Computation of Generalized Aspect of Parallel ... - Univ. Nantes

Computation of Generalized Aspect of Parallel Manipulators June 14, 2011 Daisuke ISHII, Christophe JERMANN, Alexandre GOLDSZTEJN, LINA Université de Nantes 1

l Pose Error Problem

Parallel Mechanism (Manipulator) Conclusions and Future Works

’d) • Closed loop mechanism in which the end-effector

is connected to the base by at least two independent kinematic chains End-effector Moving Platform Pi!

F1,n1

Pi

3),

! Fi,j

(1)

Fi,j

Leg 1

Leg i

Leg m

ation

rigin of

Fi,1

F1,1

Fm,1

Fixed Base

Base

Kinematic chains: Coupling of links via kinematic pairs

nding the Maximal Pose Error

2010/06/15

12 / 24

Delta robot [80] 2

Parallel vs. Serial Manipulators

Delta robot Kinematic chain(s) Workspace Accuracy Payload Stiffness

Puma robot Parallel manip. Closed Limited Good High High

Serial manip. Opened Large Low Low Low

3

Aspect Computation

• Parallel manipulators may have multiple inverse and direct kinematic solutions

- A given end-effector pose

→ several control inputs

- A given control input

→ several end-effector poses

• Domain with multiple solutions contains singular solutions

• Aspect [Chablat, 2007]:

Maximal singularity-free region within the domain

➡ Our aim: Rigorous computation of aspects 4

nitial domain ([u], [x]) ∈ IR , box width ! > 0 Example: 2-RPR Manipulator , . . . , [xp ]} • Inputs: (x ,x2) - Variables end-effector 2n

1

‣ Control Variables: u1, u2 ‣ Pose Variables: x1, x2

u1

u2

Fig. 1 BranchAndPrune algorithm. Revolute Prismatic - Initial domain: u1

[2, 6], u2

x1, x2

[3, 9],

[-20, 20]

- Model:

f (u, x) =

!

2 u1

(0,0)

2 (x1

− 2 u2 − ((x1 −

joints

joints

base

2 + x2 ) 2 2 9) + x2 )

"

(9,0)

=0 5

nitial domain ([u], [x]) ∈ IR , box width ! > 0 Example: 2-RPR Manipulator , . . . , [xp ]} • Inputs: (x ,x2) - Variables end-effector 2n

1

‣ Control Variables: u1, u2 ‣ Pose Variables: x1, x2

u1

u2

Fig. 1 BranchAndPrune algorithm. Revolute Prismatic - Initial domain: u1

[2, 6], u2

x1, x2

[3, 9],

[-20, 20]

- Model:

f (u, x) =

!

2 u1

(0,0)

2 (x1

− 2 u2 − ((x1 −

joints

joints

base

2 + x2 ) 2 2 9) + x2 )

"

(9,0)

=0 6

Example: 2-RPR Manipulator

• Safe configuration • Singular configuration

- Change of control variables

robot breakdown

Algebraic characterization of singularity: det Dx f(u,x) = 0 or det Du f(u,x) = 0 7

Singularity Free Connected Components (SFCCs)

• Consider a manipulator modeled with 2n variables (u, x)

R2n

• An SFCC is a set of boxes [u]×[x] IR2n that are u - connected f(u,x)=0 - not containing any singular configuration

- proved to contain configurations

[u]

• SFCC is an inner

approximation of aspect

- Robot can move safely within

[x]

x

the x projection of a given SFCC 8

Example: 2-RPR Manipulator

• Output: - 2 SFCCs (projected on the workspace)

- Possibly singular region - Uncertified reachable region x2 x1

9

!

"

RR-RRR u21 − (x21 +Example: x22 ) =0 2 2 2 u2 − ((x1 − 9) + x2 ) • 2-dimensional variables (x1,x2) 5

2 Initial domain: • CX i cos x3 − CY i sin x3 − AX i − Li cos ui ) +

ui [-pi, pi], sin x3 + CY i cos x3 − AY i − Li sin ui )2 − Mi2 = 0 xi [-20, 20] i ∈ {1, 2, 3} 8

• Model: 



(x1 − 8 cos u1 ) +(x2 − 8 sin u1 )2 − 52    = 0 2  (x1 − 5 cos u2 − 9)  2 2 +(x2 − 5 sin u1 ) − 8 2

u1

(0,0)

8

5

u2

(9,0)

10

− AX − L cos u ) + (x + CX sin x + CY cos x − AY − L 2

Example: RR-RRR

• Example of

parallel singularity

• Example of

serial singularity

Change of control variables robot breakdown

Change of pose variables workspace limit

det Dx f(u,x) = 0

det Du f(u,x) = 0 11

Example: RR-RRR

• Computed 10 SFCCs: - Guarantees that there exists 10 aspects

12

Overview of the Proposed Method Model, initial domain, precision

Solving process Branch-and-Prune framework

Existence proving of solutions Singularity checking Management of neighboring boxes

Post process Enumeration of connected components

SFCCs, singular regions, uncertified regions

Visualization 13

Branch-and-Prune Framework Alternates search (branch) and contraction (prune) Initial domain

Prune

constraint

Branch

Set of ε-boxes

Prune

14

Existence Proof of a Configuration Fig. 1 BranchAndPrune algorithm. 2n

• Consider boxes [u]×[x]

IR ,

a continuously differentiable 2n n function f : R → R , ! ^ 2 2 2 a real vector u [u], and

u f(u,x)=0

"

u1 − (x1 + x2 ) [u] 2 2 2 an interval u Jacobian − 9) + x2 ) 2 − ((x1matrix [Ju]

IR

n×n

that contains all

=0

Duf(u,x) for (u,x) 2 [u]×[x] 2

u +x −1=0

derivative w.r.t. u

[x]

x

• Then, x [x] u [u] (f(u, x)=0), whenever u ˆ + Γ([J], ([u] − u ˆ), f (ˆ u, [x])) ⊆ int[u]

where Γ([A],[v],[b]) is the Gauss-Seidel operator

15

!

"

2 2 2 u − (x + x ) 1 1 2 Existence Proof of a Configuration =0 2 2 2 u2 − ((x1 − 9) + x2 )

• Example: - Model:

u +x −1=0 2

2

- Computation result (ε=0.2):

u ˆ + Γ([J], ([u] − u ˆ), f (ˆ u, [x])) ⊆ int[u] Proved boxes

Unproved boxes u x 16

Singularity Checking

• Consider a manipulator modeled by f(u,x)=0, where f : R2n→Rn

• A configuration (u,x)

R

2n

exhibits

- serial singularity iff det Ju = 0 - parallel singularity iff det Jx = 0 where Ju Jx

R

n×n

is Jacobian w.r.t. u and

Rn×n is Jacobian w.r.t. x 17

Guaranteeing Regularity

• Interval of configurations [u]×[x]

IR2n is

singularity free if 0

[det Ju] and 0

[det Jx] hold

computed from an interval extension of Jacobian matrix Singularity free boxes Possibly singular boxes 0 [det Ju] 0 [det Jx] 18

Fig. 1 BranchAndPrune algorithm.

Inner Testing

• A box [u]×[x]

IR

2n

is contained in an aspect

! " existence proving succeeded 2 2 2

x1 + x2 − u1 =succeeded 0 2 free 2 check 2 and singularity (x1 − 9) + x2 − u2

• Using inner test as a search termination criterion 2 2 • Example: x + u − 1 = 0 Proved & SF boxes

u + Γ(J, ([u] − u ), f (u , [u])) ⊆ int[u] !

!

!

Unproved & SF boxes

Possibly singular boxes u x

19

Overview of the Proposed Method Model, initial domain, precision

Solving process Branch-and-Prune framework

Existence proving of solutions Singularity checking Management of neighboring boxes

Post process Enumeration of connected components

SFCCs, singular regions, uncertified regions

Visualization 20

Enumeration of SFCCs

• Management of neighboring boxes during the search by Branch-and-Prune

• After the search, we apply a graph enumeration method to the set of inner boxes SFCC 1

SFCC 2

SFCC 3

SFCC 4

u x

21

Example: 3-RPR

• 3 dimensional planer manipulator (u2)

(x1,x2)

(x3) (u3)

(u1)

22

Example: 3-RPR

• Computed spiral workspace:

23

Example: 3-RRR

• Computed with fixed orientation: x3 = 0 (u2)

(x1,x2)

(u1)

(x3)

(u3) 24

Example: 3-RRR

• Computed 25 aspects:

25

Experimental Results 2-RPR

RRR

2

2

10

2

?

0.1

0.1

0.1

0.1

0.01

2

562 (2)

1456 (10)

49882 (2)

51901 (25)

# boxes

1240

31590

87584

13677836

6081438

time (s)

0.206

13.264

35.784

7913

5050

theoretical

# aspects

prec # SFCCs (filtered)

RR-RRR 3-RPR

3-RRR (x3=0)

26

Conclusion

• We present a tool that supports - simple modeling of parallel manipulators - validated computation and visualization of workspace, working modes, and generalized aspects

• Experimental results indicate the correct number of generalized aspects

27

References • D. Chablat and P. Wenger: Working Modes and Aspects in Fully Parallel Manipulators, ICRA’98, pp. 1964-1969, 1998.

• D. Chablat and P. Wenger: The Kinematic Analysis of a

Symmetrical Three-Degree-of-Freedom Planar Parallel Manipulator, Symp. on Robot Design, Dynamics and Control, pp. 1-7, 2004.

• A. Goldsztejn and L. Jaulin: Inner Approximation of the Range of Vector-Valued Functions, Reliable Computing, vol. 14, pp. 1-23, 2010.

• A. Goldsztejn and L. Jaulin: Inner and Outer

Approximations of Existentially Quantified Equality Constraints, CP’06, pp. 198-212, 2006. 28