[PPT] - Kinematic Design of Redundant Robotic Manipulators that are PowerPoint Presentation

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Kinematic Design of Redundant Robotic

Manipulators that are Optimally Fault Tolerant

Presented by:

Khaled M. Ben-Gharbia Septmber 8th, 2014

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Why Fault Tolerance?

Applications

hazardous waste cleanup space/underwater exploration anywhere failures are likely or intervention is costly

Common failure mode

locked actuators

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Simple Redundant Robot Planar 3DOF

˙ x1 ˙ x2

= J(θ)

  ˙ θ1 ˙ θ2 ˙ θ3  

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Jacobian (J) Matrix

3R planar manipulator

Forward Kinematics:

x = f (θ) x1 = a1 cos(θ1) + a2 cos(θ1 + θ2) + a3 cos(θ1 + θ2 + θ3) x2 = a1 sin(θ1) + a2 sin(θ1 + θ2) + a3 sin(θ1 + θ2 + θ3) = ⇒ taking the derivative w.r.t. time = ⇒ ˙ x = J(θ) ˙ θ Geometrically J3×3 = [j1 j2 j3] = [z1 × p1 z2 × p2 z3 × p3] where zi is the rotation axis, and here it is [0 0 1]T

Spatial manipulator ji = vi ωi

θ1+ θ2 + θ3

x1 x2 θ1 θ1+

2

θ

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Jacobian (J) Matrix

3R planar manipulator

Forward Kinematics:

x = f (θ) x1 = a1 cos(θ1) + a2 cos(θ1 + θ2) + a3 cos(θ1 + θ2 + θ3) x2 = a1 sin(θ1) + a2 sin(θ1 + θ2) + a3 sin(θ1 + θ2 + θ3) = ⇒ taking the derivative w.r.t. time = ⇒ ˙ x = J(θ) ˙ θ Geometrically J3×3 = [j1 j2 j3] = [z1 × p1 z2 × p2 z3 × p3] where zi is the rotation axis, and here it is [0 0 1]T

Spatial manipulator ji = vi ωi

θ1+ θ2 + θ3

x1 x2 p2 j2 θ1 θ1+

2

θ

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Jacobian (J) Matrix

3R planar manipulator

Forward Kinematics:

x = f (θ) x1 = a1 cos(θ1) + a2 cos(θ1 + θ2) + a3 cos(θ1 + θ2 + θ3) x2 = a1 sin(θ1) + a2 sin(θ1 + θ2) + a3 sin(θ1 + θ2 + θ3) = ⇒ taking the derivative w.r.t. time = ⇒ ˙ x = J(θ) ˙ θ Geometrically J3×3 = [j1 j2 j3] = [z1 × p1 z2 × p2 z3 × p3] where zi is the rotation axis, and here it is [0 0 1]T

Spatial manipulator ji = vi ωi

θ1+ θ2 + θ3

x1 x2 p2 j2 θ1 θ1+

2

θ

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Kinematic Dexterity Measures:

Function of Singular Values of the Jacobian

V TV = I D = σ1 σ2

U =

cos φ − sin φ sin φ cos φ

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Approaching Kinematic Singularities

Singularity: σ2 = 0 Value of σm is a common dexterity measure

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Optimal Fault Tolerant Configurations

for Locked Joints Failures

Fault-free Jacobian: Jm×n = [j1 j2 . . . jn] Failure at joint f : f Jm×n−1 = [j1 j2 . . . jf −1 jf +1 . . . jn]

f J = m i=1 f σi f ˆ

ui

f ˆ

vT

i

Worst-case remaining dexterity: K = minn

f =1 f σm

Isotropic & Optimal J: Kopt = σ

n−m

n

for all f

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Isotropic and Optimally Fault Tolerant Jacobian

equal σi’s equal f σm for all f equal ji for all i Example: 2 × 3 isotropic and optimally fault tolerant J: J = [ j1 j2 j3 ] =  −

2

3

1

6

1

6

−

1

2

1

2

  Null space equally distributed: ˆ nJ =

1 3

1

3

1

3

T

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Example: 2 × 3 Optimally Fault Tolerant Jacobian

120° 120° 120°

j3 j1 j2

Remaining dexterity: Kopt =

f σ2 =

1

3 , for all f

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Example: 2 × 3 Optimally Fault Tolerant Jacobian

120° 120° 120°

j3 p2 p3 j1 j2 p1

Remaining dexterity: Kopt =

f σ2 =

1

3 , for all f

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Example: 2 × 3 Optimally Fault Tolerant Jacobian

120° 120° 120°

j3 p2 p3 j1 j2 p1

Remaining dexterity: Kopt =

f σ2 =

1

3 , for all f

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Different Optimally Fault Tolerant Jacobians

Sign change and permuting columns do not affect the isotropy and the optimally fault tolerant properties of J, e.g. But each new J may belong to a different manipulator

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Goal of This Research

Show that multiple different manipulators possess the same desired local properties described by a Jacobian (designed to be optimally fault-tolerant) Study optimally fault tolerant Jacobians for different task space dimensions Illustrate the difference between these manipulators in terms

f their global fault tolerant properties

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

3DOF Planar Manipulators

The total possible Jacobians are 48 (23 × 3!) All of these Jacobians correspond to only 4 different manipulators Link lengths:

Robot a1 a2 a3 1 Ls Ls Ls 2 Ls Ll Ls 3 Ll Ls Ls 4 Ll Ll Ls Ls =

2/3 and Ll =

√ 2

2 3

2
3
2
3

2 3 1

1

3 2 2 ) 3 2 , ( ) 3 2 , ( −

(a) (b)

3 2

x y [ j

j ✁ j ✂ ]

[ j

j ✂ j ✁ ]

[ j

j ✁ j ✂ ]

[ j

j ✂ j ✁]

2 3

2
3
2
3

2 3 1

1

3 2 2 ) 3 2 , ( ) 3 2 , ( − 3 2

x y [ j

✄ j

✁ j ✂ ]

[ j

✄ j

✂ j ✁]

[ j
j

✁ j ✂ ]

[ j
j

✂ j ✁ ]

(c)

2 3

2
3
2
3

2 3 1

1

3 2 2 ) 3 2 , ( ) 3 2 , ( − 3 2

x y [ j

j ✁ ✄ j ✂ ]

[ j

j ✂ -j ✁]

[ j

j ✁ ✄ j ✂ ]

[ j

j ✂ -j ✁]

(d)

2 3

2
3
2
3

2 3 1

1

3 2 2 ) 3 2 , ( ) 3 2 , ( − 3 2

x y [ j

j

✁ ✄ j ✂ ]

[ j

j

✂ -j ✁ ]

[ j
j

✁ ✄ j ✂ ]

[ j
j

✂ -j ✁ ]

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

3DOF Planar Manipulators

The total possible Jacobians are 48 (23 × 3!) All of these Jacobians correspond to only 4 different manipulators Link lengths:

Robot a1 a2 a3 1 Ls Ls Ls 2 Ls Ll Ls 3 Ll Ls Ls 4 Ll Ll Ls Ls =

2/3 and Ll =

√ 2

(a)

x

y [ j

j ✁ j ✂]

[ j

j ✂ j ✁ ]

[ j

j ✁ j ✂]

[ j

j ✂ j ✁ ]

(b) (c)

2
1

2 3

3
2

2 1 1

1
3

3 2 2

) 4 3 2 , 2 2 (− ) 4 3 2 , 2 2 ( −

3 3 2

x

y [ j

j ✂ -j ✁ ]

[ j

j

✁ j ✂]

[ j

j ✂ -j ✁ ]

[ j
j

✁ j ✂]

2
1

2 3

3
2

2 1 1

1
3

3 2 2

) 4 3 2 , 2 2 (− ) 4 3 2 , 2 2 ( −

3 3 2

x

y [ j

j

✂ j ✁ ]

[ j

j ✁ -j ✂]

[ j
j

✂ j ✁ ]

[ j

j ✁ -j ✂]

2
1

2 3

3
2

2 1 1

1
3

3 2 2

) 4 3 2 , 2 2 (− ) 4 3 2 , 2 2 ( −

3 3 2

(d)

x

y [ j

j

✂ -j ✁ ]

[ j

j

✁ -j ✂]

[ j
j

✂ -j ✁ ]

[ j
j

✁ -j ✂]

2
1

2 3

3
2

2 1 1

1
3

3 2 2

) 4 3 2 , 2 2 (− ) 4 3 2 , 2 2 ( −

3 3 2

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

3DOF Planar Manipulators

The total possible Jacobians are 48 (23 × 3!) All of these Jacobians correspond to only 4 different manipulators Link lengths:

Robot a1 a2 a3 1 Ls Ls Ls 2 Ls Ll Ls 3 Ll Ls Ls 4 Ll Ll Ls Ls =

2/3 and Ll =

√ 2

1 3

3
3

3

2
1
2

2

1

1 3 2 2

) 4 3 2 , 2 2 ( − − ) 4 3 2 , 2 2 (

(a)

x

y

3 2

[ j

j ✁ j ✂]

[ j

j ✂ j ✁]

[ j

j ✁ j ✂]

[ j

j ✂ j ✁]

1 3

3
3

3

2
1
2

2

1

1 3 2 2

) 4 3 2 , 2 2 ( − − ) 4 3 2 , 2 2 (

(b)

x

y

3 2

[ j

j ✁ -j ✂]

[ j

j

✂ j ✁]

[ j

j ✁ ✄ j ✂]

[ j
✄ j

✂ j ✁ ]

2 1

☎

3
3

3

✆ ✝

1
2

2 1 3 2 2

) 4 3 2 , 2 2 ( − − ) 4 3 2 , 2 2 (

(c)

x

y

3 2

[ j

j

✁ j ✂]

[ j

j ✂ -j ✁]

[ j
j

✁ j ✂]

[ j

j ✂ -j ✁]

2

1

1 3

3
3

3

2
1
2

✝

1 3 2 2

) 4 3 2 , 2 2 ( − − ) 4 3 2 , 2 2 (

(d)

x

y

3 2

[ j

j✁ -j

✂]

[ j

j

✂ -j ✁]

[ j
j✁ -j

✂]

[ j
j

✂ -j ✁]

2

1

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

3DOF Planar Manipulators

The total possible Jacobians are 48 (23 × 3!) All of these Jacobians correspond to only 4 different manipulators Link lengths:

Robot a1 a2 a3 1 Ls Ls Ls 2 Ls Ll Ls 3 Ll Ls Ls 4 Ll Ll Ls Ls =

2/3 and Ll =

√ 2

120° 2 3

2
3

2 3

2
3

1

(a) Robot 1 (b) Robot 2

60° 150° 150°

60°
120°

2 3

2
3

2 3

2
3

1 1

(c) Robot 3 (d) Robot 4

150° 150°

150°

60°

120°
150°

3 2 2 ) 3 2 , ( 150° 3 2

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Global Workspace Properties

There different workspace boundaries:

(2Ll + Ls) (Ll + 2Ls) 3Ls

Determine the maximum value of K as a function of distance from the base

K is not a function of θ1 There is a rotational symmetry around the base x-axis trajectory was selected

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Finding Maximum K

Use homogenous solution (Null Motion) ˙ θ = J+ ˙ x + αˆ nJ

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Finding Maximum K

Use homogenous solution (Null Motion) ˙ θ = J+ ˙ x + αˆ nJ

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Results

0.5 1 1.5 2 2.5 3 3.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Distance from Base

K

Robot 1 : [Ls,Ls,Ls] Robot 2 : [Ls,Ll,Ls] Robot 3 : [Ll,Ls,Ls] Robot 4 : [Ll,Ll,Ls]

Robot4 has a wide range of K larger than the optimal value Robot1 has only a peak at the optimal design point Robot2 has a flat region in the middle of its workspace. Robot3 has a significant dip in the maximum value of K at a distance near one unit from the base before it returns to a comparable value to that of Robot2.

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Optimally 2 × 4 Fault Tolerant Jacobian (Two Failures)

J = [ j1 j2 j3 j4 ] =

1

√ 2 1 2

− 1

2 1 2 1 √ 2 1 2

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

14 Optimal Fault Tolerant Planar 4R Manipulators

Robot a1 a2 a3 a4 1/9 La/Ld La La Lb 2/10 La/Ld La Ld Lb 3/11 La/Ld Lc La Lb 4/12 La/Ld Lc Ld Lb 5/13 La/Ld Ld La Lb 6/14 La/Ld Ld Ld Lb 7 Lc La Lc Lb 8 Lc Ld Lc Lb

La =

1 −

1 √ 2, Lb = 1 √ 2, Lc = 1,

and Ld =

1 +

1 √ 2

3 2 1 4

1
4
2
3

2 1 2 1 1 1

2 1 1

3 2 4

4
2
3

1

1

(a) Robot 1 (e) Robot 5

2 4

4
2
3

1

1

3

(b) Robot 2

3 2 4

4
2
3

1

1

(d) Robot 4

3 2

1

4 1

4
2
3

(c) Robot 3

3 2 4

4
2
3
1

1

(g) Robot 7

3 2 1 4

1
4
2
3

(i) Robot 9

3 2 4

4
2
3

1

1

(f) Robot 6

3 2 4

4
2
3

1

1

(h) Robot 8 17/45

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

14 Optimal Fault Tolerant Planar 4R Manipulators

Robot a1 a2 a3 a4 1/9 La/Ld La La Lb 2/10 La/Ld La Ld Lb 3/11 La/Ld Lc La Lb 4/12 La/Ld Lc Ld Lb 5/13 La/Ld Ld La Lb 6/14 La/Ld Ld Ld Lb 7 Lc La Lc Lb 8 Lc Ld Lc Lb

La =

1 −

1 √ 2, Lb = 1 √ 2, Lc = 1,

and Ld =

1 +

1 √ 2

3 2 4

4
2
3

1

1

(n) Robot 14

3 2 4

4
2
3

1

1

(l) Robot 12

3 2 1 4

1
4
2
3

(j) Robot 10

3 2 4

4
2
3
1

1

(m) Robot 13

3 2 4

4
2
1

1

(k) Robot 11

3

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Results

0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Distance from Base

K Robot 1 : [La, La, La, Lb] Robot 14: [Ld, Ld, Ld, Lb]

Robot1 has only a peak at the optimal design point Robot14 has a wide range of K larger than the optimal value for 80% of its total workspace

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Published Papers
K. M. Ben-Gharbia, A. A. Maciejewski, and R. G. Roberts, “An illustration of

generating robots from optimal fault-tolerant Jacobians,” 15th IASTED International Conference on Robotics and Applications, pp. 453–460, Cambridge, MA, Nov. 1–3, 2010.

K. M. Ben-Gharbia, R. G. Roberts, and A. A. Maciejewski, “Examples of planar

robot kinematic designs from optimally fault-tolerant Jacobians,” IEEE International Conference on Robotics and Automation, pp. 4710–4715, Shanghai, China, May 9–13, 2011.

K. M. Ben-Gharbia, A. A. Maciejewski, and R. G. Roberts, “A kinematic

analysis and evaluation of planar robots designed from optimally fault-tolerant Jacobians,” IEEE Transactions on Robotics, Vol. 30, No. 2, pp. 516–524, April 2014.

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Spatial Manipulators

The geometry becomes more complicated Denavit and Hartenberg (DH) parameters are used to describe the kinematic, parameters including link lengths

1

ai: Link length

2

αi: Link twist

3

di: Joint offset

4

θi: Joint value

Any spatial Jacobian consisting all rotational joints is represented by a 6 × n matrix, s.t. ji = vi ωi

=

zi−1 × pi−1 zi−1

Determining the DH parameters from a Jacobian becomes

harder than the case for a planar manipulator Computing a maximum reach and a total workspace volume is not simple either

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Spatial Manipulators

The geometry becomes more complicated Denavit and Hartenberg (DH) parameters are used to describe the kinematic, parameters including link lengths

1

ai: Link length

2

αi: Link twist

3

di: Joint offset

4

θi: Joint value

Any spatial Jacobian consisting all rotational joints is represented by a 6 × n matrix, s.t. ji = vi ωi

=

zi−1 × pi−1 zi−1

Determining the DH parameters from a Jacobian becomes

harder than the case for a planar manipulator Computing a maximum reach and a total workspace volume is not simple either

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Spatial Manipulators

The geometry becomes more complicated Denavit and Hartenberg (DH) parameters are used to describe the kinematic, parameters including link lengths

1

ai: Link length

2

αi: Link twist

3

di: Joint offset

4

θi: Joint value

Any spatial Jacobian consisting all rotational joints is represented by a 6 × n matrix, s.t. ji = vi ωi

=

zi−1 × pi−1 zi−1

Determining the DH parameters from a Jacobian becomes

harder than the case for a planar manipulator Computing a maximum reach and a total workspace volume is not simple either

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Spatial Manipulators

The geometry becomes more complicated Denavit and Hartenberg (DH) parameters are used to describe the kinematic, parameters including link lengths

1

ai: Link length

2

αi: Link twist

3

di: Joint offset

4

θi: Joint value

Any spatial Jacobian consisting all rotational joints is represented by a 6 × n matrix, s.t. ji = vi ωi

=

zi−1 × pi−1 zi−1

Determining the DH parameters from a Jacobian becomes

harder than the case for a planar manipulator Computing a maximum reach and a total workspace volume is not simple either

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

3 × 4 Optimally Fault Tolerant Jacobian

J = [ j1 j2 j3 j4 ] =      −

3

4

1

12

1

12

1

12

−

2

3

1

6

1

6

−

1

2

1

2

    

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Generating Physical Robots

Rotation axes are not defined by only the 3 × 4 J By presenting J as J6×4 =

Jv

Jω

There is some freedom in selecting Jω =

⇒ thus, having different robots Each ji =

vi

ωi

has

ωi is orthogonal to vi ωi is normalized = ⇒ ωi = f (βi) = ⇒ DH = f (β)

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Global Measurements

Maximum reach The fraction of the total workspace that is fault tolerant, denoted WK (K ≥ γKopt , 0 < γ ≤ 1)

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Monte Carlo Integration Algorithm

x2 x1 x3

ne million

random points boundary sphere for Monte Carlo approximation unknown true boundary R is 110% maximum reach nr n

10,000 uniformly distributed random samples within a sphere

f radius of 110% of the maximum reach are used

WK = nf /nr One million uniformly distributed random samples are generated in the joint space The maximum reach is computed from the largest norm

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Monte Carlo Integration Algorithm

x2 x1 x3

ne million

random points boundary sphere for Monte Carlo approximation fault tolerant workspace boundary unknown true boundary R is 110% maximum reach nf nr n

10,000 uniformly distributed random samples within a sphere

f radius of 110% of the maximum reach are used

WK = nf /nr One million uniformly distributed random samples are generated in the joint space The maximum reach is computed from the largest norm

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Monte Carlo Integration Algorithm

x2 x1 x3

ne million

random points boundary sphere for Monte Carlo approximation fault tolerant workspace boundary unknown true boundary R is 110% maximum reach nf nr n

10,000 uniformly distributed random samples within a sphere

f radius of 110% of the maximum reach are used

WK = nf /nr One million uniformly distributed random samples are generated in the joint space The maximum reach is computed from the largest norm

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Monte Carlo Integration Algorithm

x2 x1 x3

ne million

random points boundary sphere for Monte Carlo approximation range of forward kinematics inverse kinematics to find maximum reach in this direction fault tolerant workspace boundary unknown true boundary R is 110% maximum reach max ||x||

θ1 θ4 θ2 θ3

forward kinematics

10,000 uniformly distributed random samples within a sphere

f radius of 110% of the maximum reach are used

WK = nf /nr One million uniformly distributed random samples are generated in the joint space The maximum reach is computed from the largest norm

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Multiple Self-Motion Manifolds

−400 −200 200 −100 100 200 300 400 −100 −50 50 100 150 200 250 300 350 θ3 θ2 θ4 −100 −50 50 100 150 C D A B θ1

Close up of boundary calculation Unknown true fault tolerant boundary θ1 θ4 θ3 θ2 The largest K on this manifold is < γ Kmax The largest K on this manifold is > γ Kmax

(a) (b)

Some of the points have multiple self-motion manifolds On one manifold, K ≥ γKopt 5 joint configurations (whose locations are close to the point) are selected to increase the probability of K ≥ γKopt Computationally expensive, i.e., 10-30 minutes/robot

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Multiple Self-Motion Manifolds

−400 −200 200 −100 100 200 300 400 −100 −50 50 100 150 200 250 300 350 θ3 θ2 θ4 −100 −50 50 100 150 C D A B θ1

Close up of boundary calculation Unknown true fault tolerant boundary θ1 θ4 θ3 θ2 The largest K on this manifold is < γ Kmax The largest K on this manifold is > γ Kmax

(a) (b)

Some of the points have multiple self-motion manifolds On one manifold, K ≥ γKopt 5 joint configurations (whose locations are close to the point) are selected to increase the probability of K ≥ γKopt Computationally expensive, i.e., 10-30 minutes/robot

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Examples of Manipulators with Common Link Twist

Parameters

There is a design flexibility within this entire family of manipulators as DH = f (β) Setting αi’s to ±90◦, 0◦, or 180◦ is common in many commercial manipulators Recall that the parameter αi is defined as the angle between the rotation axes of joints i and i + 1, which is the same as ωi and ωi+1

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Manipulator Categories

Robot Group Relationship between joint axes i − 1 and i i = (1, 2, 3, 4) Size of Group 1 (, , , ) 2 (, , ⊥, ) 3 (⊥, , , ) 4 (, ⊥, , ) 8 5 (, ⊥, ⊥, ) 8 6 (⊥, , ⊥, ) 8 7 (⊥, ⊥, , ) 8 8 (⊥, ⊥, ⊥, ) ∞

Best out of each group:

WK [%] max reach [m] 59 5.19 30 4.96 67 3.92 48 3.93 75 5.50

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Manipulator Categories

Robot Group Relationship between joint axes i − 1 and i i = (1, 2, 3, 4) Size of Group 1 (, , , ) 2 (, , ⊥, ) 3 (⊥, , , ) 4 (, ⊥, , ) 8 5 (, ⊥, ⊥, ) 8 6 (⊥, , ⊥, ) 8 7 (⊥, ⊥, , ) 8 8 (⊥, ⊥, ⊥, ) ∞

Best out of each group:

WK [%] max reach [m] 59 5.19 30 4.96 67 3.92 48 3.93 75 5.50

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Manipulator Categories

Robot Group Relationship between joint axes i − 1 and i i = (1, 2, 3, 4) Size of Group 1 (, , , ) 2 (, , ⊥, ) 3 (⊥, , , ) 4 (, ⊥, , ) 8 5 (, ⊥, ⊥, ) 8 6 (⊥, , ⊥, ) 8 7 (⊥, ⊥, , ) 8 8 (⊥, ⊥, ⊥, ) ∞

Best out of each group:

WK [%] max reach [m] 59 5.19 30 4.96 67 3.92 48 3.93 75 5.50

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The Best Manipulator

DH parameters:

i αi [degrees] ai [m] di [m] θi [degrees] 1 90 √ 2 2

90

√ 2 1 180 3 90 √ 2

1

180 4 √ 3/2 1/2 145 28/45

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Columns Permutation and/or Sign Change Effect

Sign change for very column is equivalent to reversing the direction of the corresponding joint axis A permutation is only equivalent to either a rotation or a reflection of the original Jacobian

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Published Papers
K. M. Ben-Gharbia, A. A. Maciejewski, and R. G. Roberts, “Examples of

spatial positioning redundant robotic manipulators that are optimally fault tolerant,” IEEE International Conference on Systems, Man, and Cybernetics,

pp. 1526–1531, Anchorage, Alaska, Oct. 9–12, 2011.
K. M. Ben-Gharbia, A. A. Maciejewski, and R. G. Roberts, “Kinematic design
f redundant robotic manipulators for spatial positioning that are optimally

fault tolerant,” IEEE Transactions on Robotics, Vol. 29, No. 5, pp. 1300–1307, Oct. 2013.

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6 × 7 Optimally Fault Tolerant Jacobian

J =               −

6

7

1

42

1

42

1

42

1

42

1

42

1

42

−

5

6

1

30

1

30

1

30

1

30

1

30

−

4

5

1

20

1

20

1

20

1

20

−

3

4

1

12

1

12

1

12

−

2

3

1

6

1

6

−

1

2

1

2

              Unfortunately, it is not feasible to identify directly a rotational manipulator from this canonical J

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6 × 7 Optimally Fault Tolerant Jacobian

J =               −

6

7

1

42

1

42

1

42

1

42

1

42

1

42

−

5

6

1

30

1

30

1

30

1

30

1

30

−

4

5

1

20

1

20

1

20

1

20

−

3

4

1

12

1

12

1

12

−

2

3

1

6

1

6

−

1

2

1

2

              Unfortunately, it is not feasible to identify directly a rotational manipulator from this canonical J

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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions

Feasible 7R Jacobian

ji =

vi

ωi

=

       

cos(γi)   cos(αi) sin(βi) sin(αi) sin(βi) − cos(βi)   + sin(γi)   sin(αi) − cos(αi)     cos(αi) cos(βi) sin(αi) cos(βi) sin(βi)  

       

α γ ω z y x β

i i i i

vi

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Two Approaches

No exact solution was found Two approaches were used

1

Solving for 21 nonlinear equations that correspond to the constraints of the optimally fault tolerant Jacobian

2

Solving directly for equal and maximum f σ6 and equal magnitudes for the null vector elements

Each approach’s objective functions converges to the same values

Theoretical values 21 equations Direct optimization for K K 0.5774 0.5196 0.5714 σ1 1.5275 1.5829 1.6455 σ6 1.5275 1.4726 1.4169 |ˆ nJi |’s equal Not equal equal

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6D Volume

When looking to a 3D volume, it can be approximated by hi · ∆Ai Similarly, a 6D volume can be described as each point within a 3D workspace has its own υoi By integrating for the whole 6D volume we get: V6d ≈

Nr

i=1

∆υr · υoi = Vr Nr

Nr

i=1

υoi

Δ

}h

A i

i

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6D Volume

When looking to a 3D volume, it can be approximated by hi · ∆Ai Similarly, a 6D volume can be described as each point within a 3D workspace has its own υoi By integrating for the whole 6D volume we get: V6d ≈

Nr

i=1

∆υr · υoi = Vr Nr

Nr

i=1

υoi

Δ

}h

A i

i

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6D Volume

When looking to a 3D volume, it can be approximated by hi · ∆Ai Similarly, a 6D volume can be described as each point within a 3D workspace has its own υoi By integrating for the whole 6D volume we get: V6d ≈

Nr

i=1

∆υr · υoi = Vr Nr

Nr

i=1

υoi

υ

Δυ

r

i

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6D Volume

When looking to a 3D volume, it can be approximated by hi · ∆Ai Similarly, a 6D volume can be described as each point within a 3D workspace has its own υoi By integrating for the whole 6D volume we get: V6d ≈

Nr

i=1

∆υr · υoi = Vr Nr

Nr

i=1

υoi

υ

Δυ

r

i

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Orientation Volume

Unit quaternions q were used: q = [s , v] =

cos(θ/2) , sin(θ/2)ˆ

k

θ k

z y x

^

All of unit quaternions lay on 4D sphere surface (surface = 2π2) Only the half of quaternions are needed to represent uniquely all possible orientations By uniformly sampling on the half of 4D sphere surface, we get by using Monte Carlo integration: Voi ≈ Vomax Noi No = π2 Noi No .

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Orientation Volume

Unit quaternions q were used: q = [s , v] =

cos(θ/2) , sin(θ/2)ˆ

k

θ k

z y x

^

All of unit quaternions lay on 4D sphere surface (surface = 2π2) Only the half of quaternions are needed to represent uniquely all possible orientations By uniformly sampling on the half of 4D sphere surface, we get by using Monte Carlo integration: Voi ≈ Vomax Noi No = π2 Noi No .

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Orientation Volume

Unit quaternions q were used: q = [s , v] =

cos(θ/2) , sin(θ/2)ˆ

k

θ k

z y x

^

All of unit quaternions lay on 4D sphere surface (surface = 2π2) Only the half of quaternions are needed to represent uniquely all possible orientations By uniformly sampling on the half of 4D sphere surface, we get by using Monte Carlo integration: Voi ≈ Vomax Noi No = π2 Noi No .

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Max Reach and 3D Reachable Volume

x2 x1 x3

boundary sphere for Monte Carlo approximation range of forward kinematics inverse kinematics to find maximum reach in this direction unknown true boundary (found by performing positiong inverse kinematics) max ||x|| forward kinematics

ne million

random points

θ1 θ7 θ3 θ2 θ4 θ5 θ6

R is 110% maximum reach

Vr ≈ Nr N

Vrmax =

N3d N 4 3πR3

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Max Reach and 3D Reachable Volume

x2 x1 x3

boundary sphere for Monte Carlo approximation range of forward kinematics inverse kinematics to find maximum reach in this direction unknown true boundary (found by performing positiong inverse kinematics) max ||x|| forward kinematics

ne million

random points

θ1 θ7 θ3 θ2 θ4 θ5 θ6

R is 110% maximum reach

Vr ≈ Nr N

Vrmax =

N3d N 4 3πR3

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6D Reachable Volume

x2 x1 x3

total workspace boundary

φ ψ θ

rientation

volume of Pi calculate

rientation

volume

position (Pi)

boundary for Monte Carlo approximation

(c) (b) (a)

rientation

(Qi)

V6d ≈ Vr Nr

Nr

i=1

υoi

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6D Reachable Volume

x2 x1 x3

total workspace boundary

φ ψ θ

rientation

volume of Pi calculate

rientation

volume

position (Pi)

boundary for Monte Carlo approximation

(c) (b) (a)

rientation

(Qi)

V6d ≈ Vr Nr

Nr

i=1

υoi

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6D Fault Tolerant Volume

x2 x1 x3

total workspace boundary

φ ψ θ

rientation

volume of Pi calculate

rientation

volume

position (Pi)

boundary for Monte Carlo approximation

θ1 θ7 θ3 θ2 θ4 θ5 θ6 largest K on this manifold is greater than γ Kmax

largest K on these manifolds is less than γ Kmax

determining if this position and

rientation is

fault tolerant

fault tolerant workspace boundary (at least one orientation is fault tolerant)

(c) (b) (a)

rientation

(Qi)

VFT 3d ≈ NFT 3d N 4 3πR3 VFT ≈ VFT3d NFT3d

NFT3d

i=1

VFToi

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6D Fault Tolerant Volume

x2 x1 x3

total workspace boundary

φ ψ θ

rientation

volume of Pi calculate

rientation

volume

position (Pi)

boundary for Monte Carlo approximation

θ1 θ7 θ3 θ2 θ4 θ5 θ6 largest K on this manifold is greater than γ Kmax

largest K on these manifolds is less than γ Kmax

determining if this position and

rientation is

fault tolerant

fault tolerant workspace boundary (at least one orientation is fault tolerant)

(c) (b) (a)

rientation

(Qi)

VFT 3d ≈ NFT 3d N 4 3πR3 VFT ≈ VFT3d NFT3d

NFT3d

i=1

VFToi

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Maximizing K

Searching in all self-motion manifolds Optimizing locally using ∇K Least norm solution ˙ θ = J+ ˙ x+αˆ nJ∇K

−6 −4 −2 2 4 6 8 0.1 0.2 0.3 0.4 0.5 0.6 Distance from design point [m]

K

−6 −4 −2 2 4 6 8 1 2 3 4 5 6 Distance from design point [m] |∆ θ| [rad]

Self Motion Self Motion

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Maximizing K

Searching in all self-motion manifolds Optimizing locally using ∇K Least norm solution ˙ θ = J+ ˙ x+αˆ nJ∇K

−6 −4 −2 2 4 6 8 0.1 0.2 0.3 0.4 0.5 0.6 Distance from design point [m]

K

−6 −4 −2 2 4 6 8 1 2 3 4 5 6 Distance from design point [m] |∆ θ| [rad]

Self Motion ∇K Self Motion ∇K

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Maximizing K

Searching in all self-motion manifolds Optimizing locally using ∇K Least norm solution ˙ θ = J+ ˙ x+αˆ nJ∇K

−6 −4 −2 2 4 6 8 0.1 0.2 0.3 0.4 0.5 0.6 Distance from design point [m]

K

−6 −4 −2 2 4 6 8 1 2 3 4 5 6 Distance from design point [m] |∆ θ| [rad]

Self Motion ∇K Least Norm Self Motion ∇K Least Norm

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Columns Permutation

There are 7! Jacobians All of different found feasible solutions were evaluated to be almost equivalent All of different significant designs may be found using

nly one feasible J with its 7!

permutations

1000 2000 3000 4000 5000 6000 5 6 7 8 9 10 11 12 13 permutation index max reach J1 J2

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Examples

min Robot of J1 mid Robot of J1 Robots from J1 Robots from J2 Max reach Rmax [m] Max reach Rmax [m] min mid max min mid max 6.0 9.4 12.1 5.9 8.7 11.7 Volumes[%] Vr 99 96 62 97 95 68 VFT3d 81 81 51 81 73 59 V6d 49 46 26 49 41 28 VFT 27 24 12 26 19 13 41/45

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Examples

min Robot of J1 mid Robot of J1 Robots from J1 Robots from J2 K ≥ γKopt Max reach Rmax [m] Max reach Rmax [m] , γ ≈ 0.4 min mid max min mid max 6.0 9.4 12.1 5.9 8.7 11.7 Volumes[%] Vr 99 96 62 97 95 68 VFT3d 81 81 51 81 73 59 V6d 49 46 26 49 41 28 VFT 27 24 12 26 19 13 41/45

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Examples

min Robot of J1 mid Robot of J1 Robots from J1 Robots from J2 K ≥ γKopt Max reach Rmax [m] Max reach Rmax [m] , γ ≈ 0.4 min mid max min mid max 6.0 9.4 12.1 5.9 8.7 11.7 Volumes[%] Vr 99 96 62 97 95 68 VFT3d 81 81 51 81 73 59 V6d 49 46 26 49 41 28 VFT 27 24 12 26 19 13 41/45

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6D Volumes Plots

6D reachable volume of min Robot for J1 6D FT volume of min Robot for J1 42/45

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Published Papers
K. M. Ben-Gharbia, A. A. Maciejewski, and R. G. Roberts, “An Example of a

Seven Joint Manipulator Optimized for Kinematic Fault Tolerance,” accepted to appear in IEEE International Conference on Systems, Man, and Cybernetics,

pp. XXX–XXX, San Diego, CA, Oct. 5–8, 2014.

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Conclusions

Summary

There are multiple different robot designs that possess the same desired optimal Jacobian Global properties are different More optimal robot choices for designers

Future directions

Characterize and count self-motion manifolds Explore all of optimal 7R manipulators Study 8R optimally fault tolerant Jacobians

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Thanks!

?

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