kinematic design of redundant robotic manipulators that ... · robot3 has a signi cant dip in the...

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •

Kinematic Design of Redundant RoboticManipulators that are Optimally Fault

Tolerant

Presented by:Khaled M. Ben-Gharbia

January 15th, 2013

1/40


Why Fault Tolerance?

Applications

hazardous waste cleanupspace/underwaterexplorationanywhere failures are likely orintervention is costly

Common failure mode

locked actuators

2/40


Simple Redundant Robot Planar 3DOF

[x1x2

]= J(θ)

θ1θ2θ3

3/40


Jacobian (J) Matrix

3R planar manipulatorForward 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

J = [j1 j2 j3] = [z1 × p1 z2 × p2 z3 × p3]

where zi is the rotation axis, andhere it is [0 0 1]T

Spatial manipulator

ji =

[viωi

]

θ1+θ2 + θ3

x1

x2

θ1

θ1+ 2θ

4/40


Jacobian (J) Matrix

3R planar manipulatorForward 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

J = [j1 j2 j3] = [z1 × p1 z2 × p2 z3 × p3]

where zi is the rotation axis, andhere it is [0 0 1]T

Spatial manipulator

ji =

[viωi

]

θ1+θ2 + θ3

x1

x2

p2

j2

θ1

θ1+ 2θ

4/40


Kinematic Dexterity Measures:Function of Singular Values of the Jacobian

V TV = I D =

[σ1 0 00 σ2 0

]U =

[cosφ − sinφsinφ cosφ

]

5/40


Approaching Kinematic Singularities

Singularity: σ2 = 0

Value of σm is a common dexterity measure

6/40


Optimal Fault Tolerant Configurationsfor 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=1f σi

f uif vTi

Worst-case remaining dexterity: K = minnf=1

f σm

Isotropic & Optimal J: K = σ√

n−mn for all f

7/40


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 ] =

−√23

√16

√16

0 −√

12

√12

Null space equally distributed: nJ =

[ √13

√13

√13

]

8/40


Example: 2× 3 Optimally Fault Tolerant Jacobian

120°

120°

120°

j3

j1

j2

Remaining dexterity: K = f σ2 =√

13

9/40


Example: 2× 3 Optimally Fault Tolerant Jacobian

120°

120°

120°

j3

p2

p3

j1

j2

p1

Remaining dexterity: K = f σ2 =√

13

9/40


Different Optimally Fault Tolerant Jacobians

Sign change and permuting columns do not affect the isotropyand the optimally fault tolerant properties of J, e.g.

But each new J may belong to a different manipulator

10/40


Goal of This Research

Show that multiple different manipulators possess the samedesired local properties described by a Jacobian (designed tobe optimally fault-tolerant)

Study optimally fault tolerant Jacobians for different taskspace dimensions

Illustrate the difference between these manipulators in termsof their global fault tolerant properties

11/40


3DOF Planar Manipulators

The total possible Jacobiansare 48 (23 × 3!)

All of these Jacobianscorresponds to only 4different manipulators

Link lengths:

Robot a1 a2 a31 Ls Ls Ls

2 Ls Ll Ls

3 Ll Ls Ls

4 Ll Ll Ls

Ls =√

2/3 and Ll =√2

23

-2-3

-2 -3

23

1

-1

32

2

)32,0(

)32,0( −

(a) (b)

32

x

y

[ j� j� j� ]

[ j� j� j� ]

-[ j� j� j� ]

-[ j� j� j� ]

23

-2-3

-2 -3

23

1

-1

32

2

)32,0(

)32,0( −

32

x

y

[ j� � j� j� ]

[ j� � j� j� ]

-[ j� -j� j� ]

-[ j� -j� j� ]

(c)

23

-2-3

-2 -3

23

1

-1

32

2

)32,0(

)32,0( −

32

x

y

[ j� j� � j� ]

[ j� j� -j� ]

-[ j� j� � j� ]

-[ j� j� -j� ](d)

23

-2-3

-2 -3

23

1

-1

32

2

)32,0(

)32,0( −

32

x

y

[ j� -j� � j� ]

[ j� -j� -j� ]

-[ j� -j� � j� ]

-[ j� -j� -j� ]

Why 4?

12/40





Link lengths:


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

21

1-1

-3

32

2

)4

32

,2

2(−

)4

32

,2

2( −

3

32 x

y [ j� j� -j� ]

[ j� -j� j� ]

-[ j� j� -j� ]

-[ j� -j� j� ]

-2 -1

2

3

-3

-2

21

1-1

-3

32

2

)4

32

,2

2(−

)4

32

,2

2( −

3

32

x

y [ j� -j� j� ]

[ j� j� -j� ]

-[ j� -j� j� ]

-[ j� j� -j� ]

-2 -1

2

3

-3

-2

21

1-1

-3

32

2

)4

32

,2

2(−

)4

32

,2

2( −

3

32

(d)

x

y [ j� -j� -j� ]

[ j� -j� -j� ]

-[ j� -j� -j� ]

-[ j� -j� -j� ]

-2 -1

2

3

-3

-2

21

1-1

-3

32

2

)4

32

,2

2(−

)4

32

,2

2( −

3

32

Why 4?

12/40





Link lengths:


2 Ls Ll Ls

3 Ll Ls Ls

4 Ll Ll Ls

Ls =√

2/3 and Ll =√2

13

-3

-3

3

-2

-1

-2

2

-1

13

2

2

)4

32

,2

2(

−−

)4

32

,2

2(

(a)

x

y

32

[ j� j� j� ]

[ j� j� j� ]

-[ j� j� j� ]

-[ j� j� j� ]

13

-3

-3

3

-2

-1

-2

2

-1

1

32

2

)4

32

,2

2(

−−

)4

32

,2

2(

(b)

x

y

32

[ j� j� -j� ]

[ j� -j� j� ]

-[ j� j� � j� ]

-[ j� � j� j� ]

2

1�

-3

-3

3

��-1

-2

2

1

32

2

)4

32

,2

2(

−−

)4

32

,2

2(

(c)

x

y

32

[ j� -j� j� ]

[ j� j� -j� ]

-[ j� -j� j� ]

-[ j� j� -j� ]

2

-1

13

-3

-3

3

-2

-1

-2

� 1

32

2

)4

32

,2

2(

−−

)4

32

,2

2(

(d)

x

y

32

[ j� -j� -j� ]

[ j� -j� -j� ]

-[ j� -j� -j� ]

-[ j� -j� -j� ]

2

-1

Why 4?

12/40





Link lengths:


2 Ls Ll Ls

3 Ll Ls Ls

4 Ll Ll Ls

Ls =√

2/3 and Ll =√2

120°

23

-2 -3

23

-2 -3

1

1

(a) Robot 1 (b) Robot 2

60°

150°

150°

-60°

-120° 23

-2 -3

23

-2 -3

1

1

(c) Robot 3 (d) Robot 4

150°

150°-150°

60°

-120°

-150°

32

2

)32,0(

150°

32

Why 4?

12/40


Different Robots Using the Gram MatrixGram matrix is:

G = JT J= I − NNT when J is isotropic

N is a matrix consisting of orthonormal null vectors of JN = nJ when n −m = 1If J ′ is a rotation and/or reflection of J, then J ′T J ′ = JT J;they both belong to the same manipulator

Replacing nJ with −nJ doesn’t affect G

Only the 4 cases nJ =√

1/3[1, ± 1, ± 1] determine fourfamilies of non-equivalent Jacobians

With nJ =√

1/3[1, 1, 1] the Gram matrix is:

G =

23 −1

3 −13

−13

23 −1

3

−13 −1

323

a2i = gii + gi+1,i+1 − gi ,i+1 (and a2n = gnn)

13/40


Global Workspace properties

There different workspace boundaries:

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

Determine the maximum value of K as a function of distancefrom the base

K is not a function of θ1There is a symmetry while rotating around the basex-axis trajectory was selected

14/40


Finding Maximum K

Use homogenous solution (Null Motion)

θ = J+x + αnJ

15/40


Results

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Distance from Base

K

Robot 1 : [L

s,L

s,L

s]

Robot 2 : [Ls,L

l,L

s]

Robot 3 : [Ll,L

s,L

s]

Robot 4 : [Ll,L

l,L

s]

Robot4 has a wide range of Klarger than the optimal value

Robot1 has only a peak atthe optimal design point

Robot2 has a flat region inthe middle of its workspace.

Robot3 has a significant dipin the maximum value of Kat a distance near one unitfrom the base before itreturns to a comparable valueto that of Robot2.

16/40


Cited Papers

K. M. Ben-Gharbia, A. A. Maciejewski, and R. G. Roberts, “An illustration ofgenerating robots from optimal fault-tolerant Jacobians,” 15th IASTEDInternational 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 planarrobot kinematic designs from optimally fault-tolerant Jacobians,” IEEEInternational 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 Trans. Robot., (under review).

17/40


Optimally 2× 4 Fault Tolerant Jacobian (Two Failures)

A 2× 4 optimally fault tolerant Jacobian is:

J =

[ 1√2

12 0 −1

2

0 12

1√2

12

].

N has two orthonormal vectors, and each row norm is√

1/2

The corresponding Gram matrix:

G =

12

1

2√2

0 −1

2√2

1

2√2

12

1

2√2

0

0 1

2√2

12

1

2√2

−1

2√2

0 1

2√2

12

.The superdiagonal can be exactly one of three forms:(± 1

2√2,± 1

2√2,± 1

2√2

), (± 12√2, 0,± 1

2√2

), or (0,± 12√2, 0)

Thus the total number of different manipulators is23 + 22 + 2 = 14

18/40


14 Optimal Fault Tolerant Planar 4R Manipulators

Robot a1 a2 a3 a41/9 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

21

21

1+

1

211−

3

2

4

-4

-2

-3

-1

1

3

2

4

-4

-2

-3

-1

1

19/40


Results

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Distance from Base

K

Robot 1 : [L

a, L

a, L

a, L

b]

Robot 14: [Ld, L

d, L

d, L

b]

Robot1 has only a peak atthe optimal design point

Robot14 has a wide range ofK larger than the optimalvalue for 80% of its totalworkspace

20/40


Comparison

Robot Reach[m]

Fault tolerantworkspace [%]

Jointmotion[◦/m]

1 2.33 0.00 −2 2.79 15.14/(0.25) 107.3

3 3.10 9.99/(0.07) 117.0

4 3.10 22.16/(1.31) 98.6

5 3.10 26.96/(0.73) 372.8

6 3.25 47.76 94.1

7 3.56 46.21/(0.04) 190.2

8 3.56 49.06/(0.10) 72.2

9 3.86 57.68 79.7

10 3.86 58.02 109.8

11 3.86 58.78 71.0

12 4.01 73.03 76.5

13 4.32 77.42 74.1

14 4.63 80.23 70.7

21/40


Cited Papers

K. M. Ben-Gharbia, R. G. Roberts, and A. A. Maciejewski, “Examples of planarrobot kinematic designs from optimally fault-tolerant Jacobians,” IEEEInternational 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 Trans. Robot., (under review).

22/40


Spatial Manipulators

The geometry becomes more complicatedDenavit and Hartenberg (DH) parameters are used to describethe kinematic, parameters including link lengths

ai : Link lengthαi : Link twistdi : Joint offsetθi : Joint value

Any spatial Jacobian is represented by a 6× n matrix, s.t.

ji =

[viωi

]=

[zi−1 × pi−1

zi−1

]Even for a spatial positioning task, we still need to define theorientational part of Jacobian

Computing a maximum reach and a total workspace volume isharder than the case for a planar manipulator

Moreover, it is harder to determine the DH parameters from aJacobian

23/40


Calculating DH Parameters from a Given Desired Jacobian

1ˆ −iziz

ix

1ˆ−ix

iy

1ˆ−iy

iaid

iθ

iα ilink1−ilink

1−ip

1+iω1+iv

ip

iv

iω

ji =

[viωi

]=

[zi−1 × pi−1

zi−1

]

xi = ± ωi×ωi+1

‖ωi×ωi+1‖xi is pointing away fromzi−1

pi−1 can’t be determineddirectly so

Use p′i−1 instead:p′i−1 = ωi × vi

Only the origins arerequired to determine di

24/40


3× 4 Optimally Fault Tolerant Jacobian

J = [ j1 j2 j3 j4 ] =

−√

34

√112

√112

√112

0 −√

23

√16

√16

0 0 −√

12

√12

Null space equally distributed: nJ = 12 [ 1 1 1 1 ]

Remaining dexterity: K = f σ3 = 12

25/40


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

26/40


Characterizing All 6× 4 Fault Tolerant Jacobians

Each ji =

[vi

ωi

]has

ωi is orthogonal to viωi is normalized=⇒ ωi = f (βi ) s.t.

ω1 =

0cos(β1)sin(β1)

, ω2 =

2√2

3 cos(β2)13 cos(β2)

sin(β2)

, ω3 =

2√2

3 sin(β3)

−(√

32 cos(β3) + 1

6 sin(β3))

− 12 cos(β3) +

√36 sin(β3)

,

ω4 =

2√2

3 sin(β4)

−(√

32 cos(β4) + 1

6 sin(β4))

−(− 1

2 cos(β4) +√36 sin(β4)

)

0◦ ≤ βi ≤ 360◦

All rotate within a circle centered at origin27/40


The Relationship Between DH Parameters and βi ’s

060

120180

240300

360

0

60

120

180

240

300

3600

30

60

90

120

150

180

β1β2

α 1

DH parameters areparameterized as:αi = fαi (βi , βi+1)

ai = fai (βi , βi+1)

θi = fθi

(βi−1, βi , βi+1)

di = fdi

(βi−1, βi , βi+1)

Because the 5th coordinateframe (tool) is arbitrary,

α4 = 0

a4 =√

3/2

θ4 = fθ4

(β3, β4)

d4 = fd4

(β3, β4)

θ1 = d1 = 0 arbitrarily28/40


Global Measurements

Maximum reach

The fraction of the total workspace that is fault tolerant,denoted WK (K ≥ γKopt)

29/40


Monte Carlo Integration Algorithm

z

x

y

1 million random

points

Boundary

sphere for

Monte Carlo

approximation

Range of forward

kinematics

Inv. Kinematics

to find max

reach in this

direction

Fault tolerant

workspace

boundary

Unknown true

boundary

r = 1.1*max reach

max ||.||

θ1

θ4

θ3

θ2

Forward

kinematics

One million uniformly distributed random samples aregenerated in the joint spaceThe maximum reach is computed from the largest norm10,000 uniformly distributed random samples within a sphereof radius of 110% of the maximum reach are usedWK = nf /nr

30/40


Multiple Self-Motion Manifolds

−400

−200

0

200

−1000100200300400

−100

−50

0

50

100

150

200

250

300

350

θ3

θ

2

θ 4

−100

−50

0

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 havemultiple self-motion manifolds

On one manifold, K ≥ γKopt

Missing being on that manifoldwould fail the test of havingthis point inside the faulttolerant workspace

5 joint configurations (whoselocations are close to thepoint) are selected to increasethe probability of K ≥ γKopt

Computationally expensive,i.e., 10-30 minutes/robot

31/40


Examples of Manipulators with Common Link TwistParameters

Setting αi ’s to ±90◦, 0◦, or 180◦ is common in manycommercial manipulators

Recall that the parameter αi is defined as the angle betweenthe rotation axes of joints i and i + 1, which is the same as ωi

and ωi+1

32/40


Link Twist αi = 0◦ or 180◦

ωi · ωi+1 = 1 when αi = 0◦

ωi · ωi+1 = −1 when αi = 180◦

This yields to discrete value of βi and βi+1

i αi [degrees] (βi , βi+1) [degrees]

10 (90, 90)

180 (90, 270)

20 (30, 120), (210, 300)

180 (30, 300), (210, 120)

30 (60, 60), (240, 240)

180 (60, 240), (240, 60)

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

x axisy axis

z ax

is

ω1

ω2

αi and αi+1 can’t be both 0◦ (or 180◦)

33/40


Link Twist αi = ±90◦

ωi · ωi+1 = 0

0 ≤ β1 < 180◦ (when 180 ≤ β1 < 360◦ the resulting robotsare mirrors), but β2 = f

β2(β1), β3 = f

β3(β2), and β4 = f

β4(β3)

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

x axisy axis

z ax

is

ω1

ω2

β1 = 0◦

34/40



ωi · ωi+1 = 0


β2(β1), β3 = f

β3(β2), and β4 = f

β4(β3)

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

x axisy axis

z ax

is

ω1

ω2

β1 = 10◦

34/40



ωi · ωi+1 = 0


β2(β1), β3 = f

β3(β2), and β4 = f

β4(β3)

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

x axisy axis

z ax

is

ω1

ω2

β1 = 30◦

34/40


Manipulator Categories

RobotGroup

Relationship betweenjoint axes i − 1 and i

i = (1, 2, 3, 4)

Size ofGroup

1 (‖, ‖, ‖, ‖) 0

2 (‖, ‖, ⊥, ‖) 0

3 (⊥, ‖, ‖, ‖) 0

4 (‖, ⊥, ‖, ‖) 8

5 (‖, ⊥, ⊥, ‖) 8

6 (⊥, ‖, ⊥, ‖) 8

7 (⊥, ⊥, ‖, ‖) 8

8 (⊥, ⊥, ⊥, ‖) ∞

Best out of eachgroup:

WK[%]

maxreach[m]

59 5.19

30 4.96

67 3.92

48 3.93

75 5.50

35/40


The Best Manipulator

DH parameters:i αi [degrees] ai [m] di [m] θi [degrees]

1 90√2 0 0

2 -90√2 1 180

3 90√2 -1 180

4 0√3/2 1/2 145

36/40


Columns Permutation and/or Sign Change Effect

Sign change for very column is equivalent to reversing thedirection of the corresponding joint axisA permutation is only equivalent to either a rotation or areflection of the original Jacobian (A regular tetrahedron wasuseful to describe all permutations)

37/40


Cited Papers

K. M. Ben-Gharbia, A. A. Maciejewski, and R. G. Roberts, “An illustration ofgenerating robots from optimal fault-tolerant Jacobians,” 15th IASTEDInternational Conference on Robotics and Applications, pp. 453–460,Cambridge, MA, Nov. 1–3, 2010.

K. M. Ben-Gharbia, A. A. Maciejewski, and R. G. Roberts, “Examples ofspatial positioning redundant robotic manipulators that are optimally faulttolerant,” 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

of Redundant Robotic Manipulators for Spatial Positioning that are Optimally

Fault Tolerant,” IEEE Trans. Robot., (under review).

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Conclusions

Summary

There are multiple different robot designs that possess thesame desired optimal Jacobian

Global properties are different

More optimal robot choices for designers

Future directions

Study the case of optimally fault tolerant Jacobians for a sixdimensional task space

Extend the Jacobian to DH parameters algorithm with anynumber of prismatic joints

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Thanks!

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kinematic design of redundant robotic manipulators that ... · robot3 has a signi cant dip in the...

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