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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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
Why Fault Tolerance?
Applications
hazardous waste cleanupspace/underwaterexplorationanywhere failures are likely orintervention is costly
Common failure mode
locked actuators
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
Simple Redundant Robot Planar 3DOF
[x1x2
]= J(θ)
θ1θ2θ3
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R 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 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R 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 ] =
−√23
√16
√16
0 −√
12
√12
Null space equally distributed: nJ =
[ √13
√13
√13
]
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
Example: 2× 3 Optimally Fault Tolerant Jacobian
120°
120°
120°
j3
j1
j2
Remaining dexterity: K = f σ2 =√
13
9/40
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
Example: 2× 3 Optimally Fault Tolerant Jacobian
120°
120°
120°
j3
p2
p3
j1
j2
p1
Remaining dexterity: K = f σ2 =√
13
9/40
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
Example: 2× 3 Optimally Fault Tolerant Jacobian
120°
120°
120°
j3
p2
p3
j1
j2
p1
Remaining dexterity: K = f σ2 =√
13
9/40
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
(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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
Finding Maximum K
Use homogenous solution (Null Motion)
θ = J+x + αnJ
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
Finding Maximum K
Use homogenous solution (Null Motion)
θ = J+x + αnJ
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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.
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R 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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
Global Measurements
Maximum reach
The fraction of the total workspace that is fault tolerant,denoted WK (K ≥ γKopt)
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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◦)
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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◦
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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◦
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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 = 10◦
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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 = 30◦
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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)
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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)
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
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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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar 4R Spatial Conclusions •
Thanks!
?
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