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Identification of joint position-dependent stiffness parameters and analysis of robot milling deformation


With the changes in joint torque and driving state caused by robot postures, the stiffness properties behave differently. However, constant joint stiffness parameters cannot accurately reflect the deformation of different robot postures. To solve this problem, based on the hypothesis of flexible joints, this paper proposes a pose-dependent identification method for joint stiffness. By changing the load at the end of the robot, the laser tracker is used to monitor the slight change of the measuring point on the link near the joint, and the joint deformation monitoring is realized with the analysis of micro displacements. Combined with the external loads monitored by the dynamometer, the change of joint torque is obtained through structural analysis, and then the joint stiffness at a given joint position is identified. Based on the joint stiffness identification results of different joint positions, the joint stiffness is fitted by a polynomial function, and then the varied robot joint stiffness model is obtained. Jacobean transformation and conservative congruence transformation are combined to predict the Cartesian stiffness of the robot. The effectiveness of the stiffness model proposed in this paper is verified by the loading experiments at the end of the robot and robot deformation measurement for milling long aluminum strip.

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Data availability

The measuring data for the varied joint stiffness using the laser tracker and in our paper are available from the corresponding author by request, and other related materials can also be obtained from the corresponding author.

Code availability

The code for joint stiffness identification and analysis of stiffness distribution during the study are available from the corresponding author by request.


  1. 1.

    Dumas C, Caro S, Garnier S, Furet B (2011) Joint stiffness identification of six-revolute industrial serial robots. Robot Comput Integr Manuf 27(4):881–888

    Article  Google Scholar 

  2. 2.

    Dumas C, Caro S, ChéRif M, Garnier S (2010) A methodology for joint stiffness identification of serial robots. In: Ieee/rsj International Conference on Intelligent Robots and Systems. pp 464–469

  3. 3.

    Cordes M, Hintze W (2017) Offline simulation of path deviation due to joint compliance and hysteresis for robot machining (in English). Int J Adv Manuf Technol 90(1–4):1075–1083

    Article  Google Scholar 

  4. 4.

    Rezaei A, Akbarzadeh A, Akbarzadeht M (2012) An investigation on stiffness of a 3-PSP spatial parallel mechanism with flexible moving platform using invariant form. Mech Mach Theory 51:195–216

    Article  Google Scholar 

  5. 5.

    Cammarata A (2016) Unified formulation for the stiffness analysis of spatial mechanisms. J Mech Mach Theory 105:272–284

    Article  Google Scholar 

  6. 6.

    Deblaise D, Hernot X, Maurine P (2006) A systematic analytical method for PKM stiffness matrix calculation. In International conference on robotics and automation. pp 4213–4219

  7. 7.

    Salisbury J (1980) Active stiffness control of a manipulator in cartesian coordinates. Conf Decis Control 19(19):95–100

    Google Scholar 

  8. 8.

    Gosselin C (1990) Stiffness mapping for parallel manipulators. Int Conf Robot Automation 6(3):377–382

    Google Scholar 

  9. 9.

    Hu B, Lu Y, Tan Q, Yu J, Han J (2011) Analysis of stiffness and elastic deformation of a 2(SP+SPR+SPU) serial-parallel manipulator. Robot Comput Integr Manuf 27(2):418–425

    Article  Google Scholar 

  10. 10.

    Klimchik A, Pashkevich A, Chablat D (2019) Fundamentals of manipulator stiffness modeling using matrix structural analysis. Mech Mach Theory 133:365–394

    Article  Google Scholar 

  11. 11.

    Rezaei A, Akbarzadeh A (2018) Compliance error modeling for manipulators considering the effects of the component weights and the body and joint flexibilities. Mech Mach Theory 130:244–275

    Article  Google Scholar 

  12. 12.

    Klimchik A, Pashkevich A (2017) Serial vs. quasi-serial manipulators: comparison analysis of elasto-static behaviors. Mech Mach Theory 107:46–70

    Article  Google Scholar 

  13. 13.

    Pashkevich A, Klimchik A, Chablat D (2011) Enhanced stiffness modeling of manipulators with passive joints. Mech Mach Theory 46(5):662–679

    Article  Google Scholar 

  14. 14.

    Dumas C, Caro S, Cherif M, Garnier S, Furet B (2012) Joint stiffness identification of industrial serial robots (in English). Robotica 30:649–659

    Article  Google Scholar 

  15. 15.

    Slavkovic N, Milutinovic D, Glavonjic M (2014) A method for off-line compensation of cutting force-induced errors in robotic machining by tool path modification. Int J Adv Manuf Technol 70(9):2083–2096

    Article  Google Scholar 

  16. 16.

    Yushan C (2011) Joint stiffness identification of 6R industrial robot and experimental verification. Huazhong University of Science and Technology, Master

    Google Scholar 

  17. 17.

    Penghui H (2013) Study on the stiffness performance optimization for robot machining system. Zhejiang University, Master

    Google Scholar 

  18. 18.

    Huang C, Hung W, Kao I (2002) New conservative stiffness mapping for the Stewart-Gough platform. Int Conf Robot Automation 1:823–828

    Google Scholar 

  19. 19.

    Alici G, Shirinzadeh B (2005) Enhanced stiffness modeling, identification and characterization for robot manipulators. IEEE Trans Robot Autom 21(4):554–564

    Article  Google Scholar 

  20. 20.

    Dumas C, Caro S, Cherif M, Garnier S, Furet B (2010) A methodology for joint stiffness identification of serial robots. In Ieee/Rsj 2010 International Conference on Intelligent Robots and Systems (IEEE International Conference on Intelligent Robots and Systems. IEEE, New York, pp 464–469

  21. 21.

    Weiwei Q, Penghui H, Genjun Y, Guanping H, Fucheng Y, Xin S (2013) “Research on the stiffness performance for robot machining systems”, (in Chinese). Acta Aeronautica ET Astronautica Sinica 034(012):2823–2832

    Google Scholar 

  22. 22.

    Hoevenaars AG, Lambert P, Herder JL (2015) Jacobian-based stiffness analysis method for parallel manipulators with non-redundant legs. Proc Inst Mech Eng Part C: J Mech Eng 0954406215602283

  23. 23.

    Hoevenaars AGL, Gosselin C, Lambert P, Herder JL (2016) Experimental validation of Jacobian-based stiffness analysis method for parallel manipulators with nonredundant legs. J Mech Robot Auton Syst 8(4):041002

    Article  Google Scholar 

  24. 24.

    Klimchik A, Pashkevich A, Chablat D (2013) CAD-based approach for identification of elasto-static parameters of robotic manipulators. Finite Elem Anal Des 75:19–30

    MathSciNet  Article  Google Scholar 

  25. 25.

    Klimchik A, Wu Y, Dumas C, Caro S, Furet B, Pashkevich A (2013) Identification of geometrical and elastostatic parameters of heavy industrial robots. In International conference on robotics and automation. pp 3707–3714

  26. 26.

    Klimchik A, Furet B, Caro S, Pashkevich A (2015) Identification of the manipulator stiffness model parameters in industrial environment. Mech Mach Theory 90:1–22

    Article  Google Scholar 

  27. 27.

    Nguyen H, Zhou J, Kang H (2015) A calibration method for enhancing robot accuracy through integration of an extended Kalman filter algorithm and an artificial neural network. Neurocomputing 151:996–1005

    Article  Google Scholar 

  28. 28.

    Yang K, Yang W, Cheng G, Lu B (2018) A new methodology for joint stiffness identification of heavy duty industrial robots with the counterbalancing system. Robot Comput Integr Manuf 53:58–71

    Article  Google Scholar 

  29. 29

    Nguyen Vu L, Kuo C-H (2019) An analytical stiffness method for spring-articulated planar serial or quasi-serial manipulators under gravity and an arbitrary load. Mech Mach Theory 137:108–126

    Article  Google Scholar 

  30. 30.

    SicilianoB, Sciavicco L, Villani L, Oriolo G (2009) Robotics: modelling, planning and control (Advanced Textbooks in Control and Signal Processing). Springer-Verlag London, pp XXIV, 632

  31. 31.

    Alici G, Shirinzadeh B (2005) Enhanced stiffness modeling identification and characterization for robot manipulators (in English). IEEE Trans Robot 21(4):554–564

    Article  Google Scholar 

  32. 32.

    Denkena B, Litwinski K, Schönherr M (2013) Innovative drive concept for machining robots. Procedia CIRP 9:67–72

    Article  Google Scholar 

  33. 33.

    Vieler H, Karim A, Lechler A (2017) Drive based damping for robots with secondary encoders. Robot Comput Integr Manuf 47:117–122, 2017/10/01

  34. 34.

    Chen C, Peng F, Yan R, Tang X, Li Y, Fan Z (2020) Rapid prediction of posture-dependent FRF of the tool tip in robotic milling. Robot Comput Integr Manuf 64:101906, 2020/08/01/

  35. 35.

    Chen C et al. (2019) Stiffness performance index based posture and feed orientation optimization in robotic milling process. Robot Comput Integr Manuf 55:29–40, 2019/02/01/

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This work was supported by National Natural Science Foundation of China (Grant No. 52005201) and National Science Fund for Distinguished Young Scholars of China (Grant No. 51625502).

Author information




Zerun Zhu proposed the article’s innovative thinking, derived the core formula of the article, and completed the English writing of the article. Chen Chen and Fangyu Peng putted forward many constructive suggestions for the writing of the whole article, while Xianyin Duan putted forward some constructive suggestions for the experimental part. Dequan Wei accomplished some parts of experimental validations.

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Correspondence to Chen Chen.

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Appendix I. Kinematics of ABB IRB6660-205/1.9 robot

Appendix I. Kinematics of ABB IRB6660-205/1.9 robot

The kinematics model of ABB IRB6660 robot is shown in Fig. 17. The modified DH model is adopted here, and the homogeneous coordinate transformation matrix of adjacent motion axis coordinates is shown as follows:

$${}_i^{i - 1}T = \left[ {\begin{array}{*{20}{c}} {\operatorname{c} {\theta_i}}&{ - \operatorname{s} {\theta_i}}&0&{{a_{i - 1}}} \\ {\operatorname{s} {\theta_i}\operatorname{c} {\alpha_{i - 1}}}&{\operatorname{c} {\theta_i}\operatorname{c} {\alpha_{i - 1}}}&{ - \operatorname{s} {\alpha_{i - 1}}}&{ - {d_i}\operatorname{s} {\alpha_{i - 1}}} \\ {\operatorname{s} {\theta_i}\operatorname{s} {\alpha_{i - 1}}}&{\operatorname{c} {\theta_i}\operatorname{s} {\alpha_{i - 1}}}&{\operatorname{c} {\alpha_{i - 1}}}&{{d_i}\operatorname{c} {\alpha_{i - 1}}} \\ 0&0&0&1 \end{array}} \right],(i = 1,2,3,4,5,6)$$

where, \(\operatorname{s} {\theta_i}\) is short for \(\sin {\theta_i}\), and \(\operatorname{c} {\theta_i}\) for \(\cos {\theta_i}\). For a revolute joint, only angle \({\theta_i}\) varies in the D-H parameters, so \({}_i^{i - 1}T\) is a function of \({\theta_i}\).

Fig. 17

Kinematics of ABB IRB6660-205/1.9 robot

The modified DH model parameters of ABB IRB6660-205/1.9 robot are shown in Table 4. Under this condition, the homogeneous coordinate transformation relation between the adjacent joint axes is as follows:

$${}_1^0T = \left[ {\begin{array}{*{20}{c}} {{\text{c}}{\theta_1}}&{ - {\text{s}}{\theta_1}}&0&0 \\ {{\text{s}}{\theta_1}}&{{\text{c}}\theta }&0&0 \\ 0&0&1&{814.5} \\ 0&0&0&1 \end{array}} \right],\,{}_{31}^1T = \left[ {\begin{array}{*{20}{c}} {\operatorname{c} {\theta_3}}&{ - \operatorname{s} {\theta_3}}&0&{300} \\ 0&0&{ - 1}&0 \\ {\operatorname{s} {\theta_3}}&{\operatorname{c} {\theta_3}}&0&0 \\ 0&0&0&1 \end{array}} \right],\,$$
$${}_{32}^{31}T = \left[ {\begin{array}{*{20}{c}} {\operatorname{c} \left( { - {\theta_2} - {\theta_3}} \right)}&{ - \operatorname{s} \left( { - {\theta_2} - {\theta_3}} \right)}&0&{500} \\ {\operatorname{s} \left( { - {\theta_2} - {\theta_3}} \right)}&{\operatorname{c} \left( { - {\theta_2} - {\theta_3}} \right)}&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right],$$
$${}_{33}^{32}T = \left[ {\begin{array}{*{20}{c}} {\operatorname{c} \left( {{\theta_2} + {\theta_3} - \pi } \right)}&{ - \operatorname{s} \left( {{\theta_2} + {\theta_3} - \pi } \right)}&0&{700} \\ {\operatorname{s} \left( {{\theta_2} + {\theta_3} - \pi } \right)}&{\operatorname{c} \left( {{\theta_2} + {\theta_3} - \pi } \right)}&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right], {}_3^{33}T = \left[ {\begin{array}{*{20}{c}} 0&{ - 1}&0&{500} \\ 1&0&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}} \right],$$
$${}_4^3T = \left[ {\begin{array}{*{20}{c}} {\operatorname{c} {\theta_4}}&{ - \operatorname{s} {\theta_4}}&0&{280} \\ 0&0&{ - 1}&{ - 893} \\ {\operatorname{s} {\theta_4}}&{\operatorname{c} {\theta_4}}&0&0 \\ 0&0&0&1 \end{array}} \right], {}_5^4T = \left[ {\begin{array}{*{20}{c}} {\operatorname{c} {\theta_5}}&{ - \operatorname{s} {\theta_5}}&0&0 \\ 0&0&{ - 1}&0 \\ {\operatorname{s} {\theta_5}}&{\operatorname{c} {\theta_5}}&0&0 \\ 0&0&0&1 \end{array}} \right],$$
$${}_6^5T = \left[ {\begin{array}{*{20}{c}} {\operatorname{c} {\theta_6}}&{ - \operatorname{s} {\theta_6}}&0&0 \\ 0&0&1&{200} \\ { - \operatorname{s} {\theta_6}}&{ - \operatorname{c} {\theta_6}}&0&0 \\ 0&0&0&1 \end{array}} \right].$$
Table 4 The modified DH parameters of ABB IRB6660-205/1.9 [35]

Then the transformation relation \({}_{\text{i}}^{0}{\mathbf{T}}\) of any joint coordinate system i to base coordinate system 0 can be obtained:

$${}_{\text{i}}^{0}{\mathbf{T}} = {}_1^{0}{\mathbf{T}}{}_{31}^1{\mathbf{T}} \cdot \cdot \cdot {}_i^{i - 1}{\mathbf{T}},(i = 1,31,32,33,3,4,5,6).$$

The coordinate transformation relation between the robot end flange coordinate system (joint 6 coordinate system) and the robot base coordinate system 0 is expressed as follows:

$${}_6^{0}{\mathbf{T}} = {}_1^{0}{\mathbf{T}}{}_{31}^1{\mathbf{T}}{}_{32}^{31}{\mathbf{T}}{}_{33}^{32}{\mathbf{T}}{}_3^{33}{\mathbf{T}}{}_4^3{\mathbf{T}}{}_5^4{\mathbf{T}}{}_6^5{\mathbf{T}}.$$

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Zerun, Z., Chen, C., Fangyu, P. et al. Identification of joint position-dependent stiffness parameters and analysis of robot milling deformation. Int J Adv Manuf Technol (2021).

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  • Robotic milling
  • Varied joint stiffness identification
  • Deformation
  • Virtual joint
  • Large workpiece