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Special Configuration of Mechanisms

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Theory of Parallel Mechanisms

Part of the book series: Mechanisms and Machine Science ((Mechan. Machine Science,volume 6))

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Abstract

This chapter resumptively introduces our studies on the singularity of parallel mechanisms for the Stewart manipulator and the 3-RPS mechanism. It analyzes the singular kinematic principle and the singularity classification based on the kinematic status of the machinery and the linear-complex and focuses on discussing the structure and property of the singularity loci of 3/6- and 6/6-Stewart platform for special and general orientations. The singularity of the 3-RPS mechanism is also discussed in the latter parts. Many interesting properties, such as the remarkable intersection of all six segments of the six legs of the 6/6-Stewart platform with one common line, are discovered.

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Notes

  1. 1.

    We prefer to adopt the term “special configuration” instead of “singularity” because the latter originates from mathematics, whereas the former originates from mechanical engineering and its meaning in physics is clear and can easily be understood by mechanical students and engineers. However, the term “singularity” is also used in the paper a number of times to simplify the expression.

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Author information

Authors and Affiliations

  1. Robotics Research Center, Yanshan University, Qinhuangdao, Hebei, China

    Zhen Huang

  2. Mechatronic Department, Zhejiang Sci-Tech University, Hangzhou, China

    Qinchuan Li

  3. Robotics Research Center, Yanshan University, Qinhuangdao, China

    Huafeng Ding

Authors
  1. Zhen Huang
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  2. Qinchuan Li
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  3. Huafeng Ding
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Appendix A

Appendix A

To verify the correctness of the remarkable singularity occurring when six segments associated with the six extensible links of the 6/6-Gough-Stewart manipulator intersect one common line, the singularity point P 1 in Sect. 5.3.2.3 is taken as an example. The corresponding data cited from our calculations are given below, including the coordinates of the 12 vertices of the moving platform and the base platform, the Plücker coordinates of the six line vectors, the instantaneously moving reciprocal screw, and the corresponding reciprocal products.

$$ {{\hbox{C}}_1}:\left( {\begin{array}{llll} { - 1.41421356237309,} & { - 1.41421356237309,} & 0 \end{array}} \right) $$
$$ {{\hbox{C}}_2}:\left( {\begin{array}{llll} {1.41421356237309,} & { - 1.41421356237309,} & 0 \\\end{array} } \right) $$
$$ {{\hbox{C}}_3}:\left( {\begin{array}{llll} {1.93185165257814,} & { - 0.51763809020504,} & 0 \\\end{array} } \right) $$
$$ {{\hbox{C}}_4}:\left( {\begin{array}{llll} {0.51763809020504,} & {1.93185165257814,} & 0 \\\end{array} } \right) $$
$$ {{\hbox{C}}_5}:\left( {\begin{array}{llll} { - 0.51763809020504,} & {1.93185165257814,} & 0 \\\end{array} } \right) $$
$$ {{\hbox{C}}_6}:\left( {\begin{array}{llll} { - 1.93185165257814,} & { - 0.51763809020504,} & 0 \\\end{array} } \right) $$
$$ {{\hbox{B}}_1}:\left( {\begin{array}{llll} { - 2.09129075934452,} & { - 5.55407350116214,} & {1.44888873943360} \\\end{array} } \right) $$
$$ {{\hbox{B}}_2}:\left( {\begin{array}{llll} { - 1.75507495728150,} & { - 4.97173064968147,} & {1.06066017177982} \\\end{array} } \right) $$
$$ {{\hbox{B}}_3}:\left( {\begin{array}{llll} { - 2.88678588817939,} & { - 3.25767686730291,} & {0.53033008588991} \\\end{array} } \right) $$
$$ {{\hbox{B}}_{{4}}}:\left( {\begin{array}{llll} { - 3.63723664069157,} & { - 3.21263249098023,} & {0.72444436971680} \\\end{array} } \right) $$
$$ {{\hbox{B}}_5}:\left( {\begin{array}{llll} { - 4.55579529423526,} & { - 4.80362274864996,} & {1.78510454149662} \\\end{array} } \right) $$
$$ {{\hbox{B}}_6}:\left( {\begin{array}{llll} { - 4.14156034378610,} & { - 5.43100997645331,} & {1.97921882532351} \\\end{array} } \right) $$
$$ {{\$}_1} = {\user2{C}_1}{\user2{B}_1} = \left( \begin{array}{lllllllll} { - 0.15256,} & { - 0.93281,} & 0.32647;\quad {0.46170,} & {0.461699,} & {1.10344} \end{array} \right) $$
$$ {{\$}_2} = {\user2{C}_2}{\user2{B}_2} = \left( {\begin{array}{llll} { - 0.64930,} & { - 0.72883,} & {0.21730;\quad \begin{array}{llll} { - 0.30731,} & { - 0.30730,} & { - 1.94896} \\\end{array} } \\\end{array} } \!\right) $$
$$ {{\$}_3} = {\user2{C}_3}{\user2{B}_3} = \left( {\begin{array}{llll} { - 0.86533,} & { - 0.49206,} & {0.09524;\quad \begin{array}{llll} { - 0.04930,} & { - 0.18398,} & { - 1.39852} \\\end{array} } \\\end{array} } \right) $$
$$ {{\$}_4} = {\user2{C}_4}{\user2{B}_4} = \left( {\begin{array}{llll} { - 0.62457,} & { - 0.77333,} & {0.10890;\quad \begin{array}{llll} {0.21038,} & { - 0.05637,} & {0.80628} \\\end{array} } \\\end{array} } \!\!\right) $$
$$ {{\$}_5} = {\user2{C}_5}{\user2{B}_5} = \left( {\begin{array}{llll} { - 0.50141,} & { - 0.83633,} & {0.22165;\quad \begin{array}{llll} {0.42820,} & {0.11474,} & {1.40157} \\\end{array} } \\\end{array} } \!\!\right) $$
$$ {{\$}_6} = {\user2{C}_6}{\user2{B}_6} = \left( {\begin{array}{llll} { - 0.38500,} & { - 0.85607,} & {0.34484;\quad \begin{array}{llll} {0.17850,} & {0.66619,} & {1.45451} \\\end{array} } \\\end{array} } \!\!\right) $$
$$ {{\$}^m} = \left( {\begin{array}{llll} {0.48296,} & {0.83652,} & { - 0.25882;\quad \begin{array}{llll} {0.50000,} & { - 0.86603,} & { - 1.86603} \!\!\\\end{array} } \\\end{array} } \right) $$
$$ p1 = {{\$}_1} \circ {{\$}^{\user2{m}}} = 0.06939 \times {10^{{ - 16}}} $$
$$ p2 = {{\$}_2} \circ {{\$}^{\user2{m}}} = - 0.13878 \times {10^{{ - 16}}} $$
$$ p3 = {{\$}_3} \circ {{\$}^{\user2{m}}} = 1.939478 \times {10^{{ - 16}}} $$
$$ p4 = {{\$}_4} \circ {{\$}^{\user2{m}}} = - 0.06939 \times {10^{{ - 16}}} $$
$$ p5 = {{\$}_5} \circ {{\$}^{\user2{m}}} = - 0.13878 \times {10^{{ - 16}}} $$
$$ p6 = {{\$}_6} \circ {{\$}^{\user2{m}}} = 0.27756 \times {10^{{ - 16}}} $$

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Huang, Z., Li, Q., Ding, H. (2013). Special Configuration of Mechanisms. In: Theory of Parallel Mechanisms. Mechanisms and Machine Science, vol 6. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4201-7_7

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  • DOI: https://doi.org/10.1007/978-94-007-4201-7_7

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