Chapter 9 Parallel Manipulators Introduction Configuration Space and Singularities Singularity Classification Chapter 9 Parallel Manipulators 1 Lecture Notes for A Geometrical Introduction to Robotics and Manipulation Richard Murray and Zexiang Li and Shankar S. Sastry CRC Press Zexiang Li 1 and Yuanqing Wu 1 1 ECE, Hong Kong University of Science & Technology July 29, 2010
2nd International Summer School on Geometric Methods in Robotics, Mechanism Design and Manufacturing Research-Lecture 09 Parallel Manipulators
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Chapter9 ParallelManipulators
Introduction
ConfigurationSpace andSingularities
SingularityClassification
Chapter 9 Parallel Manipulators
1
Lecture Notes for
A Geometrical Introduction to
Robotics and Manipulation
Richard Murray and Zexiang Li and Shankar S. Sastry
CRC Press
Zexiang Li1 and Yuanqing Wu1
1ECE, Hong Kong University of Science & Technology
July 29, 2010
Chapter9 ParallelManipulators
Introduction
ConfigurationSpace andSingularities
SingularityClassification
Chapter 9 Parallel Manipulators
2
Chapter 9 Parallel Manipulators
1 Introduction
2 Con�guration Space and Singularities
3 Singularity Classi�cation
Chapter9 ParallelManipulators
Introduction
ConfigurationSpace andSingularities
SingularityClassification
9.1 IntroductionChapter 9 Parallel Manipulators
3
◻ Samples of parallel manipulators:1-DoF:
2-DoF:
3-DoF:
Chapter9 ParallelManipulators
Introduction
ConfigurationSpace andSingularities
SingularityClassification
9.1 IntroductionChapter 9 Parallel Manipulators
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◻ Samples of parallel manipulators:4-DoF:
5-DoF:
6-DoF:
Chapter9 ParallelManipulators
Introduction
ConfigurationSpace andSingularities
SingularityClassification
9.2 Configuration Space and SingularitiesChapter 9 Parallel Manipulators
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k limbs, with SE(2) as task space.Limb i:
θ i = (θ i1 , . . . , θ ini) ∈ Ei
gi ∶ Ei ↦ SE(2) ∶ θ i ↦ gi(θ i)
n =k
∑i=1
ni Vst = J1(θ1)θ1 = ⋯ = Jk(θk)θk
Ambient Space:
E = E1 ×⋯ × Ek
Loop equations or Structure constraints:
g1(θ1) = ⋯ = gk(θk)
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9.2 Configuration Space and SingularitiesChapter 9 Parallel Manipulators
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Define
H ∶ E ↦ SE(2) × ⋯ × SE(2)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
k−1
= SEk−1(2)
θ ↦ (g1(θ1)g−12 (θ2), . . . , g1(θ1)g
−1k (θk))
Configuration Space (CS)
Q = {θ ∈ E∣H(θ) = I}
Jacobian of H at θ ∈ Q:
DθH ≜ J(θ) =
⎡⎢⎢⎢⎢⎢⎣J1(θ1) −J2(θ2) 0 ⋯ 0⋮ 0 −J3(θ3) 0 ⋮⋮ ⋮ ⋱ 0
J1(θ1) 0 ⋯ 0 −Jk(θk)⎤⎥⎥⎥⎥⎥⎦∈ R3(k−1)×n
Chapter9 ParallelManipulators
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9.2 Configuration Space and SingularitiesChapter 9 Parallel Manipulators
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Property 1: If ∀θ ∈ Q, J(θ) ∈ R3(k−1)×n is of constant rank 3(k − 1),then Q is a differentiable manifold of dimension d = n − 3(k − 1).
Definition:If J(θ) is of full rank, constraints H are said to be linearlyindependent.
Grubler Fromula for predicting dimension of Q:
n = Number of joints
fi = DoF of the ith joint
m = Number of links
d =
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
3m −n
∑i=1
(3 − fi) = 3(m − n) + n
∑i=1
fi ⇒ (planar)6(m − n) + n
∑i=1
fi ⇒ (spatial)
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◇ Example: Planar mechanism & Delta manipulator
a n = 4, fi = 1,m = 3
d = 3(3 − 4) + 4 = 1
b n = 5, fi = 1,m = 4
d = 3(4 − 5) + 5 = 2
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9.2 Configuration Space and SingularitiesChapter 9 Parallel Manipulators
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c n = 9, fi = 1,m = 7
d = 3(7 − 9) + 9 ⋅ 1 = 3
d n = 7 × 3 = 21, fi = 1,m = 5 × 3 + 1 = 16d = 6 × (16 − 21) + 21 = −9(?)
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Definition: CS SingularityA point θ ∈ Q is a config. space singularity if DθH drops rank.
Definition: q ∈ Q is a criticalpoint of h2 if ∀v ∈ TpQ, ⟨dh2 , v⟩∣q= 0. δ = h2(q) is called a criticalvalue.
As Q = h−11 (0), ⟨dh1 , v⟩∣0 = 0⇒ dh1 ∧ dh2∣q = 0⇒ q is a CS Singularity.
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SingularityClassification
9.2 Configuration Space and SingularitiesChapter 9 Parallel Manipulators
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◻ Morse Theory:Let a < b and Qa = h−12 (−∞, a] = {q ∈ Q∣h2(q) ≤ a} contains nocritical points of h2, then Qa is diffeomorphic to Qb. (Qa is adeformation retract of Qb) ⇒ δ should be lie in [a, b]If D2
qh2(q) is non-degenerate, then q is an isolated critical point.
where xa = (x1 , x2), xp = x3.Implicit Function Theorem ⇒ ∃ψ ∶ R2 ↦ R s.t. xp = ψ(xa).Property 5: Let ψi ∶ E↦ R be a set of local coordinatefunctions on Q. A point p ∈ Q is a P-singularity iff