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

Abstract

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.

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Funding

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).

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Contributions

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:

$${}_i^{i - 1}T = \left[ {\begin{array}{*{20}{c}} {\operatorname{c} {\theta_i}}&{ - \operatorname{s} {\theta_i}}&0&{{a_{i - 1}}} \\ {\operatorname{s} {\theta_i}\operatorname{c} {\alpha_{i - 1}}}&{\operatorname{c} {\theta_i}\operatorname{c} {\alpha_{i - 1}}}&{ - \operatorname{s} {\alpha_{i - 1}}}&{ - {d_i}\operatorname{s} {\alpha_{i - 1}}} \\ {\operatorname{s} {\theta_i}\operatorname{s} {\alpha_{i - 1}}}&{\operatorname{c} {\theta_i}\operatorname{s} {\alpha_{i - 1}}}&{\operatorname{c} {\alpha_{i - 1}}}&{{d_i}\operatorname{c} {\alpha_{i - 1}}} \\ 0&0&0&1 \end{array}} \right],(i = 1,2,3,4,5,6)$$
(30)

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
figure17

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:

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

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:

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

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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). https://doi.org/10.1007/s00170-021-08090-3

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Keywords

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