2nd International Summer School on Geometric Methods in Robotics, Mechanism Design and Manufacturing Research-Lecture 09 Parallel Manipulators

1. Chapter 9 Parallel Manipulators
1
Chapter Lecture Notes for
Parallel
Manipulators
A Geometrical Introduction to
Introduction
Configuration
Robotics and Manipulation
Space and
Singularities
Singularity
Richard Murray and Zexiang Li and Shankar S. Sastry
Classification CRC Press
Zexiang Li and Yuanqing Wu
ECE, Hong Kong University of Science & Technology
July ,

2. Chapter 9 Parallel Manipulators
2
Chapter
Parallel
Chapter 9 Parallel Manipulators
Manipulators
Introduction
Configuration
Introduction
Space and
Singularities
Singularity
Classification Con guration Space and Singularities
Singularity Classi cation

3. Chapter 9 Parallel Manipulators
9.1 Introduction
3
◻ Samples of parallel manipulators:
1-DoF:
Chapter
Parallel
Manipulators
Introduction
Configuration
Space and
Singularities 2-DoF:
Singularity
Classification
3-DoF:

4. Chapter 9 Parallel Manipulators
9.1 Introduction
4
◻ Samples of parallel manipulators:
4-DoF:
Chapter
Parallel
Manipulators
Introduction
Configuration
Space and
Singularities 5-DoF:
Singularity
Classification
6-DoF:

5. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
5
Chapter
Parallel k limbs, with SE( ) as task space.
Manipulators Limb i:
Introduction
Configuration
θ i = (θ i , . . . , θ ini ) ∈ Ei
Space and
Singularities gi ∶ Ei ↦ SE( ) ∶ θ i ↦ gi (θ i )
Singularity k
Classification n= ni ˙ ˙
Vst = J (θ )θ = ⋯ = Jk (θ k )θ k
i=
Ambient Space:
E = E × ⋯ × Ek
Loop equations or Structure constraints:
g (θ ) = ⋯ = gk (θ k )

6. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
6
Define
Chapter
Parallel
Manipulators H ∶ E ↦ SE( ) × ⋯ × SE( ) = SEk− ( )
Introduction
k−
Configuration
− −
Space and
Singularities
θ ↦ (g (θ )g (θ ), . . . , g (θ )gk (θ k ))
Singularity
Classification
Configuration Space (CS)
Q = {θ ∈ E H(θ) = I}
Jacobian of H at θ ∈ Q:
⎡ J (θ ) −J (θ ) ⎤
⎢ ⋯ ⎥
⎢ −J (θ ) ⎥
Dθ H ≜ J(θ) = ⎢ ⎥∈R
⎢ ⎥
(k− )×n
⎢ J (θ ) ⎥
⋱
⎣ ⋯ −Jk (θ k ) ⎦

7. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
7
Property 1: If ∀θ ∈ Q, J(θ) ∈ R (k− )×n is of constant rank (k − ),
then Q is a differentiable manifold of dimension d = n − (k − ).
Chapter
Parallel
Manipulators Definition:
Introduction If J(θ) is of full rank, constraints H are said to be linearly
Configuration independent.
Space and
Singularities
Singularity Gr¨bler Fromula for predicting dimension of Q:
u
Classification
n = Number of joints
fi = DoF of the ith joint
m = Number of links
⎧ n n
⎪ m − ( − fi ) = (m − n) + fi ⇒ (planar)
⎪
⎪
⎪
⎪
⎪
d=⎨
i= i=
⎪
⎪
⎪ (m − n) + fi ⇒ (spatial)
n
⎪
⎪
⎪
⎩ i=

8. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
8
Example: Planar mechanism & Delta manipulator
Chapter
Parallel a n = , fi = , m =
d = ( − )+ =
Manipulators
Introduction
Configuration
Space and
Singularities
Singularity
Classification
b n = , fi = , m =
d= ( − )+ =

9. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
9
Chapter c n = , fi = , m =
d= ( − )+ ⋅ =
Parallel
Manipulators
Introduction
Configuration
Space and
Singularities
Singularity
Classification d n= × = , fi = , m = × + =
d = ×( − )+ = − (?)

10. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
10
Definition: CS Singularity
A point θ ∈ Q is a config. space singularity if Dθ H drops rank.
Chapter
Parallel
Manipulators
Review: Differential forms(independent of
Introduction coordinates)
Configuration
n
θ ∶ (θ , . . . , θ n ) ∈ E, h ∶ E ↦ R, dh =
Space and ∂h ∗
Singularities dθ i ∈ Tθ E
Singularity i= ∂θ i
Classification
dh ∶ Tθ E ↦ R, v ↦ dh(v) =
d
h(θ(t))
dt t=
where θ( ) = θ, θ( ) = v
˙
dθ i ( ) = δ ij , dθ i ∧ dθ j = −dθ j ∧ dθ i
∂
∂θ j
dθ i ∧ dθ j ∶ Tθ E × Tθ E ↦ R
(dθ i ∧ dθ j )(v, w)
= dθ i (v)dθ j (w) − dθ i (w)dθ j (v)

11. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
11
Given h , h :
Chapter ∂h ∂h
⋯ ∂h
dh = ∂θ ∂θ ∂θ n
⋯
Parallel
Manipulators ∂h ∂h ∂h
∂θ ∂θ ∂θ n
× :
Introduction
Configuration Principal minors of
Space and
Singularities
( ⋅ − ⋅ )dθ ∧ dθ
Singularity
∂h ∂h ∂h ∂h
Classification ∂θ ∂θ ∂θ ∂θ
( ⋅ − ⋅ )dθ ∧ dθ
∂h ∂h ∂h ∂h
∂θ ∂θ ∂θ ∂θ
us, dh ∧ dh = ( ⋅ − ⋅ )dθ i ∧ dθ j
∂h ∂h ∂h ∂h
i<j ∂θ i ∂θ j ∂θ j ∂θ i

12. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
12
Chapter Definition:
h and h are said to be linearly independent if dh (θ) and
Parallel
dh (θ) are linearly independent at θ ∈ E
Manipulators
Introduction
Configuration
Space and
Singularities
Property 3: h , h linearly independent ⇔ dh ∧ dh
Singularity
Classification θ ≠
Property 4: A set of functions hi , i = , . . . , n are linearly
independent iff
dh ∧ dh ∧ ⋯ ∧ dhm θ ≠

13. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
13
Example: 4-bar mechanism
Chapter
Parallel
θ = (θ , θ , θ ) ∈ E
Manipulators
Introduction Loop equations:
Configuration
Space and
Singularities
H ∶E ↦ R
Singularity
Classification
θ ↦ H(θ) = l sin θ + l sin θ − l sin θ h (θ)
l cos θ + l cos θ − l cos θ − δ ≜
h (θ)
CS Singularities:
dh (θ) ∧ dh (θ) = l l sin(θ − θ )dθ ∧ dθ
− l l sin(θ − θ )dθ ∧ dθ
− l l sin(θ − θ )dθ ∧ dθ
dh (θ) ∧ dh = ⇔ sin(θ i − θ j ) = , i < j

14. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
14
Assume: l − l > l , l > l
Singularities Parameter Relation Parameter Value
Chapter
p = ( , , π) l +l +l ≜ δ δ=δ
p =( , , ) l +l −l ≜ δ
Parallel
Manipulators δ=δ
Introduction p = ( , π, π) l −l +l ≜ δ δ=δ
Configuration p = ( , π, ) l −l −l ≜ δ δ=δ
Space and
Singularities Case 1 Case 2
Singularity l1 l2 l3 l1 l2
Classification 1 2 3 2
1
3
3
l3
4
Case 3 Case 4
l2
l1 l1
1
2 1 3 2
3
1 l3 l2
2
l3

15. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
15
Example: SNU manipulator
gi (θ i ) = eξi, θ i, ⋯eξi, gi ( ), i = , ,
Chapter
ˆ ˆ θ i,
Parallel
(gi− dgi )∧ =
Manipulators
Introduction Ad ˆ
ξ i,j θ i,j ˆ ξi,j dθ i,j
e ⋯e ξ i, θ i, gi ( )
Configuration
j=
∈ se ( ) ∶ Maurer-Cartan form
∗
Space and
Singularities
Singularity
Classification
V= Ad−ξˆi,j θ i,j ˆ
˙
ξi,j θ i,j
j= e ⋯e ξ i, θ i,
gi ( )
= Ji (θ i )θ i , i = , ,
˙

16. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
16
Chapter
Parallel Loop constraint:
Manipulators
Introduction
Configuration
Space and
g (θ ) = g (θ ) = g (θ )
ω θ, (g − dg )∧ − (g − dg )∧ J dθ − J dθ
Singularities
ωθ = ≜
(g − dg )∧ − (g − dg )∧
= J dθ − J dθ
Singularity
Classification ω θ,
ω θ, ∧ ⋯ ∧ ω θ, = , at home con g.

17. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
17
◻ Relation between Q and δ:
Q = h− ( ): A -dimensional torus
˜
Chapter
Parallel
Morse function:
h ∶ Q ↦ R ∶ θ ↦ h (δ)
Manipulators
Introduction
˜ ˜
Configuration
Space and
= l c + l c − l c , Q = h− (δ)
˜
Singularities
˜
Definition: q ∈ Q is a critical
Singularity
Classification
˜ if ∀v ∈ Tp Q, ⟨dh , v⟩ q
point of h ˜ ˜
= . δ = h (q) is called a critical
˜
value.
As Q = h− ( ), ⟨dh , v⟩ =
˜
⇒ dh ∧ dh q =
⇒ q is a CS Singularity.

18. Chapter 9 Parallel Manipulators
9.2 Configuration Space and Singularities
18
◻ Morse Theory:
Let a < b and Qa = h− (−∞, a] = {q ∈ Q h (q) ≤ a} contains no
¯ ˜ ˜
Chapter
Parallel ˜ ˜ ˜ ˜
critical points of h , then Qa is diffeomorphic to Qb . (Qa is a
Manipulators
deformation retract of Q ˜ b ) ⇒ δ should be lie in [a, b]
Introduction
Configuration If Dq h (q) is non-degenerate, then q is an isolated critical point.
˜
Parameterize Q by (θ , θ )
Space and
Singularities ˜
Singularity
Classification
⇒ θ = θ (θ , θ ),
∂θ ∂h ∂h
=− ,i = ,
∂θ i ∂θ i ∂θ
⎡ ∂h ⎤
⎢ ∂θ ⎥
˜ ˜
∂ h
h (θ , θ ) = h (θ , θ , θ (θ , θ )) ⇒ D h = ⎢ ∂ h ⎢
⎥
⎥
˜ ˜ ∂θ ∂θ
⎢ ∂θ ∂θ ⎥
˜ ˜
⎣ ⎦
∂ h
∂θ
⎡ ⎤
⎢ −l c − l s s + l c ⎥
⎢ ⎥
ll cc
=⎢ ⎥
c l c l c
⎢ ⎥
⎢ ⎥
ll cc l c
−l c − l s s + l c
⎣ l c c ⎦

19. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
19
Chapter
◻ Configuration space versus geometric
Parallel parameter δ:
Manipulators
Introduction
Parameter Value Description of Q Morse Index
Configuration
δ=δ a single point Mi =
δ ∈ (δ , δ )
Space and
Singularities Unit circle
δ=δ Figure Mi =
δ ∈ (δ , δ )
Singularity
Classification
Two separate circles
δ=δ Figure Mi =
δ ∈ (δ , δ ) Unit circle
δ=δ A single point Mi =
δ ∈ ( , δ ), (δ , ∞) Empty set

20. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
20
◻ Parametrization Singularity:
Consider
Chapter H ∶ R ↦ R ∶ (x , x , x ) ↦ x + x + x −
Q = H − ( ) ∶ unit sphere
Parallel
Manipulators
Local coordinates:
ψ (x)
Introduction
ψ∶Q↦R ∶x↦ x
ψ (x)
Configuration = x
Tp ψ drops rank on Q ⇔ ∃v ∈ Tp Q s.t. ⟨dψ i , v⟩ =
Space and
Singularities
Singularity However, ⟨dH, v⟩ =
⇒ dψ , dψ , dH are linearly dependent.
Classification
⇒ dH ∧ dψ ∧ dψ =
=( dx ) ∧ dx ∧ dx
∂H ∂H ∂H
dx + dx +
∂x ∂x ∂x
∂H
= dx ∧ dx ∧ dx = x dx ∧ dx ∧ dx
∂x
⇒ x = , Points of the equator are P-singularity.

21. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
21
Alternatively, p ∈ Q is a P-singularity iff ∂xp of (∗) drops rank
∂H
(∗)
∂H ∂H
Chapter dxa + xp =
∂xa ∂xp
where xa = (x , x ), xp = x .
Parallel
Manipulators
Implicit Function Theorem ⇒ ∃ψ ∶ R ↦ R s.t. xp = ψ(xa ).
Introduction
Configuration
Property 5: Let ψi ∶ E ↦ R be a set of local coordinate
Space and
Singularities
Singularity functions on Q. A point p ∈ Q is a P-singularity iff
Classification
dh ∧ ⋯ ∧ dhm ∧ dψ ∧ ⋯ ∧ dψn−m p =
Actuator Singularity:
ψ ∶ (θ , . . . , θ n ) ↦ θ a
dh ∧ ⋯ ∧ dhm ∧ dθ a, ∧ ⋯ ∧ dθ a,n−m p =
End-e ector Singularity:
ψ ∶ Q ↦ SE( ) ∶ θ ↦ x
dh ∧ ⋯ ∧ dhm ∧ dx ∧ ⋯ ∧ dxn−m p =

22. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
22
◻ Irregular P-singularity:
If p ∈ Q is also a CS-singularity, then
Chapter
Parallel
Manipulators
dh ∧ ⋯ ∧ dhm ∧ dψ ∧ ⋯ ∧ dψn−m p =
Introduction
Configuration
Space and
holds automatically. Q is not a manifold,
Singularities gains dimension by or more, thus:
Singularity
Classification
Nominally actuated ⇒ under-actuated
◻ Redundant actuation:
S = U(x ,x ) ∪ U(x ,x ) ∪ U(x ,x )
P-Singularity: {x = } ∩ {x = } ∩ {x = } = ∅

23. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
23
◻ Singularity classification:
A-singularity of redundantly actuated manipulator
Chapter
l redundant actuators: θ a = (θ a, , . . . , θ a,n−m+l ) ∈ Rn−m+l , l >
Parallel
Manipulators
Introduction
p ∈ Q: A-singularity if
Configuration
Space and
Singularities dh ∧⋯∧dhm ∧dθ a,i ∧⋯∧dθ a,in−m p = , ≤ i ≤ ⋯ ≤ in−m ≤ n−m+l
Singularity
Classification
E ect: eliminate A-singularity
2 l2 Act or
uat s l2
l1 l3 2
l3
c
l1
1 3
1 c 3
Eliminate A-singularity by 3
Eliminate A-singularity by 1

24. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
24
◻ Stratified structure of A-singularities :
Chapter
Parallel Singular set Qs ⊂ Q: all A-singularities
Manipulators
Introduction
Qs = {p ∈ Q dh ∧ ⋯ ∧ dhm ∧ dθ a,i ∧ ⋯ ∧ dθ a,in−m p = ,
dh ∧ ⋯ ∧ dhm p ≠ , ≤ i < ⋯ < in−m ≤ n − m + }
Configuration
Space and
Singularities
Singularity
Classification
Annihilation space
Tp V = {v ∈ Tp Q ⟨v, dθ a,j ⟩ = ⟨v, dhi ⟩ = ,
, i = , ⋯, m, j = , ⋯, n − m + l}
Degree of de ciency: d = dim(Tp V)
d ≤ d = min(n − m, m − l)

25. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
25
Strati ed structure
Chapter
Qsk = {p ∈ Qs dim(Tp V) = k}, k = , ⋯, kmax
Parallel
Manipulators kmax kmax
Introduction ∆sk = Tp V, Qs = Qsk , ∆s = ∆sk
p∈Qsk k= k=
Configuration
Space and
Singularities
Definition:
Singularity
p ∈ Qsk is a first-order singularity iff there does not exist v ∈ ∆sk
Classification that is also tangent to Qsk .Otherwise, p is a second-order
singularity.
Definition:
A second-order singularity p ∈ Qsk is degenerate iff ∃ constant
rank k (k < k) sub-distribution ∆sk ⊂ ∆sk s.t.
¯
∆sk (p) ⊂ Tp Qsk ,
¯
∆sk (p) is involutive.
¯

26. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
26
Chapter
Parallel
Manipulators
Introduction
Configuration
Space and
Singularities
Singularity
Classification

27. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
27
◻ Degeneracy of P-singularity:
Chapter
Parallel
Degenerate: allow continuous motion with xed parameters
Manipulators
Non-degenerate: allow instantaneous motion with xed
Introduction
parameters
Configuration
Space and
Singularities
Singularity
Classification
Fig. 19. Nondegenerate P-singularity. Fig. 20. Degenerate P-singularity.

28. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
28
◻ Conditions for degenerate A-singularity:
Chapter Basis of ∆sk :
Parallel
∆sk = span{Y}, Y = {Y , ⋯, Yk }
Manipulators
Introduction
Configuration
Space and Annihilation vector:
Singularities
Singularity ˙
θa
Classification
vs = ˙ = ˙
θp = Y α ∈ ∆sk , α ∈ Rk
θp
θ s ∈ Qsk : degenerate A-singularity i
hi (θ s + εvs ) − hi (θ s ) = , i = , ⋯, m
Conditions on coe cients of Taylor series
(Y α, Y α) = α T Y T
∂ hi ∂ hi
Y α = , ⋯, i = , ⋯, m
∂θ θ s ∂θ θ s

29. Chapter 9 Parallel Manipulators
9.3 Singularity Classification
29
◻ Classification diagram:
Chapter
Parallel
Manipulators
Introduction
Configuration
Space and
Singularities
Singularity
Classification
Fig. 15. A hierarchic diagram of singularities, A-sing.: actuator singularity.
E-sing.: end-effector singularity. P-sing.: parametrization singularity. N.
Degenerate: Nondegenerate.