beta_{n}\end{bmatrix}  ^{T}\); \(\mathbf{M}(\mathbf{\alpha},\mathbf{\beta})\in\mathbf{R}^{n\times n}\)--an inertia matrix; \(\mathbf{C}\big{(}\mathbf{\alpha},\dot{\mathbf{\alpha}},\mathbf{\beta}\big{)}\dot{\mathbf{\alpha}} \in\mathbf{R}^{n}\)--a vector of centrifugal and Coriolis forces (moments of forces); \(\mathbf{D}\in\mathbf{R}^{n\times n}\)--a matrix of joints damping; \(\mathbf{K}\in\mathbf{R}^{n\times n}\)--a matrix of joint stiffness; \(\mathbf{u}\in\mathbf{R}^{n}\)--a control input vector; \(\mathbf{\tau}_{ext}\in\mathbf{R}^{n}\)--an external excitation vector. Equation (3) is a system of second-order nonlinear coupled differential equations.