# Research on Friction Compensation Method of Electromechanical Actuator Based on Improved Active Disturbance Rejection Control

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of the PMSM Established Using the FOC Method

_{d}and i

_{q}are the d-axis and q-axis currents, u

_{d}and u

_{q}are the d-axis and q-axis voltages, L

_{d}and L

_{q}are the d-axis and q-axis inductances, ψ

_{d}and ψ

_{q}are, respectively, the d-axis and q-axis magnetic linkages, p

_{n}is the number of pole pairs, T

_{e}is the electromagnetic torque, J is the rotational inertia, T

_{L}is the load torque, B is the damping coefficient, ψ

_{f}is the permanent magnet magnetic flux, ω is to the electrical angular velocity, and ω

_{r}is the mechanical angular velocity.

_{d}= L

_{q}. When i

_{d}= 0 or L

_{d}= L

_{q}, Equation (3) can be simplified as

## 3. Transmission Model of the Harmonic Reducer

_{f}

_{1}is the friction generated by the wave generator, T

_{f2}refers to the friction between the flexspline and the circular spine, T

_{f}

_{3}refers to the friction generated by the flexspline, θ

_{m}and T

_{m}are, respectively, the rotor position and torque, θ

_{ng}and T

_{ng}are, respectively, the output positions and moments of the wave generator, θ

_{nin}and T

_{nin}are, respectively, the input angle and input torque of the flexspline torsion spring model, θ

_{nout}and T

_{nout}are, respectively, the displacement and output torque of the flexspline, Tk and Ts are, respectively, the torsion spring force and damping force of the flexspline torsion spring model, T

_{L}is the output torque of the flexspline, and θ

_{L}is the position of the flexspline.

_{f}acting on the harmonic reducer is T

_{f}= T

_{f}

_{1}+ T

_{f}

_{2}+ T

_{f}

_{3}. The friction torque T

_{f}

_{3}acting on the load end under low speeds and heavy loads is much smaller than the friction torque T

_{f}

_{1}acting on the motor and wave generator end under high speeds and light loads and can be ignored; that is, T

_{f}

_{3}≈ 0. T

_{L}= T

_{nout}= T

_{k}+ T

_{s}, and T

_{nout}= f(Δθ, K

_{L}). f(Δθ, K

_{L}) = T

_{k}+ T

_{s}= T

_{L}. k is the stiffness coefficient of the harmonic drive. Thus, the relationship between the input torque, friction torque, and output torque of the harmonic reducer can be expressed as follows:

_{L}is the equivalent stiffness coefficient of the harmonic reducer, neglecting the rotational inertia of the reducer, J is the rotational inertia of the motor rotor, J

_{L}is the rotational inertia of the load end, and f(Δθ, K

_{L}) = T

_{L}. The relationship between the motor torque Te and the wave generator torque T

_{m}is

_{f}of the harmonic reducer can be expressed as

_{q}. During no-load and constant speed operation, the change rule of friction torque can be obtained by measuring the torque current i

_{q}at different speeds and fitting the i

_{q}change curve.

## 4. Improved ADRC Control Principle

#### 4.1. Composite Second-Order ADRC Control Principle

_{1}converges to the input signal x

_{in}, x

_{2}is the derivative of the input signal, r is the speed factor and determines the tracking speed, T is the sampling time, and h

_{1}is the filtering factor. fhan(·) is the fastest synthesis function, represented as

_{1}(k) is the tracking signal of y(k), z

_{2}(k) is the tracking signal of z

_{1}(k), z

_{3}(k) is the total disturbance of the system, z

_{3}(k) is feed back to the control variable u(k) for compensation, and b is the compensation factor. β

_{01}, β

_{02}, and β

_{03}are the output error correction gains, α

_{01}and α

_{02}are the nonlinear factors, and δ is the filtering factor.

_{1}(k) and e

_{2}(k) are error signals, and β

_{1}and β

_{2}are, respectively, the error gain and differential gain. When 0 < α < 1, fal(·) achieves a mathematical fitting of “small error with large gain, large error with a small gain.” Fuzzy control, variable gain PID, and intelligent control are based on the control concept of “small error with a large gain, large error with small gain” to adjust the output. fal(·) is a nonlinear feedback function and can be expressed as follows:

_{θ}is the total disturbance in position mode, and w

_{ω}is the total disturbance in velocity mode. θ is the measured angle of the rotor, and ω is the speed of the rotor.

_{01}, β

_{02}, and β

_{03}in ESO; and β

_{1}and β

_{2}in NLSEF. Although there are many parameters that must be adjusted, the three stages have their own engineering significance, and the principle of separate directional adjustment can be used to adjust the parameters of each stage.

#### 4.2. Fuzzy ADRC Control Principle

_{1}and the change rate e

_{2}of the deviation to achieve online adjustment of the NLSEF coefficients and achieve adaptive ability:

_{1}and e

_{2}, and the outputs are Δβ

_{0}, Δβ

_{1}, and Δβ

_{2}. In fuzzy PID control, based on the variation of e

_{1}and e

_{2}, fuzzy subsets of five language variables, namely {“Negative Big (NB),” “Negative Small (NS),” “Zero (ZO),” “Positive Small (PS),” and “Positive Big (PB)”} are often used, or fuzzy subsets of seven language variables, namely {“Negative Big (NB),” “Negative Medium (NM),” “Negative Small (NS),” “Zero (ZO),” “Positive Small (PS),” “Positive Medium (PM),” and “Positive Big (PB)”} are often used. The control accuracy of seven fuzzy subsets is better than that of five fuzzy subsets. Here, β

_{0}, β

_{1}, and β

_{2}have the same control effect as k

_{i}, k

_{p}, and k

_{d}, so seven subsets are selected here. Common membership functions include triangle membership, Z/S membership, trapezoid membership, and Gaussian membership, and in order to reduce the workload of operations, triangular membership functions are used for each fuzzy variable. The established fuzzy rules are presented in Table 1.

_{0}, β

_{1}, and β

_{2}obtained from the domain of each variable and fuzzy reasoning are shown in Figure 5.

_{0}, Δβ

_{1}, and Δβ

_{2}are obtained using the fuzzy rule table and the deblurring algorithm. The control parameters in NLSEF are obtained after correction by using Equation (20). Thus, ADRC parameter self-tuning is realized, and the adaptive ability of the system can be improved by adjusting and controlling the control parameters in NLSEF in real time. β

_{00}, β

_{10}, and β

_{20}are the initial values; select the initial value according to the empirical method:

## 5. EMA Control System Design

## 6. Experimental Analysis

_{q}at a constant speed without load. This article performed experimental analysis on frictional forces in the counter-clockwise rotation direction. The inertia of the reducer was disregarded, and it was assumed that the torque during no-load operation equals the friction torque during uniform motion. A friction model was developed by measuring torque values at various speeds and fitting the data. This model was incorporated into the control system through feedforward compensation, effectively eliminating friction disturbances. Friction torque testing was performed on the RT-Cube platform, which is capable of achieving a minimum control cycle for the motor within 100 µs. Moreover, this platform allows for the online modification of any control parameter and the online monitoring of any system variable during the control process. The tests were made at a room temperature of approximately 25 °C and a relative humidity ranging from 40% to 70%RH. The experimental platform and the test results obtained using the Gaussian fitting method are shown in Figure 11 and Figure 12, respectively.

_{1}. The r affects the tracking effect. A larger r corresponds to a shorter transition time and thus a faster tracking response. However, very large r leads to overshoot and oscillation. When the r is constant and the h

_{1}is large, the tracking signal error is large; when the h

_{1}is small, the noise suppression is more prominent. However, when the h

_{1}is too small, the ability of the TD to suppress noise will be weakened. The disturbance compensation factor b

_{0}mainly affects the disturbance compensation capacity. If the system disturbance is significant, b

_{0}should be slightly larger; if the system disturbance is small, b

_{0}should be marginally lower. Directional adjustment is adopted. When we set α

_{01}= α

_{02}= 1, fal(e,α,δ) can be linearized to fal(e,α,δ) = e. The values for the parameters β

_{01}, β

_{02,}and β

_{03}need to be adjusted in practical applications according to the system output. The tuning rules for these parameters are listed in Table 4. Notably, when one parameter is tuned, the other two remain constant.

^{2}”. The ITAE calculation result within 0–1 s of IADRC was 15.445 (rev/min)*s

^{2}, thus indicating the optimal control performance of IADRC. The number of encoder lines is 2880, and after fourfold frequency, it is 11,520. The position input signal is y = 115,200*sin(0.05*pi*t), and the unit of y is the carving line number of the encoder (LNE). The main parameters in the experimental are shown in Table 6.

## 7. Conclusions

_{q}was analyzed. Furthermore, on the RT-Cube platform, the torque current i

_{q}at different speeds was measured and then added to the current loop control through feedforward compensation, determining controller parameters through empirical methodologies. In addition, speed-mode and position-mode experiments were conducted in the PI control mode, ADRC control mode, and IADRC control mode. Moreover, the experimental results of the speed step response were analyzed using the IATAE criteria. The IADRC control mode yielded the smallest calculation result and the best control performance. Neglecting the inertia of the reducer, assuming that the no-load running torque is equal to the friction torque during uniform motion, the experimental results of sinusoidal position tracking were analyzed, and the results were evaluated using RMSE and peak-to-peak values. Under conditions of pure inertial load, the integration of friction feedforward compensation combined with the implementation of the IADRC control method enhances the accuracy of EMA transmission.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Zhang, Y.; Zhao, C.; Dai, B.; Li, Z. Dynamic simulation of permanent magnet synchronous motor (PMSM) electric vehicle based on Simulink. Energies
**2022**, 15, 1134. [Google Scholar] [CrossRef] - Zhou, D.; Luo, K.; Shen, Z.; Zou, J. Vector-Space-Decomposition-Based Power Flow Control of Single-Stage-Multiport-Inverter-Fed PMSM Drive for Hybrid Electric Vehicles. IEEE Trans. Ind. Electron.
**2023**, 1–11. [Google Scholar] [CrossRef] - Chuyen, T.D.; Van Hoa, R.; Co, H.D.; Huong, T.T.; Ha, P.T.T.; Linh, B.T.H.; Nguyen, T.L. Improving control quality of PMSM drive systems based on adaptive fuzzy sliding control method. Int. J. Power Electron. Drive Syst.
**2022**, 13, 835–845. [Google Scholar] [CrossRef] - Assoun, I.; Idkha, L.; Nahid-Mobarakeh, B.; Meibody-Tabar, F.; Monmasson, E.; Pacault, N. Wide-Speed Range Sensorless Control of Five-Phase PMSM Drive under Healthy and Open Phase Fault Conditions for Aerospace Applications. Energies
**2022**, 16, 279. [Google Scholar] [CrossRef] - Li, P.; Xu, X.; Yang, S.; Jiang, X. Open circuit fault diagnosis strategy of PMSM drive system based on grey prediction theory for industrial robot. Energy Rep.
**2023**, 9, 313–320. [Google Scholar] [CrossRef] - Spong, M.; Khorasani, K.; Kokotovic, P. An integral manifold approach to the feedback control of flexible joint robots. IEEE J. Robot. Autom.
**1987**, 3, 291–300. [Google Scholar] [CrossRef] - Gandhi, P.S.; Ghorbel, F.H.; Dabney, J. (Eds.) Modeling, Identification, and Compensation of Friction in Harmonic Drives. In Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, USA, 10–13 December 2002; pp. 160–166. [Google Scholar]
- Taghirad, H.D.; Belanger, P.R. Modeling and parameter identification of harmonic drive systems. ASME J. Dyn. Syst. Meas. Control
**1998**, 120, 439–444. [Google Scholar] [CrossRef] - Maré, J.C. Friction modelling and simulation at system level: Considerations to load and temperature effects. Proc. Inst. Mech. Eng. Part I J. Syst. Control. Eng.
**2015**, 229, 27–48. [Google Scholar] [CrossRef] - Martineau, J.P.; Chedmail, P. Caractérisation Expérimentale et Modélisation du Comportement des Réducteurs Harmonic Drive. In Proceedings of the Congrès Mondial des Engrenages et des Transmissions de Puissance, Paris, France, 16–17 March 1999; pp. 1089–1100. [Google Scholar]
- Martineau J, P. Modelisation Experimentale des Reducteurs Harmonic Drive, Application a la Determination des Parametres Minimaux d’un Robot Souple Trois Axes[D]. Ph.D. Thesis, Nantes University, Nantes, France, 1996. [Google Scholar]
- Marc, M.J.C.B. Conception Préliminaire des Actionneurs Électromagnétiques Basée sur les Modèles: Lois d’estimations et Règles de Conception pour la Transmission de Puissance Mécanique; INSA: Toulouse, France, 2012. [Google Scholar]
- Yadav, D.; Verma, A. Comperative performance analysis of PMSM drive using MPSO and ACO techniques. Int. J. Power Electron. Drive Syst.
**2018**, 9, 1510–1522. [Google Scholar] [CrossRef] - Salem, W.A.A.; Osman, G.F.; Arfa, S.H. (Eds.) Adaptive Neuro-Fuzzy Inference System Based Field Oriented Control of PMSM & Speed Estimation. In Proceedings of the 2018 Twentieth International Middle East Power Systems Conference (MEPCON), Cairo, Egypt, 18–20 December 2018; pp. 626–631. [Google Scholar]
- Liu, Z.; Wei, H.; Zhong, Q.-C.; Liu, K.; Xiao, X.-S.; Wu, L.-H. Parameter estimation for VSI-fed PMSM based on a dynamic PSO with learning strategies. IEEE Trans. Power Electron.
**2016**, 32, 3154–3165. [Google Scholar] [CrossRef] - Barkat, S.; Tlemçani, A.; Nouri, H. Noninteracting adaptive control of PMSM using interval type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst.
**2011**, 19, 925–936. [Google Scholar] [CrossRef] - Han, J. From PID to active disturbance rejection control. IEEE Trans. Ind. Electron.
**2009**, 56, 900–906. [Google Scholar] [CrossRef] - Gao, S.; Wei, Y.; Zhang, D.; Qi, H.; Wei, Y. A modified model predictive torque control with parameters robustness improvement for PMSM of electric vehicles. Actuators
**2021**, 10, 132. [Google Scholar] [CrossRef] - Wang, X.; Zhu, H. Active disturbance rejection control of bearingless permanent magnet synchronous motor based on genetic algorithm and neural network parameters dynamic adjustment method. Electronics
**2023**, 12, 1455. [Google Scholar] [CrossRef] - Bao, H.; He, D.; Zhang, B.; Zhong, Q.; Hong, H.; Yang, H. Research on dynamic performance of independent metering valves controlling concrete-placing booms based on fuzzy-LADRC controller. Actuators
**2023**, 12, 139. [Google Scholar] [CrossRef] - Jin, K.; Song, J.; Li, Y.; Zhang, Z.; Zhou, H.; Chang, X. Linear active disturbance rejection control for the electro-hydraulic position servo system. Sci. Prog.
**2021**, 104, 00368504211000907. [Google Scholar] [CrossRef] - Hu, X.; Han, S.; Liu, Y.; Wang, H. Two-axis optoelectronic stabilized platform based on active disturbance rejection controller with LuGre friction model. Electronics
**2023**, 12, 1261. [Google Scholar] [CrossRef] - Sira-Ramírez, H.; Linares-Flores, J.; García-Rodríguez, C.; Contreras-Ordaz, M.A. On the control of the permanent magnet synchronous motor: An active disturbance rejection control approach. IEEE Trans. Control. Syst. Technol.
**2014**, 22, 2056–2063. [Google Scholar] [CrossRef] - Li, S.; Li, J. Output predictor-based active disturbance rejection control for a wind energy conversion system with PMSG. IEEE Access
**2017**, 5, 5205–5214. [Google Scholar] [CrossRef] - Du, B.; Wu, S.; Han, S.; Cui, S. Application of linear active disturbance rejection controller for sensorless control of internal permanent-magnet synchronous motor. IEEE Trans. Ind. Electron.
**2016**, 63, 3019–3027. [Google Scholar] [CrossRef] - Lu, Y.S.; Hwang, C.S. Tracking Control of a Harmonic Drive Actuator with Sliding-Mode Disturbance Observers. In Proceedings of the 2009 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Singapore, 14–17 July 2009; pp. 1798–1803. [Google Scholar]
- Huang, Y.; Xue, W. Active disturbance rejection control: Methodology and theoretical analysis. ISA Trans.
**2014**, 53, 963–976. [Google Scholar] [CrossRef] [PubMed] - Jingqing, H. Active Disturbance Rejection Control Technique—The Technique for Estimating and Compensating the Uncertainties. Master’s Thesis, National Defense Industry Press, Beijing, China, 2008. [Google Scholar]
- Wang, Y.; Fang, S.; Hu, J. Active disturbance rejection control based on deep reinforcement learning of PMSM for more electric aircraft. IEEE Trans. Power Electron.
**2022**, 38, 406–416. [Google Scholar] [CrossRef] - Rong, Z.; Huang, Q. A new PMSM speed modulation system with sliding mode based on active-disturbance-rejection control. J. Cent. South Univ.
**2016**, 23, 1406–1415. [Google Scholar] [CrossRef]

**Figure 1.**Structure of the harmonic reducer [26].

e_{1} | e_{2} | ||||||
---|---|---|---|---|---|---|---|

NB | NM | NS | ZO | PS | PM | PB | |

NB | NB/PB/PS | NB/PB/NS | NM/PM/NB | NM/PM/NB | NS/PS/NB | ZO/ZO/NM | ZO/ZO/PS |

NM | NB/PB/PS | NB/PB/NS | NM/PM/NB | NS/PS/NM | NS/PS/NM | ZO/ZO/NS | ZO/NS/ZO |

NS | NB/PM/ZO | NM/PM/NS | NS/PM/NM | NS/PS/NM | ZO/ZO/NS | PS/NS/NS | PS/NS/ZO |

ZO | NM/PM/ZO | NM/PM/NS | NS/PS/NS | ZO/ZO/NS | PS/NS/NS | PM/NM/NS | PM/NM/ZO |

PS | NM/PS/ZO | NS/PS/ZO | ZO/ZO/ZO | PS/NS/ZO | PS/NS/ZO | PM/NM/ZO | PB/NM/ZO |

PM | ZO/PS/PB | ZO/ZO/NS | PS/NS/PS | PS/NM/PS | PM/NM/PS | PB/NM/PS | PB/NB/PB |

PB | ZO/ZO/PB | ZO/ZO/PM | PS/NM/PM | PM/NM/PM | PM/NM/PS | PB/NB/PS | PB/NB/PB |

Reduction Ratio | Transmission Direct Efficiency at Rated Load | Max Torque (N.m) | Max Input Speed (rev/min) | Theoretical Lifespan (h) | Weight (kg) |
---|---|---|---|---|---|

100 | 0.69 | 49 | 7000 | 15,000 | 0.8 |

Resistance (Ω) | Inductance (mH) | Rated Torque (N.m) | Peak Torque (N.m) | Max Speed (rev/min) | Peak Current (A) | Inertia |
---|---|---|---|---|---|---|

100 | 0.65 | 0.72 | 3.8 | 3100 | 27 | 3.04 ∗ 10^{−5} kgm |

Constant Parameters | System Response Phenomena | Tuning Rules |
---|---|---|

β_{02}, β_{03} | Oscillation occurs | Decrease β_{01} |

Divergence occurs | Decrease β_{01} | |

Steady-state high-frequency oscillation occurs | Increase β_{01} | |

β_{01}, β_{03} | High-frequency oscillation occurs | Decrease β_{02} |

Disturbance rejection performance decrease | Increase β_{02} | |

Oscillation amplitude increase | Increase β_{02} | |

β_{01}, β_{02} | Overshoot occurs | Increase β_{02} |

Response time is long | Increase β_{03} | |

Large oscillation occurs | Decrease β_{03} |

Control Mode | PI | ADRC | IADRC |
---|---|---|---|

ITAE | 47.714 (rev/min)*s ^{2} | 11.559 (rev/min)*s ^{2} | 5.727 (rev/min)*s ^{2} |

Name | Parameter Value |
---|---|

Encoder | 2880 PPR |

Reduction ratio | 50 |

Counter weight | 25 N |

Disc radius | 0.1 m |

Load | 2.5 N·m |

Control Mode | RMSE | Peak-to-Peak Values |
---|---|---|

PID | 4165.1 | 13685 LNE |

ADRC | 1201.4 | 6218 LNE |

IADRC | 1040.8 | 4780 LNE |

Control Mode | RMSE | Peak-to-Peak Values |
---|---|---|

PID | 4183.6 | 13,773 LNE |

ADRC | 1636.4 | 8242 LNE |

IADRC | 1046.8 | 4869 LNE |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, P.; Shi, Z.; Yu, B.
Research on Friction Compensation Method of Electromechanical Actuator Based on Improved Active Disturbance Rejection Control. *Actuators* **2023**, *12*, 445.
https://doi.org/10.3390/act12120445

**AMA Style**

Zhang P, Shi Z, Yu B.
Research on Friction Compensation Method of Electromechanical Actuator Based on Improved Active Disturbance Rejection Control. *Actuators*. 2023; 12(12):445.
https://doi.org/10.3390/act12120445

**Chicago/Turabian Style**

Zhang, Pan, Zhaoyao Shi, and Bo Yu.
2023. "Research on Friction Compensation Method of Electromechanical Actuator Based on Improved Active Disturbance Rejection Control" *Actuators* 12, no. 12: 445.
https://doi.org/10.3390/act12120445