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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


Introduction




2


1 Introduction

2 Con�guration Space and Singularities

3 Singularity Classi�cation


Introduction



9.1 IntroductionChapter 9 Parallel Manipulators

3

◻ Samples of parallel manipulators:1-DoF:

2-DoF:

3-DoF:


Introduction



9.1 IntroductionChapter 9 Parallel Manipulators

4

◻ Samples of parallel manipulators:4-DoF:

5-DoF:

6-DoF:


Introduction



9.2 Configuration Space and SingularitiesChapter 9 Parallel Manipulators

5

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)


Introduction




6

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


Introduction




7

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)


Introduction




8

◇ 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


Introduction




9

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(?)


Introduction




10

Definition: CS SingularityA point θ ∈ Q is a config. space singularity if DθH drops rank.

♢ Review: Differential forms(independent ofcoordinates)

θ ∶ (θ1 , . . . , θn) ∈ E, h ∶ E↦ R, dh =n

∑i=1

∂h

∂θ idθ i ∈ T∗θ E

dh ∶ TθE↦ R, v↦ dh(v) = d

dt∣t=0

h(θ(t))where θ(0) = θ , θ(0) = v

dθ i( ∂

∂θ j) = δij , dθ i ∧ dθ j = −dθ j ∧ dθ i

dθ i ∧ dθ j ∶ TθE × TθE↦ R

(dθ i ∧ dθ j)(v,w)= dθ i(v)dθ j(w) − dθ i(w)dθ j(v)


Introduction




11

Given h1 , h2:

dh = [ ∂h1∂θ1

∂h1∂θ2

⋯ ∂h1∂θn

∂h2∂θ1

∂h2∂θ2

⋯ ∂h2∂θn

]Principal minors of 2 × 2:

( ∂h1∂θ1⋅ ∂h2∂θ2− ∂h1∂θ2⋅ ∂h2∂θ1)dθ1 ∧ dθ2

( ∂h1∂θ1⋅ ∂h2∂θ3− ∂h1∂θ3⋅ ∂h2∂θ1)dθ1 ∧ dθ3

�us, dh1 ∧ dh2 = ∑i<j

(∂h1∂θi⋅ ∂h2∂θj− ∂h1∂θj⋅ ∂h2∂θi)dθi ∧ dθj


Introduction




12

Definition:h1 and h2 are said to be linearly independent if dh1(θ) anddh2(θ) are linearly independent at θ ∈ E

Property 3: h1 , h2 linearly independent ⇔ dh1 ∧ dh2∣θ ≠ 0Property 4: A set of functions hi , i = 1, . . . , n are linearlyindependent iff

dh1 ∧ dh2 ∧⋯∧ dhm∣θ ≠ 0


Introduction




13

◇ Example: 4-bar mechanism

θ = (θ1 , θ2, θ3) ∈ ELoop equations:

H ∶E ↦ R2

θ ↦ H(θ) = [ l1 sin θ1 + l2 sin θ2 − l3 sin θ3l1 cos θ1 + l2 cos θ2 − l3 cos θ3 − δ ] ≜ [ h1(θ)

h2(θ) ]CS Singularities:

dh1(θ) ∧ dh2(θ) = l1l2 sin(θ1 − θ2)dθ1 ∧ dθ2− l1l3 sin(θ1 − θ3)dθ1 ∧ dθ3− l2l3 sin(θ2 − θ2)dθ2 ∧ dθ3

dh1(θ) ∧ dh2 = 0⇔ sin(θi − θj) = 0, i < j


Introduction




14

Assume: l1 − l2 > l3, l2 > l3Singularities Parameter Relation Parameter Valuep1 = (0, 0, π) l1 + l2 + l3 ≜ δ4 δ = δ4p2 = (0, 0, 0) l1 + l2 − l3 ≜ δ3 δ = δ3p3 = (0, π , π) l1 − l2 + l3 ≜ δ2 δ = δ2p1 = (0, π , 0) l1 − l2 − l3 ≜ δ1 δ = δ1

21

3

3

21 3

4

1l

2l

3l

1l

2l

3l

2

1

3

2

21

3

1

1l

2l

3l

1l

2l

3l

Case 1 Case 2

Case 3 Case 4


Introduction




15

◇ Example: SNU manipulator

gi(θi) = eξi,1θ i,1⋯eξi,5θ i,5gi(0), i = 1, 2, 3(g−1i dgi)∧ = 5

∑j=1

Adeξi,j θi,j⋯e ξi,5 θi,5 gi(0)

ξi,jdθi,j

∈ se∗(3) ∶ Maurer-Cartan form

V =5

∑j=1

Ad−1eξi,j θi,j⋯e ξi,5 θi,5 gi(0)

ξi,jθi,j

= Ji(θi)θi , i = 1, 2, 3


Introduction




16

Loop constraint:

g1(θ1) = g2(θ2) = g3(θ3)ωθ = [ ωθ ,1⋮

ωθ ,12] ≜ [ (g−11 dg1)∧ − (g−12 dg2)∧(g−11 dg1)∧ − (g−13 dg3)∧ ] = [ J1dθ1 − J2dθ2

J1dθ1 − J3dθ3 ]ωθ ,1 ∧⋯∧ ωθ ,12 = 0, at home con�g.


Introduction




17

◻ Relation between Q and δ:Q = h−11 (0): A 2-dimensional torus

Morse function:h2 ∶ Q↦ R ∶ θ ↦ h2(δ)= l1c1 + l2c2 − l3c3,Q = h−12 (δ)

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.


Introduction




18

◻ 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.

Parameterize Q by (θ1 , θ2)⇒ θ3 = θ3(θ1 , θ2), ∂θ3

∂θi= −

∂h1∂θi/ ∂h1∂θ3

, i = 1, 2

h2(θ1 , θ2) = h2(θ1 , θ2, θ3(θ1 , θ2))⇒ D2h2 =⎡⎢⎢⎢⎢⎣

∂2 h2∂θ21

∂2 h2∂θ1∂θ2

∂2 h2∂θ1∂θ2

∂2 h2∂θ22

⎤⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎣−l1c1 − l1s1s2

c3+ l21 c

21

l3c33

l1 l2c1c2l3c33

l1 l2c1c2l3c33

−l2c2 − l2s2s3c3+ l22c

22

l3c33

⎤⎥⎥⎥⎥⎥⎦


Introduction



9.3 Singularity ClassificationChapter 9 Parallel Manipulators

19

◻ Configuration space versus geometricparameter δ:

Parameter Value Description of Q Morse Indexδ = δ4 a single point Mi = 2

δ ∈ (δ3 , δ4) Unit circleδ = δ3 Figure 8 Mi = 1

δ ∈ (δ2 , δ3) Two separate circlesδ = δ2 Figure 8 Mi = 1

δ ∈ (δ1 , δ2) Unit circleδ = δ1 A single point Mi = 0

δ ∈ (0, δ1), (δ4 ,∞) Empty set


Introduction




20

◻ Parametrization Singularity:Consider

H ∶ R3 ↦ R ∶ (x1 , x2 , x3) ↦ x21 + x22 + x

23 − 1

Q = H−1(0) ∶ unit sphereLocal coordinates:

ψ ∶ Q↦ R2 ∶ x↦ [ ψ1(x)

ψ2(x) ] = [ x1x2 ]Tpψ drops rank on Q⇔ ∃v ∈ TpQ s.t. ⟨dψi , v⟩ = 0

However, ⟨dH , v⟩ = 0⇒ dψ1 , dψ2 , dH are linearly dependent.

⇒ dH ∧ dψ1 ∧ dψ2 = 0

= (∂H∂x1

dx1 +∂H

∂x2dx2 +

∂H

∂x3dx3) ∧ dx1 ∧ dx2

=∂H

∂x3dx1 ∧ dx2 ∧ dx3 = 2x3dx1 ∧ dx2 ∧ dx3

⇒ x3 = 0, Points of the equator are P-singularity.


Introduction




21

Alternatively, p ∈ Q is a P-singularity iff ∂H∂xp

of (∗) drops rank∂H

∂xadxa +

∂H

∂xpxp = 0 (∗)

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

dh1 ∧⋯∧ dhm ∧ dψ1 ∧⋯∧ dψn−m∣p = 0Actuator Singularity:

ψ ∶ (θ1 , . . . , θn)↦ θadh1 ∧⋯ ∧ dhm ∧ dθa,1 ∧⋯∧ dθa,n−m∣p = 0

End-e�ector Singularity:

ψ ∶ Q↦ SE(2) ∶ θ ↦ xdh1 ∧⋯∧ dhm ∧ dx1 ∧⋯∧ dxn−m∣p = 0


Introduction




22

◻ Irregular P-singularity:If p ∈ Q is also a CS-singularity, then

dh1 ∧⋯ ∧ dhm ∧ dψ1 ∧⋯∧ dψn−m∣p = 0holds automatically. Q is not a manifold,gains dimension by 1 or more, thus:

Nominally actuated ⇒ under-actuated

◻ Redundant actuation:

S2 = U(x1 ,x2) ∪U(x2 ,x3) ∪U(x1 ,x3)

P-Singularity: {x3 = 0} ∩ {x1 = 0} ∩ {x2 = 0} = ∅


Introduction




23

◻ Singularity classification:A-singularity of redundantly actuated manipulator

l redundant actuators: θa = (θa,1 , . . . , θa,n−m+l) ∈ Rn−m+l , l > 0

p ∈ Q: A-singularity if

dh1∧⋯∧dhm∧dθa,i1∧⋯∧dθa,in−m ∣p = 0, 1 ≤ i1 ≤ ⋯ ≤ in−m ≤ n−m+lE�ect: eliminate A-singularity

2

13

1l

3l

c

Actuators 2l2

1 3

1l 3

l

c

2l

Eliminate A-singularity by3

Eliminate A-singularity by1


Introduction




24

◻ Stratified structure of A-singularities :

Singular set Qs ⊂ Q: all A-singularities

Qs = {p ∈ Q∣dh1 ∧⋯ ∧ dhm ∧ dθa,i1 ∧⋯∧ dθa,in−m ∣p = 0,dh1 ∧⋯ ∧ dhm∣p ≠ 0, 1 ≤ i1 < ⋯ < in−m ≤ n −m + 1}

Annihilation space

TpV = {v ∈ TpQ∣⟨v, dθa,j⟩ = ⟨v, dhi⟩ = 0,, i = 1,⋯,m, j = 1,⋯, n −m + l}

Degree of de�ciency: d = dim(TpV)d ≤ d0 =min(n −m,m − l)


Introduction




25

Strati�ed structure

Qsk = {p ∈ Qs∣dim(TpV) = k}, k = 1,⋯, kmax

∆sk = ⋃p∈Qsk

TpV , Qs =kmax

⋃k=1

Qsk , ∆s =kmax

⋃k=1

∆sk

Definition:p ∈ Qsk is a first-order singularity iff there does not exist v ∈ ∆sk

that is also tangent to Qsk.Otherwise, p is a second-ordersingularity.

Definition:A second-order singularity p ∈ Qsk is degenerate iff ∃ constantrank k1(k1 < k) sub-distribution ∆sk1 ⊂ ∆sk s.t.

1 ∆sk1(p) ⊂ TpQsk,

2 ∆sk1(p) is involutive.


Introduction




26


Introduction




27

◻ Degeneracy of P-singularity:

Degenerate: allow continuous motion with �xed parameters

Non-degenerate: allow instantaneous motion with �xed

parameters

Fig. 19. Nondegenerate P-singularity. Fig. 20. Degenerate P-singularity.


Introduction




28

◻ Conditions for degenerate A-singularity:

Basis of ∆sk:

∆sk = span{Y},Y = {Y1,⋯,Yk}Annihilation vector:

vs = [ θaθp] = [ 0

θp] = Yα ∈ ∆sk , α ∈ R

k

θs ∈ Qsk: degenerate A-singularity i�

hi(θs + εvs) − hi(θs) = 0, i = 1,⋯,mConditions on coe�cients of Taylor series

[∂2hi∂θ2]θs

(Yα ,Yα) = αTYT[∂2hi∂θ2]θs

Yα = 0,⋯, i = 1,⋯,m


Introduction




29

◻ Classification diagram:

Fig. 15. A hierarchic diagram of singularities, A-sing.: actuator singularity.E-sing.: end-effector singularity. P-sing.: parametrization singularity. N.Degenerate: Nondegenerate.

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