# A Study of Friction Nonlinearity and Compensation for Turntable Servo Systems
## 3 Compensation Based on the Friction Model
### Simulation and Experimental Validation  
At 0.5 s, a 90-degree step position signal was introduced. Subsequent analysis delineated the system responses and friction torque manifestations under varying conditions, specifically the no-load and 20% rated load scenarios, which are presented in Figures 10 and 11. Notably, the incurred steady-state errors amounted to 0.01\({}^{\circ}\) and 0.001\({}^{\circ}\), respectively. This observation result underscores the mitigative influence of loading on the system, a phenomenon corroborated by prior scholarly discourse. The utilization of a second-order extended state observer (ESO) engenders a discernible delay in the observation of friction torque, a feature congruent with theoretical conjecture. Figure 10: System response and friction torque measurement under no-load conditions. Figure 9: Experimental platform. By altering the friction parameter to \(T_{c}^{\prime}=1.5T_{c}\), the system's response and the observation of the friction torque are illustrated in Figure 12. The steady-state error of \(0.0055^{\circ}\) indicates that the compensation algorithm remained effective even when the friction varied over time. Figure 13 compares the system responses of two control methods when there was a mismatch in the friction parameters. Utilizing only \"PI control + state feedback control\" resulted in a steady-state error of \(0.34^{\circ}\) due to the absence of a friction model's feedforward compensation. In contrast, when employing \"enhanced LADRC + state feedback control\", the steady-state error was significantly reduced to just \(0.014^{\circ}\). Figure 11: System response and friction torque observation at 20% rated load conditions. Figure 12: System response and friction torque observation at 20% rated load with a friction parameter of \(\text{T}_{c}^{\prime}=1.5\text{T}_{c}\). dynamic response performance of the conventional ADRC controller in the presence of a gap, it could not meet the demand of control performance. Therefore, the performance of ADRC was improved by combining it with the state feedback method, and the improved controller gain was calculated to correct the steady-state static differential equation. The simulations and experiments showed that this method was able to compensate for the gap and friction nonlinearity in an integrated way, and the following conclusions are drawn:  
1. Feedforward compensation based on a friction model is highly dependent on the accuracy of parameter identification, and inaccuracies in the motor parameters trigger system oscillations. 2. With the application of the traditional ADRC algorithm in friction compensation, a small steady-state error and a rapid dynamic response can be reached. However, the compensation effect of the gap is not obvious, and the dynamic performance is seriously degraded when there is a gap in the transmission mechanism. 3. Introducing state feedback compensation on the basis of the traditional ADRC can effectively improve the positioning accuracy of the load side of the transmission system. The experimental steady state error was reduced from 0.55\({}^{\circ}\) to 1/3 (0.19\({}^{\circ}\)) after compensation. Moreover, the robustness of the improved compensation algorithm to the parameters is further improved for different load conditions. Due to the experimental conditions, the dynamic model was not chosen to describe the friction in this project. In subsequent research, the friction can be described more accurately based on the dynamic model, and the suppression algorithm of the friction disturbance can be further improved on this basis to obtain a better suppression effect. Conceptualization, methodology, writing--original draft, M.Y. ; resources, supervision, writing--review and editing, K.L. ; validation, R.M.S. ; visualization, R.J.; project administration, H.Y. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The original contributions presented in the study are included in the article.