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Dynamic modeling and identification of low impact docking mechanism based on symmetric excitation trajectory

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Abstract

Inertial and nonlinear friction parameters precise identification of Stewart platform (SP) is a pending problem owing to the strong coupling and small workspace. This paper proposes a novel identification model based on symmetric excitation trajectory (SET), which is verified on low impact docking mechanism (LIDM, a typical SP). Firstly, the SET is defined, and its decoupling and filtering characteristics of the SP are analyzed. The dynamic equation of the LIDM is derived from Boltzmann–Hamel–d’alembert formula, and a typical dynamic parameter identification model (TM) is obtained. Then, a symmetric dynamic parameter identification model (SM) with the decoupling characteristic is derived from the SET. Moreover, the SM is improved according to the nonlinear friction phenomenon, and a nonlinear dynamic parameter identification model (SM_Non) including acceleration term is established. Besides, the speed-span index is given to ensure sufficient excitation of friction at different velocities during the identification process. Finally, experiments on the LIDM demonstrate the validity and accuracy of the proposed methods. The results of joint current prediction experiments show that the root mean square error of the SM_Non is 92.789% lower than that of the TM.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Funding

This research work was supported by National Natural Science Foundation of China (NSFC) [grant numbers U21B6002] and National key research and development program[grant numbers 2019YFB1312700].

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Authors and Affiliations

  1. State Key Laboratory of Robotic and Systems, Harbin Institute of Technology, Yikuang, Harbin, 150080, Heilongjiang, China

    Congcong Xu, Gangfeng Liu, Changle Li, Yanhe Zhu & Jie Zhao

  2. School of Mechatronic Engineering, Harbin Institute of Technology, XidaZhi, Harbin, Heilongjiang, China

    Congcong Xu, Gangfeng Liu, Changle Li, Yanhe Zhu & Jie Zhao

Authors
  1. Congcong Xu
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  3. Changle Li
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  4. Yanhe Zhu
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Congcong Xu, Gangfeng Liu, Changle Li, Yanhe Zhu and Jie Zhao. The first draft of the manuscript was written by Congcong Xu and Gangfeng Liu. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Gangfeng Liu.

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Appendices

Appendix A: Inertia matrix, Coriolis and centrifugal terms, gravity vector

Inertial parameters of the moving platform at the coordinate system \({{O}_{B}}-{{X}_{B}}{{Y}_{B}}{{Z}_{B}}\):

$$\begin{aligned} \left\{ \begin{array}{*{35}{l}} {\varvec{M}_{P}}=\varvec{J}_{C}^{T}{\varvec{M}_{C}}{\varvec{J}_{C}} \\ {\varvec{C}_{P}}=\varvec{J}_{C}^{T}{\varvec{M}_{C}}{\varvec{{\dot{J}}}_{C}}+\varvec{J}_{C}^{T}{\varvec{C}_{C}}{\varvec{J}_{C}} \\ {\varvec{G}_{P}}=\varvec{J}_{C}^{T}{\varvec{G}_{C}} \\ \end{array} \right. , \end{aligned}$$
(A1)

where \({\varvec{M}_{C}}=\left[ \begin{matrix} m{\varvec{I}_{3\times 3}} &{} {\varvec{0}_{3\times 3}} \\ {\varvec{0}_{3\times 3}} &{} {\varvec{I}_{C}} \\ \end{matrix} \right] \), \({\varvec{C}_{C}}=\left[ \begin{matrix} {\varvec{0}_{3\times 3}} &{} {\varvec{0}_{3\times 3}} \\ {\varvec{0}_{3\times 3}} &{} \varvec{\omega } \times {\varvec{I}_{C}} \\ \end{matrix} \right] \), \({\varvec{G}_{C}}=\left[ \begin{matrix} m\varvec{g} \\ {\varvec{0}_{3\times 1}} \\ \end{matrix} \right] \) and \({\varvec{J}_{C}}=\left[ \begin{matrix} {\varvec{I}_{3\times 3}} &{} -{{({}^{A}{\varvec{R}_{B}}{}^{B}\varvec{c})}_{\times }} \\ {\varvec{0}_{3\times 3}} &{} {\varvec{I}_{3\times 3}} \\ \end{matrix} \right] \). The inertial parameters of joint (the sum of the upper and lower parts ):

$$\begin{aligned} \left\{ \begin{array}{*{35}{l}} {\varvec{M}_{i}}=-{{m}_{{{c}_{e}}}}\hat{s}_{{{i}_{\times }}}^{2}+{{m}_{2}}{{{\hat{s}}}_{i}}\hat{\varvec{s}}_{i}^{T}-\frac{1}{l_{i}^{2}}\left( {{I}_{c1}}+{{I}_{c2}} \right) \hat{\varvec{s}}_{{{i}_{\times }}}^{2} \\ {\varvec{C}_{i}}=-\frac{1}{l_{i}^{2}}{{m}_{2}}{{c}_{2}}{{{\hat{\varvec{s}}}}_{i}}\dot{\varvec{x}}_{i}^{T}\hat{\varvec{s}}_{{{i}_{\times }}}^{2}+\frac{2}{{{l}_{i}}}{{m}_{{{c}_{o}}}}{{{\dot{l}}}_{i}}\hat{\varvec{s}}_{{{i}_{\times }}}^{2} \\ {\varvec{G}_{i}}=\left( {{m}_{{{g}_{e}}}}\hat{\varvec{s}}_{{{i}_{\times }}}^{2}-{{m}_{2}}{{{\hat{\varvec{s}}}}_{i}}\hat{\varvec{s}}_{i}^{T} \right) \varvec{g} \\ \end{array} \right. , \end{aligned}$$
(A2)

where \({{m}_{{{c}_{e}}}}=\frac{1}{l_{i}^{2}}\left( {{m}_{1}}c_{1}^{2}+{{m}_{2}}{{\left( {{l}_{i}}-{{c}_{{{i}_{2}}}} \right) }^{2}} \right) \), \({{m}_{{{c}_{o}}}}={{m}_{{{c}_{e}}}}+\frac{1}{{{l}_{i}}}{{m}_{2}}\left( {{l}_{i}}-{{c}_{2}} \right) +\frac{1}{l_{i}^{2}}\left( {{I}_{c1}}+{{I}_{c2}} \right) \) and \({{m}_{{{g}_{e}}}}=\frac{1}{{{l}_{i}}}\left( {{m}_{1}}{{c}_{1}}+{{m}_{2}}\left( {{l}_{i}}-{{c}_{2}} \right) \right) \).

Appendix B: Definitions of symbol and their operation

Given \(\varvec{\omega } ={{[\begin{matrix} {{\omega }_{x}} &{} {{\omega }_{y}} &{} {{\omega }_{z}} \\ \end{matrix}]}^{T}}\) and \(\varvec{I}=\left[ \begin{matrix} {{I}_{11}} &{} {{I}_{12}} &{} {{I}_{13}} \\ {{I}_{12}} &{} {{I}_{22}} &{} {{I}_{23}} \\ {{I}_{13}} &{} {{I}_{23}} &{} {{I}_{33}} \\ \end{matrix} \right] \), define the following operations:

$$\begin{aligned} \tilde{\varvec{\omega }}=\left[ \begin{matrix} {{\omega }_{x}} &{} {{\omega }_{y}} &{} {{\omega }_{z}} &{} 0 &{} 0 &{} 0 \\ 0 &{} {{\omega }_{x}} &{} 0 &{} {{\omega }_{y}} &{} {{\omega }_{z}} &{} 0 \\ 0 &{} 0 &{} {{\omega }_{x}} &{} 0 &{} {{\omega }_{y}} &{} {{\omega }_{z}} \\ \end{matrix} \right] \text { }\\ and\text { }\tilde{\varvec{I}}={{[\begin{matrix} {{I}_{11}}&{{I}_{12}}&{{I}_{13}}&{{I}_{22}}&{{I}_{23}}&{{I}_{33}} \nonumber \end{matrix}]}^{T}}. \end{aligned}$$
(B3)

Then,

$$\begin{aligned} \left\{ \begin{array}{*{35}{l}} \varvec{\Gamma }_1=\dot{\varvec{w}}_i \times \hat{\varvec{s}}_i-\left\| \varvec{w}_i\right\| _2^2 \hat{\varvec{s}}_i \\ \varvec{\Gamma }_2=\varvec{g}-2 \dot{l}_i\left( \varvec{w}_i\times \hat{\varvec{s}}_i\right) -\ddot{l}_i \hat{\varvec{s}}_i-l_i \varvec{\Gamma }_1 \\ \varvec{\Gamma }_3=\tilde{\dot{\varvec{w}}}_{i \times } + \varvec{w}_{i \times } \tilde{\varvec{w}}_{i \times } \\ \end{array} \right. , \end{aligned}$$
(B4)

In addition, \(\varvec{R}_C^*\) is solved by \({ }^A \varvec{R}_B{ }^B \varvec{I}_C{ }^A \varvec{R}_B^T=\varvec{R}_C^* \tilde{\varvec{I}}_C; \)\(\varvec{R}_i^*\) is solved by \({ }^A \varvec{R}_B{ }^B \varvec{I}_i{ }^A \varvec{R}_B^T=\varvec{R}_i^* \tilde{\varvec{I}}_i\) .

Appendix C: Parameters of the excitation trajectories S1, S2 and S3 in the form of Fourier series

 

\(\varvec{a}_k^i(\times 10^3)\)

\(\varvec{a}_k^i(\times 10^3)\)

S1

\(\left[ \begin{array}{lllll} 1.049 &{} 2.066 &{} 0.692 &{} -4.296 &{} 0.489 \\ 0.036 &{} 1.226 &{} 0.831 &{} 5.863 &{} -7.956 \\ -2.24 &{} -9.559 &{} -11.217 &{} -59.974 &{} 82.99 \\ 2.426 &{} -19.403 &{} -72.245 &{} 57.12 &{} 32.102 \\ -7.298 &{} -3.106 &{} 12.935 &{} -4.227 &{} 1.695 \\ 3.444 &{} 5.996 &{} -45.783 &{} -11.371 &{} 47.714 \\ \end{array} \right] \)

\(\left[ \begin{array}{lllll} -0.692 &{} -0.899 &{} 1.847 &{} 7.196 &{} -6.366 \\ 0.091 &{} -1.017 &{} 5.756 &{} -9.873 &{} 4.833 \\ -13.363 &{} 37.725 &{} -19.755 &{} 13.317 &{} -11.218 \\ 3.521 &{} -25.813 &{} -13.796 &{} 115.605 &{} -74.585 \\ 10.032 &{} -30.001 &{} -66.107 &{} 189.692 &{} -102.095 \\ 3.924 &{} 4.425 &{} 36.202 &{} -148.313 &{} 94.375 \\ \end{array} \right] \)

S2

\(\left[ \begin{array}{lllll} -0.074 &{} -2.097 &{} 0.753 &{} -1.557 &{} 2.976 \\ -0.222 &{} 0.618 &{} -1.201 &{} 0.047 &{} 0.758 \\ 5.832 &{} -26.593 &{} 9.476 &{} -11.087 &{} 22.371 \\ 6.807 &{} -19.542 &{} -36.893 &{} -5.058 &{} 54.686 \\ 0.41 &{} -15.493 &{} -12.091 &{} 106.989 &{} -79.816 \\ 13.178 &{} -8.695 &{} -64.457 &{} 9.248 &{} 50.725 \\ \end{array} \right] \)

\(\left[ \begin{array}{lllll} 0.15 &{} 0.311 &{} 4.392 &{} -13.461 &{} 7.979 \\ -0.612 &{} 0.901 &{} -4.36 &{} 12.662 &{} -7.751 \\ 1.669 &{} 27.299 &{} -24.375 &{} -87.416 &{} 73.305 \\ -5.242 &{} 27.762 &{} -74.773 &{} 103.593 &{} -48.067 \\ 1.658 &{} -8.953 &{} 60.478 &{} -119.257 &{} 62.369 \\ -25.785 &{} 8.812 &{} 56.707 &{} 99.507 &{} -111.997 \\ \end{array} \right] \)

S3

\(\left[ \begin{array}{lllll} -0.48 &{} -0.015 &{} 1.199 &{} 3.046 &{} -3.751 \\ 0.198 &{} -0.05 &{} -0.264 &{} -0.62 &{} 0.737 \\ 4.808 &{} -2.556 &{} 1.539 &{} 63.499 &{} -67.29 \\ 9.172 &{} 6.122 &{} 15.399 &{} -57.871 &{} 27.179 \\ -0.433 &{} -21.87 &{} 6.677 &{} -18.875 &{} 34.501 \\ -1.737 &{} 12.78 &{} 34.642 &{} -1.949 &{} 43.737 \\ \end{array} \right] \)

\(\left[ \begin{array}{lllll} -0.616 &{} -1.048 &{} -1.318 &{} 14.589 &{} -10.338 \\ 1.173 &{} 0.587 &{} -1.265 &{} -12.252 &{} 10.091 \\ -10.006 &{} 28.794 &{} 6.641 &{} -43.38 &{} 21.203 \\ -22.781 &{} 10.125 &{} 21.725 &{} 144.248 &{} -127.927 \\ -0.783 &{} 22.299 &{} 24.338 &{} -153.4 &{} 99.354 \\ 1.861 &{} 58.009 &{} -57.457 &{} -154.361 &{} 134.387 \\ \end{array} \right] \)

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Xu, C., Liu, G., Li, C. et al. Dynamic modeling and identification of low impact docking mechanism based on symmetric excitation trajectory. Nonlinear Dyn 111, 9919–9937 (2023). https://doi.org/10.1007/s11071-023-08341-w

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