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, Goldberg, Baker, Wohlhart, Dietmaier, . . .
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
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
Closed n-bar movable linkages Many n-bar (R, H, P) movable linkages introduced by Bennett, Bricard, Goldberg, Baker, Wohlhart, Dietmaier, . . .
Closed n-bar movable linkages Many n-bar (R, H, P) movable linkages introduced by Bennett, Bricard, Goldberg, Baker, Wohlhart, Dietmaier, . . . Problem
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 .
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.
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 = m 0 xG u 63 WH 0
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.
SE 3 and DH SE 3 ◮ Special Euclidean group SE 3 ( R ) is defined as the group of all maps from R 3 to itself preserving distance and orientation. DH ◮ DH ( dual quaternions ): 8-dimensional real vector space generated by 1 , ǫ, i , j , k , ǫ i , ǫ j , ǫ k . ◮ Study quadric S = { h ∈ DH | h ¯ h ∈ R } and E = { h ∈ S | h ¯ h = 0 } . ◮ The complement S − E can be identified with SE 3 by an isomorphism : α : ( S − E ) / R ∗ → SE 3 .
Closed 6-bar (R, P) linkages with mobility one Let L = [ h 1 , h 2 , h 3 , h 4 , h 5 , h 6 ] denote a closed 6-bar linkages where h 2 i = − 1 or 0 for i = 1 , . . . , 6. Remark : The group parametrized by ( t − h i ) t ∈ P 1 ( 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 ( t 1 − h 1 )( t 2 − h 2 )( t 3 − h 3 )( t 4 − h 4 )( t 5 − h 5 )( t 6 − h 6 ) ∈ R \{ 0 } Definition : Configuration set ( t i ) ∈ ( P 1 ) 6 | ( t i ) fulfilling the closure condition � � K L = Mobility one means that the K L is a one dimensional set.
Bonds theory for 6-bar (R,P) linkages Bonds C ) n be the Zariski closure of K . We set Let K C ⊂ ( P 1 B := { ( t 1 , . . . , t n ) ∈ K C | ( t 1 − h 1 )( t 2 − h 2 ) · · · ( t n − h n ) = 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 ( t 1 , . . . , t n ), there exist indices i , j ∈ [ n ], i < j , such that t 2 i = − 1 or 0, t 2 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 .
Main lemmas and theorems Lemma Assume that j i is a P-joint, and j i +1 and j i +2 are R-joints. (a) The joints j i and j i +1 cannot be connected by a bond. (b) If the joints j i and j i +2 are connected by a bond, then the axes h i +1 and h i +2 are parallel.
Main lemmas and theorems Lemma Assume that j i is a P-joint, and j i +1 and j i +2 are R-joints. (a) The joints j i and j i +1 cannot be connected by a bond. (b) If the joints j i and j i +2 are connected by a bond, then the axes h i +1 and h i +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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