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Closed-form dynamic modeling and performance analysis of an over-constrained 2PUR-PSR parallel manipulator with parasitic motions

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Abstract

This paper presents an systematic dynamic modeling and performance analysis method of an over-constrained 2PUR-PSR parallel manipulator with parasitic motions, where P, U, S, and R represent the prismatic joint, universal joint, spherical joint, and revolute joint, respectively. The process of the deduction of the over-constrained forces/moment is given. The Newton–Euler approach and natural orthogonal complement method are adopted to establish two types of dynamic models with and without constrained forces/moments with the consideration of the over-constrained forces/moments. The dynamic manipulability ellipsoid, which measures the uniformity of changing the position and orientation of the manipulator’s moving platform, is adopted to evaluate the dynamic performance of the parallel manipulator. To show the feasibility of the proposed method, numerical simulations are conducted to investigate the dynamic models and performance of the 2PUR-PSR manipulator. The actuation forces, constrained forces/moments, and the compatible deformation are calculated. Distribution of the DME index is also obtained. The proposed modeling approach provides a fundamental basis for the structural optimization and control scheme design of the over-constrained parallel manipulator.

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Acknowledgements

This research was supported in part by the Natural Science Foundation of China under Grant No. 51525504, the Natural Science Foundation of Zhejiang under Grant No. LY18E050019, the Excellent Talent Cultivation Foundation under Grant No. ZSTUME02B09, and the Starting Foundation of Zhejiang Sci-Tech University under Grant No. 17022051-Y.

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Authors and Affiliations

  1. Mechatronic Institute, Zhejiang Sci-Tech University, Hangzhou, 310018, Zhejiang Province, China

    Zhengsheng Chen, Lingming Xu, Weizhong Zhang & Qinchuan Li

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  1. Zhengsheng Chen
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  2. Lingming Xu
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  3. Weizhong Zhang
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  4. Qinchuan Li
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Correspondence to Qinchuan Li.

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Appendix

Appendix

$$\begin{aligned}&{{\varvec{C}}}_\lambda ^O =\left( {{\begin{array}{c@{\quad }c} {\frac{{l}_3^3 {{\varvec{e}}}_3^{{\varvec{T}}} {{\varvec{R}}}_{{L,}1}^{-1} {{\varvec{L}}}_{{B}_{{1,T}} }^{{\varvec{c}}} }{3}+\frac{{l}_3^2 {{\varvec{e}}}_1^{{\varvec{T}}} {{\varvec{R}}}_{{L,}1}^{-1} {{\varvec{L}}}_{{B}_{1{,{{R}}}} }^{{\varvec{c}}} }{2}}&{} {-\frac{{l}_3^3 {{\varvec{e}}}_3^{{\varvec{T}}} {{\varvec{R}}}_{{L,}2}^{-1} {{\varvec{L}}}_{{B}_{2{,T}} }^{{\varvec{c}}} }{3}-\frac{{l}_3^2 {{\varvec{e}}}_1^{{\varvec{T}}} {{\varvec{R}}}_{{L,}2}^{-1} {{\varvec{L}}}_{{B}_{{2,{{R}}}} }^{{\varvec{c}}} }{2}} \\ {{l}_3 {{\varvec{e}}}_4^T {{\varvec{R}}}_{{L,1}}^{-1} {{\varvec{L}}}_{{B}_{{1,{{R}}}} }^{{\varvec{c}}} }&{} {{-l}_3 {{\varvec{e}}}_4^T {{\varvec{R}}}_{{L,2}}^{-1} {{\varvec{L}}}_{{B}_{{2,{{R}}}} }^{{\varvec{c}}} } \\ \end{array} }} \right) \\&{{\varvec{C}}}_{M}^{O} =\left( {{\begin{array}{c@{\quad }c} {\left[ {\frac{\left( {{l}_3 -{r}_{{L,}2,1} } \right) ^{3}}{3}+\frac{{r}_{{L,}2,1} \left( {{l}_3 -{r}_{{L,}2,1} } \right) ^{2}}{2}} \right] {{\varvec{e}}}_3^{{\varvec{T}}} {{\varvec{R}}}_{{L,}1}^{-1} {m}_{{L,}1} }&{} {\mathbf{0}_{1\times 3} } \\ {\mathbf{0}_{1\times 3} }&{} {{{\varvec{e}}}_4^{{\varvec{T}}} {{\varvec{R}}}_{{L,}1}^{-1} {{\varvec{R}}}_{{L,}1} {{\varvec{I}}}_{{L,}1}^{l} {{\varvec{R}}}_{{L,}1}^{T} } \\ \end{array} }} \right. \\&\quad \left. {{\begin{array}{c@{\quad }c} {-\left[ {\frac{\left( {{l}_3 -{r}_{{L,}2,2} } \right) ^{3}}{3}+\frac{{r}_{{L,}2,2} \left( {{l}_3 -{r}_{{L,}2,2} } \right) ^{2}}{2}} \right] {{\varvec{e}}}_3^{{\varvec{T}}} {{\varvec{R}}}_{{L,2}}^{-1} {m}_{{L,2}} }&{} {\mathbf{0}_{1\times 3} } \\ {\mathbf{0}_{1\times 3} }&{} {-{{\varvec{e}}}_4^{{\varvec{T}}} {{\varvec{R}}}_{{L,}2}^{-1} {{\varvec{R}}}_{{L,}2} {{\varvec{I}}}_{{L,}2}^{l} {{\varvec{R}}}_{{L,}2}^{T} } \\ \end{array} }} \right) \\&{{\varvec{C}}}_G^{{\varvec{O}}} =\left( {\begin{array}{c} \left[ {\frac{\left( {{l}_3 -{r}_{{L}_{2,1} } } \right) ^{3}}{3}+\frac{{r}_{{L}_{2,1} } \left( {{l}_3 -{r}_{{L}_{2,1} } } \right) ^{2}}{2}} \right] e_3^{{\varvec{T}}} {{\varvec{R}}}_{{L,}1}^{-1} {m}_{{L,}1} -\left[ {\frac{\left( {{l}_3 -{r}_{{L}_{2,2} } } \right) ^{3}}{3}+\frac{{r}_{{L}_{2,2} } \left( {{l}_3 -{r}_{{L}_{2,2} } } \right) ^{2}}{2}} \right] {{\varvec{e}}}_3^{{\varvec{T}}} {{\varvec{R}}}_{{L,}2}^{-1} {m}_{{L,}2} \\ \mathbf{0}_{1\times 3} \\ \end{array}} \right) \\&{{\varvec{C}}}_W^{{\varvec{O}}} =\left( {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0&{} 0&{} 0&{} 0 \\ 0&{} {{{\varvec{e}}}_4^{{\varvec{T}}} {{\varvec{R}}}_{{L,}1}^{-1} {\tilde{{\varvec{\omega }}}}_{{L,}1} {{\varvec{R}}}_{{L,}1} {{\varvec{I}}}_{{L,}1}^{l} {{\varvec{R}}}_{{L,}1}^{T} }&{} 0&{} {-{{\varvec{e}}}_4^{{\varvec{T}}} {{\varvec{R}}}_{{L,}2}^{-1} {\tilde{{\varvec{\omega }}}}_{{L,}2} {{\varvec{R}}}_{{L,}2} {{\varvec{I}}}_{{L,}2}^{l} {{\varvec{R}}}_{{L,}2}^{T} } \\ \end{array} }} \right) \end{aligned}$$
$$\begin{aligned}&{{\varvec{C}}}_{\lambda ,1}^{{\varvec{O}}} =\left[ {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} {{{\varvec{C}}}_{\lambda ,1}^{{\varvec{O}}} \left( {:,1} \right) }&{} {{{\varvec{C}}}_{\lambda ,1}^{{\varvec{O}}} \left( {:,5} \right) } \\ \end{array} }} \right] \\&{{\varvec{C}}}_{\lambda ,2}^{{\varvec{O}}} =\left[ {{\begin{array}{l@{\quad }l} {{{\varvec{C}}}_{\lambda ,1}^{{\varvec{O}}} \left( {:,2:4} \right) }&{} {{{\varvec{C}}}_{\lambda ,1}^{{\varvec{O}}} \left( {:,6:10} \right) } \\ \end{array} }} \right] \\&{{\varvec{C}}}_{\lambda ,1,2}^{{\varvec{O}}} =-\left( {{{\varvec{C}}}_{\lambda ,1}^{{\varvec{O}}} } \right) ^{-1}{{\varvec{C}}}_{\lambda ,2}^{{\varvec{O}}} , \quad {{\varvec{C}}}_{{M},1,2}^{O} =\left( {{{\varvec{C}}}_{\lambda ,1}^{O} } \right) ^{-1}{{\varvec{C}}}_{M}^{O} , {{\varvec{C}}}_{{W},1,2}^{O}\\&\quad =\left( {{{\varvec{C}}}_{\lambda ,1}^{O} } \right) ^{-1}{{\varvec{C}}}_{W}^{O} , {{\varvec{C}}}_{{G},1,2}^{O} =\left( {{{\varvec{C}}}_{\lambda ,1}^{O} } \right) ^{-1}{{\varvec{C}}}_{G}^{O}\\&\Delta {{\varvec{G}}}=\left( {{\begin{array}{l@{\quad }l} {{{\varvec{C}}}_\lambda }&{} {{{\varvec{w}}}_{{CS}}^{a} } \\ \end{array} }} \right) \left( {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l@{\quad }l} {\mathbf{0}_{9\times 3} }&{} {{{\varvec{C}}}_{{G},1,2}^{{{\varvec{o}}}\mathrm{T}} \left( {:,1} \right) }&{} {\mathbf{0}_3 }&{} {{{\varvec{C}}}_{{G},1,2}^{{{\varvec{o}}}\mathrm{T}} \left( {:,2} \right) }&{} {\mathbf{0}_{3\times 30} } \\ \end{array} }} \right) ^{\mathrm{T}}\\&\Delta {{\varvec{M}}}=\left( {{\begin{array}{l@{\quad }l} {{{\varvec{C}}}_\lambda }&{} {{{\varvec{w}}}_{{CS}}^{a} } \\ \end{array} }} \right) \\&\left( {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {\mathbf{0}_{9\times 6} }&{} {\mathbf{0}_{9\times 6} }&{} {\mathbf{0}_{9\times 6} }&{} {\mathbf{0}_{9\times 6} }&{} {\mathbf{0}_{9\times 18} } \\ {\mathbf{0}_{1\times 6} }&{} {{{\varvec{C}}}_{{M},1,2}^{{\varvec{O}}} \left( {1,1:6} \right) }&{} {\mathbf{0}_{1\times 6} }&{} {{{\varvec{C}}}_{{M},1,2}^{{\varvec{O}}} \left( {1,7:12} \right) }&{} {\mathbf{0}_{1\times 18} } \\ {\mathbf{0}_{3\times 6} }&{} {\mathbf{0}_{3\times 6} }&{} {\mathbf{0}_{3\times 6} }&{} {\mathbf{0}_{3\times 6} }&{} {\mathbf{0}_{3\times 18} } \\ {\mathbf{0}_{1\times 6} }&{} {{{\varvec{C}}}_{{M},1,2}^{{\varvec{O}}} \left( {2,1:6} \right) }&{} {\mathbf{0}_{1\times 6} }&{} {{{\varvec{C}}}_{{M},1,2}^{{\varvec{O}}} \left( {2,7:12} \right) }&{} {\mathbf{0}_{1\times 18} } \\ {\mathbf{0}_{30\times 6} }&{} {\mathbf{0}_{30\times 6} }&{} {\mathbf{0}_{30\times 6} }&{} {\mathbf{0}_{30\times 6} }&{} {\mathbf{0}_{30\times 18} } \\ \end{array} }} \right) ,\\&\Delta {{\varvec{C}}}_\lambda = \left( {{\begin{array}{l@{\quad }l} {{{\varvec{C}}}_\lambda }&{} {{{\varvec{w}}}_{{CS}}^{a} } \\ \end{array} }} \right) \\&\quad \left( {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {E_9 }&{} {\mathbf{0}_{9\times 3} }&{} {\mathbf{0}_{9\times 9} }&{} {\mathbf{0}_{9\times 5} }&{} {\mathbf{0}_{9\times 16} } \\ {\mathbf{0}_{1\times 9} }&{} {{{\varvec{C}}}_{\lambda ,1,2}^{{\varvec{O}}} \left( {1,1:3} \right) }&{} {\mathbf{0}_{1\times 9} }&{} {{{\varvec{C}}}_{\lambda ,1,2}^{{\varvec{O}}} \left( {1,4:8} \right) }&{} {\mathbf{0}_{1\times 16} } \\ {\mathbf{0}_{3\times 9} }&{} {E_3 }&{} {\mathbf{0}_{3\times 9} }&{} {\mathbf{0}_{3\times 5} }&{} {\mathbf{0}_{3\times 16} } \\ {\mathbf{0}_{1\times 9} }&{} {{{\varvec{C}}}_{\lambda ,1,2}^{{\varvec{O}}} \left( {2,1:3} \right) }&{} {\mathbf{0}_{1\times 9} }&{} {{{\varvec{C}}}_{\lambda ,1,2}^{{\varvec{O}}} \left( {2,4:8} \right) }&{} {\mathbf{0}_{1\times 16} } \\ {\mathbf{0}_{30\times 2} }&{} {\mathbf{0}_{30\times 3} }&{} {\mathbf{0}_{30\times 2} }&{} {\mathbf{0}_{30\times 5} }&{} {{{\varvec{E}}}_{30} } \\ \end{array} }} \right) \,,\\&\Delta {{\varvec{W}}} =\left( {{\begin{array}{l@{\quad }l} {{{\varvec{C}}}_\lambda }&{} {{{\varvec{w}}}_{{CS}}^{a} } \\ \end{array} }} \right) \\&\quad \left( {{\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} {\mathbf{0}_{9\times 6} }&{} {\mathbf{0}_{9\times 6} }&{} {\mathbf{0}_{9\times 6} }&{} {\mathbf{0}_{9\times 6} }&{} {\mathbf{0}_{9\times 18} } \\ {\mathbf{0}_{1\times 6} }&{} {{{\varvec{C}}}_{{W},1,2}^{{\varvec{O}}} \left( {1,1:6} \right) }&{} {\mathbf{0}_{1\times 6} }&{} {{{\varvec{C}}}_{{W},1,2}^{{\varvec{O}}} \left( {1,7:12} \right) }&{} {\mathbf{0}_{1\times 18} } \\ {\mathbf{0}_{3\times 6} }&{} {\mathbf{0}_{3\times 6} }&{} {\mathbf{0}_{3\times 6} }&{} {\mathbf{0}_{3\times 6} }&{} {\mathbf{0}_{3\times 18} } \\ {\mathbf{0}_{1\times 6} }&{} {{{\varvec{C}}}_{{W},1,2}^{{\varvec{O}}} \left( {2,1:6} \right) }&{} {\mathbf{0}_{1\times 6} }&{} {{{\varvec{C}}}_{{W},1,2}^{{\varvec{O}}} \left( {2,7:12} \right) }&{} {\mathbf{0}_{1\times 18} } \\ {\mathbf{0}_{30\times 6} }&{} {\mathbf{0}_{30\times 6} }&{} {\mathbf{0}_{30\times 6} }&{} {\mathbf{0}_{30\times 6} }&{} {\mathbf{0}_{30\times 18} } \\ \end{array} }} \right) \,, \end{aligned}$$

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Chen, Z., Xu, L., Zhang, W. et al. Closed-form dynamic modeling and performance analysis of an over-constrained 2PUR-PSR parallel manipulator with parasitic motions. Nonlinear Dyn 96, 517–534 (2019). https://doi.org/10.1007/s11071-019-04803-2

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