# An Accurate Dynamic Model Identification Method of an Industrial Robot Based on Double-Encoder Compensation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Linearization of Dynamic Model and Identification of Friction

_{ci}

_{1}and k

_{vi}

_{1}represent the friction coefficients during forward motion, and k

_{ci}

_{2}and k

_{vi}

_{2}represent the friction coefficients during backward motion. However, accurately defining static and low-speed friction poses a significant challenge. A suitable threshold $\lambda $ is set with Equation (9) to make the joint’s low-speed and high-speed movements smoother, and friction model accuracy is ensured by considering the joint velocity squared and velocity cubed in the calculations. tanh(·) is the hyperbolic tangent function, and eps is the transition accuracy typically set to 0.0001. This method overcomes the discontinuity problem of the sign(·) function near the switching point at 0 and avoids the estimation errors in friction force caused by identification errors or switching the direction of movement. ${\mathit{\tau}}_{estfi}$ is the friction torque of the i-th joint determined through measuring torque and inner-layer identification, while ${\mathit{\tau}}_{fi}$ is the estimated friction torque of the i-th joint obtained through the friction model.

## 3. Optimization Index Based on the Condition Number of Block Regression Matrix

_{f}= 2$\pi $f

_{f}; a

_{l,i}and b

_{l,i}are the amplitudes of the trigonometric functions. Considering joint limits, velocity, and acceleration limits, the following objectives and constraints are given, where t

_{s}and t

_{e}are the start and end times of the sampling time:

## 4. Dynamic Model Identification of Link Based on WLS

**E**(·) represents mathematical expectation, and ${o}_{\delta}^{2}$ represents the variance of $\delta $. Assuming that each joint’s noise error is independent of each other,

**e**is a unit diagonal matrix. ${o}_{\delta}^{2}$

**e**represents the variance of the noise error of the driving torque of the six joints. Directly using the traditional standard LS (least squares) method for identification can only minimize the 2-norm of the error between the collected torque and the estimated torque of the linear part, without minimizing the 2-norm error $\delta $. This leads to suboptimal optimization of the minimum parameter set variance during identification. To overcome this limitation, we recommend using the WLS (weighted least square) method. First, calculate the torque error, define the collected torque as ${\mathit{\tau}}_{sample}$, the data number is 6 m, and estimate the torque through LS as ${\mathit{\tau}}_{LS}$.

**e**can be a diagonal matrix. However, in reality, the joint noise is correlated. Hence, the weight can be calculated by computing the non-diagonal covariance matrix.

## 5. Nonlinear Joint Dynamics Compensation

_{m}and J

_{l}refer to the rotational inertia of the motor and the link, respectively. K

_{w}is the transmission coupling stiffness coefficient while D

_{w}is the transmission shaft damping coefficient. ${\mathit{\theta}}_{m}$ and ${\mathit{\theta}}_{l}$ are, respectively, the theoretical angle calculated to the link side and the actual angle of the link side. Due to the challenging friction modeling and accurate modeling of the harmonic link and torque transmission error, errors occur. Also, the acceleration is measured inaccurately prone to fluctuations, and filtering causes errors. Therefore, the torque generated by J

_{m}${\ddot{\mathit{\theta}}}_{m}$ is not taken into account, and the following text will incorporate it into the error. Based on the traditional dynamic model, this paper calculates the difference between the model torque and the actual torque, and analyzes it using an error model and data model analysis. To utilize dynamic compensation, the torque ${\mathit{\tau}}_{w}$ is no longer used, and the torque ${\mathit{\tau}}_{u}$ is defined.

_{i}is

_{target}= [f

_{target}(1), f

_{target}(2),…f

_{target}(M)]; then, the weight identification is

**W**

_{r}= [w

_{r}

_{1}, w

_{r2}, …, w

_{rN}]; the actual process needs to [${\theta}_{m}$−${\theta}_{l}$, ${\dot{\theta}}_{m}$−${\dot{\theta}}_{l}$] as the input data, and torque ${\mathit{\tau}}_{u}$ as the expected training.

## 6. Simulation and Experiment

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Vandanjon, P.; Gautier, M.; Desbats, P. Identification of robot inertial parameters by means of spectrum analysis. In Proceedings of the 1995 IEEE International Conference on Robotics and Automation (ICRA), Nagoya, Japan, 21–27 May 1995; pp. 3033–3038. [Google Scholar]
- Wu, J.; Wang, J.; You, Z. An overview of dynamic parameter identification of robots. Robot. Comput. Integr. Manuf.
**2010**, 26, 414–419. [Google Scholar] [CrossRef] - Gautier, M.; Khalil, W. Direct calculation of minimum set of inertial parameters of serial robots. IEEE Trans. Robot. Autom.
**1990**, 6, 368–373. [Google Scholar] [CrossRef] - Swevers, J.; Verdonck, W.; De Schutter, J. Dynamic model identification for industrial robots. IEEE Control Syst. Mag.
**2007**, 27, 58–71. [Google Scholar] - Venture, G.; Ayusawa, K.; Nakamura, Y. A numerical method for choosing motions with optimal excitation properties for identification of biped dynamics-An application to human. In Proceedings of the 2009 IEEE International Conference on Robotics and Automation, Kobe, Japan, 12–17 May 2009; pp. 1226–1231. [Google Scholar]
- Huang, Y.; Ke, J.; Zhang, X.; Ota, J. Dynamic parameter identification of serial robots using a hybrid approach. IEEE Trans. Robot.
**2023**, 39, 1607–1621. [Google Scholar] [CrossRef] - Zhuang, C.; Yao, Y.; Shen, Y.; Xiong, Z. A convolution neural network based semi-parametric dynamic model for industrial robot. J. Mech. Eng. Sci.
**2022**, 236, 3683–3700. [Google Scholar] [CrossRef] - Huang, S.; Chen, J.; Zhang, J.; Zhu, Z.; Zhou, H.; Li, F.; Zhou, X. Robust estimation for an extended dynamic parameter set of serial manipulators and unmodeled dynamics compensation. IEEE/ASME Trans. Mechatron.
**2022**, 27, 962–973. [Google Scholar] [CrossRef] - Gautier, M.; Briot, S. New method for global identification of the joint drive gains of robots using a known payload mass. In Proceedings of the 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, San Francisco, CA, USA, 25–30 September 2011; pp. 3728–3733. [Google Scholar]
- Briot, S.; Gautier, M. Global identification of joint drive gains and dynamic parameters of parallel robots. Multibody Syst. Dyn.
**2015**, 33, 3–26. [Google Scholar] [CrossRef] - Albu Schäffer, A.; Ott, C.; Hirzinger, G. A unified passivity-based control framework for position, torque and impedance control of flexible joint robots. Int. J. Robot. Res.
**2007**, 26, 23–39. [Google Scholar] [CrossRef] - Spong, M.W. Modeling and control of elastic joint robots. J. Dyn. Sys. Meas. Control
**1987**, 109, 310–319. [Google Scholar] [CrossRef] - Han, Y.; Wu, J.; Liu, C.; Xiong, Z. Static model analysis and identification for serial articulated manipulators. Robot. Comput. Integr. Manuf.
**2019**, 57, 155–165. [Google Scholar] [CrossRef] - Gautier, M. Dynamic identification of robots with power model. In Proceedings of the International Conference on Robotics and Automation, Albuquerque, NM, USA, 25 April 1997; pp. 1922–1927. [Google Scholar]
- Wolf, S.; Iskandar, M. Extending a dynamic friction model with nonlinear viscous and thermal dependency for a motor and harmonic drive gear. In Proceedings of the 2018 IEEE International Conference on Robotics and Automation (ICRA), Brisbane, QLD, Australia, 21–25 May 2018; pp. 783–790. [Google Scholar]
- Iskandar, M.; Wolf, S. Dynamic friction model with thermal and load dependency: Modeling, compensation, and external force estimation. In Proceedings of the 2019 International Conference on Robotics and Automation (ICRA), Montreal, QC, Canada, 20–24 May 2019; pp. 7367–7373. [Google Scholar]
- Ji, Y.; Jiang, X.; Wan, L. Hierarchical least squares parameter estimation algorithm for two-input Hammerstein finite impulse response systems. J. Frankl. Inst.
**2020**, 357, 5019–5032. [Google Scholar] [CrossRef] - Kammerer, N.; Garrec, P. Dry friction modeling in dynamic identification for robot manipulators: Theory and experiments. In Proceedings of the 2013 IEEE International Conference on Mechatronics (ICM), Vicenza, Italy, 27 February–1 March 2013; pp. 422–429. [Google Scholar]
- Swevers, J.; Verdonck, W.; Naumer, B.; Pieters, S.; Biber, E. An experimental robot load identification method for industrial application. Int. J. Robot. Res.
**2002**, 21, 701–712. [Google Scholar] [CrossRef] - Zhang, L.; Wang, J.; Chen, J.; Chen, K.; Lin, B.; Xu, F. Dynamic modeling for a 6-DOF robot manipulator based on a centrosymmetric static friction model and whale genetic optimization algorithm. Adv. Eng. Softw.
**2019**, 135, 102684. [Google Scholar] [CrossRef] - Deng, J.; Shang, W.; Zhang, B.; Zhen, S.; Cong, S. Dynamic Model Identification of Collaborative Robots Using a Three-Loop Iterative Method. In Proceedings of the 2021 6th IEEE International Conference on Advanced Robotics and Mechatronics (ICARM), Chongqing, China, 3–5 July 2021; pp. 937–942. [Google Scholar]
- Shi, X.; Han, Y.; Wu, J.; Xiong, Z. Servo system identification based on curve fitting to phase-plane trajectory. J. Dyn. Sys. Meas. Control
**2020**, 142, 031001. [Google Scholar] [CrossRef] - Shi, X.; Han, Y.; Wu, J.; Xiong, Z. An FFT-based method for analysis, modeling and identification of kinematic error in Harmonic Drives. In Proceedings of the International Conference on Intelligent Robotics and Applications (ICIRA), Shenyang, China, 8–11 August 2019; pp. 191–202. [Google Scholar]
- Han, Y.; Wu, J.; Liu, C.; Xiong, Z. An Iterative Approach for Accurate Dynamic Model Identification of Industrial Robots. IEEE Trans. Robot.
**2020**, 36, 1577–1594. [Google Scholar] [CrossRef] - Herzog, A.; Righetti, L.; Grimminger, F.; Pastor, P.; Schaal, S. Momentum-based balance control for torque-controlled humanoids. Comput. Res. Repos.
**2013**, 1, 1–7. [Google Scholar] - Niku, S.B. Introduction to Robotics: Analysis, Systems, Applications; Prentice Hall: Upper Saddle River, NJ, USA, 2001. [Google Scholar]
- Lu, Y.; Shen, Y.; Zhuang, C. External force estimation for industrial robots using configuration optimization. Automatika
**2023**, 64, 365–388. [Google Scholar] [CrossRef] - Hamon, P.; Gautier, M.; Garrec, P.; Janot, A. Dynamic modeling and identification of joint drive with load-dependent friction model. In Proceedings of the 2010 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Montreal, QC, Canada, 6–9 July 2010; pp. 902–907. [Google Scholar]
- Hamon, P.; Gautier, M.; Garrec, P. New dry friction model with load and velocity-dependence and dynamic identification of multi-DoF robots. In Proceedings of the 2011 IEEE International Conference on Robotics and Automation (ICRA), Shanghai, China, 9–13 May 2011; pp. 1077–1084. [Google Scholar]
- Ijspeert, A.J.; Nakanishi, J.; Hoffmann, H.; Pastor, P.; Schaal, S. Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors. Neural Comput.
**2013**, 25, 328–373. [Google Scholar] [CrossRef] [PubMed] - Performance Real-Time Test System. Available online: https://www.speedgoat.com/products-services/real-time-target-machines/performance-real-time-target-machine (accessed on 2 November 2022).

**Figure 4.**The experimental robotic system: (

**a**) 6-DoF industrial robot; (

**b**) SpeedGoat simulation platform.

Limits | Joint 1 | Joint 2 | Joint 3 | Joint 4 | Joint 5 | Joint 6 | |
---|---|---|---|---|---|---|---|

Joint limit ${\theta}_{I,max}$ (deg) | Max | 120 | 90 | 60 | 120 | 120 | 120 |

Min | −120 | −90 | −60 | −120 | −120 | −120 | |

Joint velocity limit ${\dot{\theta}}_{I,max}$ (deg/s) | Max | 100 | 60 | 60 | 80 | 80 | 80 |

Min | −100 | −60 | −60 | −80 | −80 | −80 | |

Joint acceleration limit ${\ddot{\theta}}_{I,max}$ (deg/s^{2}) | Max | 120 | 120 | 120 | 120 | 120 | 120 |

Optimization Method | Condition Number |
---|---|

WLS | 189.4012 |

Ours | 162.2440 |

Optimization Parameters | Joint 1 | Joint 2 | Joint 3 | Joint 4 | Joint 5 | Joint 6 |
---|---|---|---|---|---|---|

a_{l,i} | 0.0046 −0.0074 0.4845 −0.5103 0.0286 | −0.0406 0.0597 0.2492 0.0678 −0.3361 | −0.1772 0.0661 0.1019 0.0189 −0.0097 | −0.1704 0.1161 −0.1703 0.1426 0.0820 | −0.0210 −0.1317 0.3614 −0.1994 −0.0094 | 0.0673 0.2685 0.1682 −0.4241 −0.0763 |

b_{l,i} | −0.0053 −0.0182 0.6963 −0.3354 −0.1411 | −0.0234 −0.2122 0.1619 −0.1082 0.0789 | −0.0001 −0.0583 0.0450 0.1273 −0.1056 | 0.0088 −0.3357 0.1996 0.1731 −0.1257 | 0.1460 −0.3491 0.3639 −0.1008 −0.0273 | 0.1335 0.0677 0.0942 −0.2907 0.1227 |

${\theta}_{i0}\left(\mathrm{rad}\right)$ | 0.3361 | −0.2762 | −0.0112 | −0.2369 | 0.1976 | 0.4777 |

Identification Method | Joint 1 | Joint 2 | Joint 3 | Joint 4 | Joint 5 | Joint 6 |
---|---|---|---|---|---|---|

WLS | 2.3376 | 2.1278 | 1.5395 | 0.3361 | 0.7101 | 0.2578 |

Ours | 1.9594 | 1.7345 | 1.1609 | 0.3027 | 0.5852 | 0.2096 |

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**

Liu, X.; Xu, Y.; Song, X.; Wu, T.; Zhang, L.; Zhao, Y.
An Accurate Dynamic Model Identification Method of an Industrial Robot Based on Double-Encoder Compensation. *Actuators* **2023**, *12*, 454.
https://doi.org/10.3390/act12120454

**AMA Style**

Liu X, Xu Y, Song X, Wu T, Zhang L, Zhao Y.
An Accurate Dynamic Model Identification Method of an Industrial Robot Based on Double-Encoder Compensation. *Actuators*. 2023; 12(12):454.
https://doi.org/10.3390/act12120454

**Chicago/Turabian Style**

Liu, Xun, Yan Xu, Xiaogang Song, Tuochang Wu, Lin Zhang, and Yanzheng Zhao.
2023. "An Accurate Dynamic Model Identification Method of an Industrial Robot Based on Double-Encoder Compensation" *Actuators* 12, no. 12: 454.
https://doi.org/10.3390/act12120454