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Types of self-motions of planar Stewart Gough platforms

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Abstract

We show that the self-motions of general planar Stewart Gough platforms can be characterized in the complex extension of the Euclidean 3-space by the movement of three platform points in planes orthogonal to the planar base (3-point Darboux motion) and a simultaneous sliding of three planes orthogonal to the planar platform through points of the base (3-plane Mannheim motion). Based on this consideration, we prove that all one-parametric self-motions of a general planar Stewart Gough platform can be classified into two types (type I DM and type II DM, where DM abbreviates Darboux Mannheim). We also succeed in presenting a set of 24 equations yielding a type II DM self-motion that can be computed explicitly and that is of great simplicity seen in the context of self-motions. These 24 conditions are the key for the complete classification of general planar Stewart Gough platforms with type II DM self-motions, which is an important step in solving the famous Borel Bricard problem.

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Notes

  1. Until now, a complete list of the special cases is missing. For known non-trivial special cases see [7].

  2. This example is initialized in [15] by the following coordinates of the first four pairs of anchor points: A 1=B 1=B 2=a 1=b 1=b 2=0, a 2=b 3=2, A 2=a 3=b 4=3, A 3=a 4=1, B 3=5, A 4=−21155/1872 and B 4=165/8.

  3. Note that the numbering of types is done with Roman numerals.

  4. Trivially, this sentence also holds for the excluded case, where all platform or base anchor points are collinear.

  5. Note that e 0=0 is preserved by translations of the reference frames.

  6. After plugging the solutions for the f i ’s of the linear system Ψ,Δ2,Δ3,Δ4 into Δ5 and Δ6, these two equations are fulfilled identically under consideration of e 0=0.

  7. The solution i=3 yields the same result, just for another indexing.

  8. A recent publication of Selig and Husty [29] can also be used for reasoning the correctness of Theorem 3.

  9. This is the reason why the congruent cubics C and c are even given by the same equation.

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Acknowledgements

This research is supported by Grant No. I 408-N13 of the Austrian Science Fund FWF within the project “Flexible polyhedra and frameworks in different spaces”, an international cooperation between FWF and RFBR, the Russian Foundation for Basic Research.

Moreover, the author would like to thank the reviewers for their useful comments and suggestions which have helped to improve the quality of the paper.

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Authors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstrasse 8-10/104, Vienna, 1040, Austria

    Georg Nawratil

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  1. Georg Nawratil
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Correspondence to Georg Nawratil.

Appendix

Appendix

In the following, we list the items 3,4,5,6,7,8,10 of Theorem 3 given by Karger [13]:

  1. 3.

    m 1,…,m 5 and M 1,…,M 5 are collinear,

  2. 4.

    m 1=m 2=m 3, m 1,…,m 5 are collinear, M 4=M 5,

  3. 5.

    m 1=m 2=m 3=m 4,

  4. 6.

    m 1=m 2=m 3, M 1,M 2,M 3 are collinear,

  5. 7.

    M 1,M 2,M 3 and M 3,M 4,M 5 are collinear, m 1=m 2, m 4=m 5, m 1,…,m 5 are collinear,

  6. 8.

    m 1,…,m 4 and M 1,…,M 4 are collinear and

    (16)
  7. 10.

    Points m 1,…,m 5 are collinear and pairwise distinct, points M 1,…,M 5 are coplanar (no three of M 1,…,M 5 are collinear) and two equations remain,

    $$ \begin{array}{@{}lll} B_3B_4(a_4-a_3)(a_5A_2-A_5a_2) \cr\noalign{\vspace{3pt}} \quad {}+B_3B_5(a_3-a_5)(a_4A_2-A_4a_2) \cr\noalign{\vspace{3pt}} \quad {}+B_4B_5(a_5-a_4)(a_3A_2-A_3a_2)=0, \cr\noalign{\vspace{3pt}} B_3B_4A_5(a_4-a_3)(a_5-a_2) \cr\noalign{\vspace{3pt}} \quad {}+B_3B_5A_4(a_4-a_2)(a_3-a_5) \cr\noalign{\vspace{3pt}} \quad {}+B_4B_5A_3(a_3-a_2)(a_5-a_4)=0. \end{array} $$
    (17)

    If m 1=m 2, points M 3,M 4,M 5 must be collinear and (17) yield one equation, a degenerated case.

Note that the formulas (16) and (17) are given with respect to the coordinate systems with A 1=B 1=B 2=a 1=b 1=b 2=0.

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Nawratil, G. Types of self-motions of planar Stewart Gough platforms. Meccanica 48, 1177–1190 (2013). https://doi.org/10.1007/s11012-012-9659-6

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