# Analysis of Bifurcation Vibrations of an Industrial Robot Arm System with Joints Compliance
## 1 Introduction  
Bifurcations in mechanical systems have been the subject of numerous studies in the field of nonlinear dynamics. Researchers have paid particular attention to systems such as pendulums [1; 2; 3], aeroelastic systems [4; 5; 6], plates [7; 8] and beams [9]. However, important bifurcation phenomena also occur in other types of mechanical systems, such as brakes [10] or gears [11]. Bifurcation vibrations are mainly analysed on the basis of resonance curves, Poincare maps or bifurcation diagrams. However, the analysis of bifurcation vibrations in industrial robot systems has not been the subject of significant research interest. There is little theoretical work on selected stability problems and bifurcation phenomena in robotic arm systems [12; 13]. However, the models used in the above articles did not take into account the important phenomenon of joint compliance. In [14], a robotic arm model was adopted that took into account the elasticity of the joints, and the behaviour of the robotic arm was analysed assuming stiffness and damping coefficients as bifurcation parameters. Such a small amount of research in this area seems to be due to the approach to robot modelling, which usually uses the simplest models to analyse the main dynamic phenomena related to the movement of robot arms. The second reason is the emphasis on solving other problems in the field of robotics, related to control, sensors, interaction with the environment, industrial applications, etc. Therefore, the aim of this paper is to try to fill the existing gap in the field of analysis of nonlinear dynamics of robots and to draw attention to the importance of nonlinear vibration phenomena. There are three basic approaches to the mathematical modelling of robot dynamics. In the first, simplest approach, the robot arm is treated as a system of rigid elements connected by joints [15; 16]. The advantage of this modelling method is the relative simplicity of the equations of motion, and it is mainly used in motion control synthesis. However, it should be noted that the stiffness of the joints is relatively low compared to the stiffnessof the links. Therefore, mathematical models of robots are sometimes formulated taking into account the flexibility of the joints, while maintaining the assumption of the stiffness of the links [17; 18; 19]. The advantage of such models is that they are more accurate, but unfortunately they are also more complex due to the additional degrees of freedom. The third class are robot models that take into account the flexibility of the links [20; 21]. They are only used in cases where this flexibility is a very important phenomenon, because such models are extremely complicated, since they contain partial differential equations [22]. In the case of modelling industrial robots, the first two approaches are used because such robots have very rigid links. Robots with flexible links are in most cases special designs that have not yet been widely used in industry. An important phenomenon in the use of robots in industry is the vibration that accompanies the movement of the robot. Robot vibrations can be divided into low-frequency and high-frequency vibrations. There are two main aspects to high-frequency vibrations. Firstly, they can be natural vibrations of very stiff links, characterised by very small amplitudes. Secondly, they can be forced vibrations that occur, for example, in robotic machining processes [23; 24; 25] and result from both the imbalance in the masses of rotating cutting tools and the cutting forces. Sometimes, the frequencies of the forced vibrations synchronise with the natural vibrations, resulting in high-frequency resonant vibrations of the robot components. Although these types of vibrations have small amplitudes, they cause negative phenomena such as noise and, in the case of mechanical machining, a negative impact on the quality of the machined surface, mainly in terms of roughness [26]. Low-frequency vibrations are related to the flexibility of the joint drive system [27].