# Analysis of Bifurcation Vibrations of an Industrial Robot Arm System with Joints Compliance
## 1 Introduction
In principle, all industrial robots have a joint stiffness that is many times lower than the stiffness of the links and exhibit significant low-frequency vibrations. Such vibrations can occur both during robotic machining processes, when the robot is in contact with the environment (the workpiece being machined), and when it is not in contact with the environment, e.g., during fast movements with sudden changes in direction. In robotic mechanical processing, low-frequency vibrations result in low dimensional and shape accuracy of the processed workpieces [28]. The isolation of low-frequency vibrations is a current research problem, as described in the article in [29]. When robots are used for tasks that do not require physical contact with the environment, low-frequency vibrations are a factor that limits the speed of movement. Another important aspect is the vibration damping time, especially when robots are used for measurement tasks. This paper deals with the analysis of low-frequency vibration phenomena in an industrial robot. The tests were carried out on a fragment of the arm of the industrial robot IRB 1600. In this case, the problem of low-frequency vibrations is important because the amplitudes of these vibrations can be many times greater than the repeatability value of the robot. In practice, this means that the vibration phenomena cause the repeatability to deteriorate many times over. In the case of the robot mentioned above, the repeatability is 0.02 mm, while the amplitude of the vibrations in the robot caused by a force pulse of 100 N is 0.1 mm. This shows the importance of the problem of low-frequency vibrations in industrial robots. This article presents a mathematical model of robot vibrations that takes into account the compliance of the joints, which results in a nonlinear model. Due to the high complexity of the problem, the analysis in this article is limited to a two-degree-of-freedom system. Focus has been paid to the analysis of nonlinear phenomena depending on the position of the robot arm determined by the configuration coordinates. First, analytical solutions to simplified (linear) equations of motion describing the free, undamped vibrations in the robot are presented. Then, the complete equations for the motion of the robot are subjected to numerical analysis, focusing on nonlinear phenomena. Harmonic forcing was used and the bifurcation of the solutions under the influence of changes in the amplitude and frequency of the forcing as a function of the arm position was analysed. The main contribution of the presented research is the analysis of the influence of selected bifurcation parameters on the vibrations of the robot. The bifurcation parameters included in the analysis correspond to practical situations in which the amplitude and frequency of force acting on the system may have different values depending on external factors, e.g., the type and structural parameters of the robot machining. This article is a continuation and extension of the work presented in [30], which studied the influence of the position of an industrial robot on the resonance frequencies, without a detailed analysis of the bifurcation phenomena.