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RCCR Linkage Zijia Li (DK9) Johann Radon Institute for - - PowerPoint PPT Presentation
RCCR Linkage Zijia Li (DK9) Johann Radon Institute for - - PowerPoint PPT Presentation
RCCR Linkage Zijia Li (DK9) Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria Joint work with Josef Schicho Closed n-bar movable linkages Many n-bar (R, H, P) movable linkages introduced by Bennett, Bricard,
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Closed n-bar movable linkages
Many n-bar (R, H, P) movable linkages introduced by Bennett, Bricard, Goldberg, Baker, Wohlhart, Dietmaier, . . .
Figure: Open and closed linkages
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Closed n-bar movable linkages
Many n-bar (R, H, P) movable linkages introduced by Bennett, Bricard, Goldberg, Baker, Wohlhart, Dietmaier, . . .
Figure: Bricard 6R linkage
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Closed n-bar movable linkages
Many n-bar (R, H, P) movable linkages introduced by Bennett, Bricard, Goldberg, Baker, Wohlhart, Dietmaier, . . .
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Closed n-bar movable linkages
Many n-bar (R, H, P) movable linkages introduced by Bennett, Bricard, Goldberg, Baker, Wohlhart, Dietmaier, . . .
Problem
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Closed n-bar movable linkages
Many n-bar (R, H, P) movable linkages introduced by Bennett, Bricard, Goldberg, Baker, Wohlhart, Dietmaier, . . .
Problem
The classification of 6-bar linkages is still open.
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Why we focus on case n = 6?
Reasons
◮ n-bar linkages are always movable when n ≥ 7. ◮ n-bar linkages are always unmovable when n ≤ 3. ◮ The classification of movable n-bar linkages are solved when
n = 4, 5.
◮ Now we have over 30 kinds of 6-bar linkages. No one knows
the classification.
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RCCR linkage
As a C joint is equal to an R and a P joint, then RCCR linkage is equal to a special 6-bar of RRRRPP linkage.
Figure: www.youtube.com/watch?v = m0xG u63WH0
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RCCR linkage
As a C joint is equal to an R and a P joint, then RCCR linkage is equal to a special 6-bar of RRRRPP linkage.
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SE3 and DH
SE3
◮ Special Euclidean group SE3(R) is defined as the group of
all maps from R3 to itself preserving distance and orientation.
DH
◮ DH (dual quaternions): 8-dimensional real vector space
generated by 1, ǫ, i, j, k, ǫi, ǫj, ǫk.
◮ Study quadric S = {h ∈ DH| h¯
h ∈ R} and E = {h ∈ S| h¯ h = 0}.
◮ The complement S − E can be identified with SE3 by an
isomorphism : α : (S − E)/R∗ → SE3 .
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Closed 6-bar (R, P) linkages with mobility one
Let L = [h1, h2, h3, h4, h5, h6] denote a closed 6-bar linkages where h2
i = −1 or 0 for i = 1, . . . , 6.
Remark: The group parametrized by (t − hi)t∈P1(R) - the
parameter t determines the rotation angle or the translation distance- is the group of the (i + 1)-th link relative to the i-th link. Closure condition (t1 − h1)(t2 − h2)(t3 − h3)(t4 − h4)(t5 − h5)(t6 − h6) ∈ R\{0}
Definition: Configuration set
KL =
- (ti) ∈ (P1)6 | (ti) fulfilling the closure condition
- Mobility one means that the KL is a one dimensional set.
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Bonds theory for 6-bar (R,P) linkages
Bonds
Let KC ⊂ (P1
C)n be the Zariski closure of K. We set
B := {(t1, . . . , tn) ∈ KC | (t1 − h1)(t2 − h2) · · · (tn − hn) = 0}.
Remark 1: If K is a nonsingular curve, then we define a bond as a
point of B.
Remark 2: Let β be a bond (t1, . . . , tn), there exist indices
i, j ∈ [n], i < j, such that t2
i = −1 or 0, t2 j = −1 or 0 . If there
are exactly two coordinates of β with values ±I or 0, then we say that β connects joints i and j.
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Main lemmas and theorems
Lemma
Assume that ji is a P-joint, and ji+1 and ji+2 are R-joints. (a) The joints ji and ji+1 cannot be connected by a bond. (b) If the joints ji and ji+2 are connected by a bond, then the axes hi+1 and hi+2 are parallel.
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Main lemmas and theorems
Lemma
Assume that ji is a P-joint, and ji+1 and ji+2 are R-joints. (a) The joints ji and ji+1 cannot be connected by a bond. (b) If the joints ji and ji+2 are connected by a bond, then the axes hi+1 and hi+2 are parallel.
Theorem [Woldron 1974]
A RCCR linkage L is able to move with one degree of freedom iff (if and only if) the cylindrical (C) and revolute (R) joints of each pair are parallel.
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