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Trajectory planning of parallel kinematic manipulators for the maximum dynamic load-carrying capacity

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Abstract

The objective of this paper is to identify the trajectory that accomplishes the assigned motion with the maximum dynamic load-carrying capacity (DLCC) which is subject to constraints imposed by the kinematics and dynamics of a manipulator structure. In this study, the possible trajectories of the manipulator are modeled using a parametric path representation, the optimal trajectory is then obtained using a two-loop of optimization process, in which the inner-loop optimization process based on the Simplex-type linear programming method is used to determine the dynamic loading at each discrete point along the presumed trajectory and then to formulate the DLCC; the outer-loop optimization process based on the particle swarm optimization algorithm is to solve the controlled points by maximizing the formulated DLCC. The numerical results confirm the feasibility of the optimized trajectories and demonstrate the effectiveness of the proposed algorithm for the maximum dynamic load-carrying trajectory planning of a parallel kinematic manipulator.

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Acknowledgments

This work was supported by the National Science Council of the ROC under the Grant NSC 101-2221-E-003-002.

Author information

Authors and Affiliations

  1. Department of Mechatronic Engineering, National Taiwan Normal University, 162, Section 1, He-ping East Road, Taipei, 10610, Taiwan

    Chun-Ta Chen

  2. Department of Mechanical Engineering, Far East University, No. 49, Chung Hua Road, Hsin-Shih, Tainan, 744, Taiwan

    Te-Tan Liao

Authors
  1. Chun-Ta Chen
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  2. Te-Tan Liao
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Corresponding author

Correspondence to Chun-Ta Chen.

Appendix

Appendix

Inertia matrix, Coriolis and centrifugal terms, gravity vector

$${\varvec{M}}_{\mathbf{1}} = \left[ {\begin{array}{*{20}c} {m_{B} \varvec{I}_{{\varvec{3} \times \varvec{3}}} } & {{}_{B}^{N} {\varvec{R}}\tilde{{\varvec{mc}_{\varvec{B}} }}^{T} } \\ {\tilde{{\varvec{mc}_{\varvec{B}} }}{}_{B}^{N} {\varvec{R}}^{T} } & {\varvec{I}_{\varvec{B}} } \\ \end{array} } \right],$$
$$\varvec{M}_{2} = \left[ {\begin{array}{*{20}c} {{\mathbf{diag}}\left[ {m_{12} , \ldots ,m_{62} } \right]} & {{\mathbf{diag}}\left[ {m_{12} \varvec{s}_{{\mathbf{1}}}^{T} \tilde{\varvec{d}}_{{{\mathbf{12}}}}^{T} , \ldots ,m_{62} \varvec{s}_{6}^{T} \tilde{\varvec{d}}_{{{\mathbf{62}}}}^{T} } \right]} \\ {{\mathbf{diag}}\left[ {m_{12} \tilde{\varvec{d}}_{{{\mathbf{12}}}} \varvec{s}_{\varvec{1}} , \ldots ,m_{62} \tilde{\varvec{d}}_{{{\mathbf{62}}}} \varvec{s}_{\varvec{6}} } \right]} & {{\mathbf{diag}}\left[ {\varvec{I}_{{\mathbf{1}}} , \ldots ,\varvec{I}_{{\mathbf{6}}} } \right]} \\ \end{array} } \right]$$

and \({\varvec{I}_{\varvec{i}}} ={\varvec{I}_{\varvec{i}{\mathbf{1}}}} + {\varvec{I}_{\varvec{i}{\mathbf{2}}}} - m_{{i2}} l_{{\varvec{i}{1}}}^{2} \tilde{\varvec{s}_{i}}^{{2}} + m_{i2} l_{i1} \left( {\tilde{\varvec{s}}_{\varvec{i}}^{T} \tilde{{\varvec{d}}}_{\varvec{i}\mathbf{2}} + \tilde{{\varvec{d}}}_{\varvec{i}\mathbf{2}}^{T} \tilde{\varvec{s}}_{\varvec{i}} } \right),\quad i = 1, \ldots ,6,\)

$${\varvec{D}}_{\mathbf{1}} = \left[ {\begin{array}{*{20}c} \varvec{0} & {{}_{B}^{N} {\varvec{R}}\tilde{{\varvec{\omega }_{\varvec{B}} }}\tilde{{\varvec{mc}_{\varvec{B}} }}^{T} } \\ \varvec{0} & {(\varvec{I}_{\varvec{B}} \otimes \varvec{\omega }_{\varvec{B}} )^{T} } \\ \end{array} } \right],$$
$$\varvec{D}_{{\mathbf{2}}} = \left[ {\begin{array}{*{20}c} 0 & {{\mathbf{diag}}\left[ {m_{12} \varvec{\omega}_{{\mathbf{1}}}^{T} \tilde{\varvec{s}}_{{\mathbf{1}}} (l_{11} \tilde{\varvec{s}}_{{\mathbf{1}}} + \tilde{\varvec{d}}_{{{\mathbf{12}}}} ), \ldots ,m_{62} \varvec{\omega}_{{\mathbf{6}}}^{T} \tilde{\varvec{s}}_{{\mathbf{6}}} (l_{61} \tilde{\varvec{s}}_{{\mathbf{6}}} + \tilde{\varvec{d}}_{{{\mathbf{62}}}} )} \right]} \\ {{\mathbf{diag}}\left[ {m_{12} (l_{11} \tilde{\varvec{s}}_{{\mathbf{1}}}^{T} + \tilde{\varvec{d}}_{{{\mathbf{12}}}}^{T} )\tilde{\varvec{s}}_{{\mathbf{1}}}\varvec{\omega}_{{\mathbf{1}}}^{{}} , \ldots ,m_{62} (l_{61} \tilde{\varvec{s}}_{6}^{T} + \tilde{\varvec{d}}_{{{\mathbf{62}}}}^{T} )\tilde{\varvec{s}}_{{\mathbf{6}}} \varvec{\omega}_{{\mathbf{6}}}^{{}} } \right]} & {{\mathbf{diag}}\left[ {\varvec{h}_{{\mathbf{1}}} , \ldots ,\varvec{h}_{{\mathbf{6}}} } \right]} \\ \end{array} } \right],$$

where \(\varvec{h}_{\varvec{i}} = {m}_{{{i}2}} {\dot{l}}_{{{i}1}} ({l}_{{{i}1}} \tilde{\varvec{s}}_{\varvec{i}}^{T} + \tilde{{\varvec{d}}}_{{\varvec{i}2}}^{T} )\tilde{\varvec{s}}_{\varvec{i}}\,+((\varvec{I}_{{\varvec{i}\varvec{1}}} + \varvec{I}_{{\varvec{i}\varvec{2}}} ) \otimes \varvec{\omega }_{\varvec{i}} )^{T} + {m}_{{{i}2}} {l}_{{{i}1}} (\varvec{\omega }_{\varvec{i}}^{T} {\varvec{d}}_{{\varvec{i}\varvec{2}}} )\tilde{\varvec{s}}_{\varvec{i}}^{T} ,\quad i = 1, \ldots, 6\)

$$\varvec{G}_{\varvec{1}} = \left[ {\begin{array}{*{20}c} { - m_{B} \varvec{g}} \\ { - {{\varvec{mc}_{\varvec{B}} }}{}_{B}^{N} {\varvec{R}}^{T} \varvec{g}} \\ \end{array} } \right],$$
$$\varvec{G}_{\mathbf{2}} = \left[ {\begin{array}{*{20}c}{\left[{\begin{array}{*{20}c} { -m_{{1{\mathbf{2}}}}\varvec{g}^{T}{}_{\varvec{1}}^{N}\varvec{Rs}_{\varvec{1}} } &\ldots& { -m_{{6{\mathbf{2}}}}\varvec{g}^{T}{}_{6}^{N} \varvec{Rs}_{\varvec{6}}} \\\end{array} } \right]^{{\text{T}}} } \\ {\left[{\begin{array}{*{20}c} {- \varvec{g}^{T}{}_{1}^{N} {\varvec{R}}(m_{{1{\mathbf{1}}}}\tilde{{\varvec{d}}}_{{\varvec{11}}} + m_{{1{\mathbf{2}}}}l_{{1{\mathbf{1}}}} \tilde{\varvec{s}}_{\varvec{i}} +m_{{1{\mathbf{2}}}} \tilde{{\varvec{d}}}_{{\varvec{12}}} )^{T} }& \ldots & { -\varvec{g}^{\text{T}}{}_{\varvec{6}}^{N} {\varvec{R}}(m_{{6{\mathbf{1}}}}\tilde{{\varvec{d}}}_{{\varvec{61}}} + m_{{6{\mathbf{2}}}}l_{{6{\mathbf{1}}}} \tilde{\varvec{s}}_{\varvec{i}} +m_{{6{\mathbf{2}}}} \tilde{{\varvec{d}}}_{{\varvec{62}}} )^{T} } \\\end{array} } \right]^{T} } \\ \end{array} } \right].$$

The actuating torques: τ = [τ 1 … τ 6].

It is noted that the symbol “⊗ ” in D 1 and h i denotes a skew-symmetric matrix formed from the matrix–vector product. The skew-symmetric matrix has the form like Eq. (5).

figure a

L = diag[p 1/2π … p 6/2π].

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Chen, CT., Liao, TT. Trajectory planning of parallel kinematic manipulators for the maximum dynamic load-carrying capacity. Meccanica 51, 1653–1674 (2016). https://doi.org/10.1007/s11012-015-0308-8

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