# A Study of Friction Nonlinearity and Compensation for Turntable Servo Systems
## 1 Introduction  
Friction is a ubiquitous phenomenon in transmission mechanisms, influenced by a multitude of factors, including relative movement speed, external forces, contact surface smoothness, ambient temperature, and the condition of lubricants [1]. The complex interplay of these factors affects the magnitude of frictional forces encountered. Unlike mechanical gaps, friction cannot be mitigated through structural optimization alone, presenting a significant challenge in enhancing the dynamic performance and steady-state accuracy of servo drive systems [2]. Therefore, the accurate modeling of and compensation for friction is crucial in high-precision servo drive systems. Previous studies have often neglected the impact of friction due to its relatively minor influence under certain conditions. However, during positioning tasks, even small frictional forces can lead to an increase in relative position errors and static friction torque, potentially causing motor stalling [3]. Especially in low-temperature application scenarios, the viscosity of the lubricant medium increases, which in turn increases the frictional torque. To address this issue, a comprehensive approach is adopted, beginning with the offline identification and modeling of the friction characteristics of the experimental platform. This process involves a detailed analysis of the frictional behavior under varying conditions [4]. Since 1519, scholars from various countries have carried out in-depth studies on friction, and various friction characteristics have been summarized over hundreds of years, including (1) static friction; (2) Coulomb friction; (3) Stribeck friction; (4) pre-sliding displacement; (5) variable static friction; (6) friction hysteresis, etc. The first three are static friction characteristics and the last three are dynamic friction characteristics. A friction model was established to reflect the above characteristics. In 1785, [PERSON] made a clear division between static friction and kinetic friction, but the jump of the Coulomb model at zero speed easily leads to instability of the kinetic model. 1866, [PERSON] was inspired by the development of fluid dynamics and introduced the concept of viscous hysteresis, which formed the'static friction + Coulomb friction + viscous friction', which is still widely used today. In 1902, [PERSON] explored the relationships among relative speed, viscosity, and friction and proposed a smooth exponential curve for the transition from static friction to viscous friction. In 1985, [PERSON]'s model was proposed to solve the problems of zero-speed detection and switching between states. The above models do not reflect the dynamic friction characteristics. In order to reflect the characteristics of pre-slip and variable static friction, [PERSON] and [PERSON] developed a seven-parameter model with three independent equations to describe the friction at each stage, but the identification of its multiple parameters is difficult and the application of the model is limited. Although the seven-parameter model reflects some of the dynamic properties, a truly dynamic model usually requires the introduction of additional state variables to provide more degrees of freedom in the description of the friction characteristics. The earliest dynamic model was proposed by [PERSON] [5], which introduced the concept of average deformation to describe the pre-slip displacement and frictional hysteresis, but ignored the static friction and Stribeck effect. [PERSON] and [PERSON], from a microscopic point of view, regarded the stochastic behavior of the two contacting surfaces as the contact of a large number of elastic bristles, which is called the bristle model. Subsequently, in order to reduce the complexity of the calculations. The two scholars also proposed an improved reset integral model [6].