kinematic design of redundant robotic manipulators that
TRANSCRIPT
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Kinematic Design of Redundant RoboticManipulators 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 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 To Spatial 4R Spatial 7R Spatial Conclusions �
Simple Redundant Robot Planar 3DOF
[x1x2
]= J(θ)
θ1θ2θ3
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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 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
J3×3 = [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 To Spatial 4R Spatial 7R 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
J3×3 = [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
4/45
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R 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
J3×3 = [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 To Spatial 4R Spatial 7R 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
]
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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 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: Kopt = σ√
n−mn 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 ] =
−√23
√16
√16
0 −√
12
√12
Null space equally distributed: nJ =
[ √13
√13
√13
]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 =√
13 , 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 =√
13 , 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 =√
13 , 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 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 To Spatial 4R Spatial 7R 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 To Spatial 4R Spatial 7R Spatial Conclusions �
3DOF Planar Manipulators
The total possible Jacobiansare 48 (23 × 3!)
All of these Jacobianscorrespond to only 4 differentmanipulators
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� ]
12/45
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
3DOF Planar Manipulators
The total possible Jacobiansare 48 (23 × 3!)
All of these Jacobianscorrespond to only 4 differentmanipulators
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
12/45
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
3DOF Planar Manipulators
The total possible Jacobiansare 48 (23 × 3!)
All of these Jacobianscorrespond to only 4 differentmanipulators
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
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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 Jacobiansare 48 (23 × 3!)
All of these Jacobianscorrespond to only 4 differentmanipulators
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
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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 distancefrom the base
K is not a function of θ1There is a rotational symmetry around the basex-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 + αnJ
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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 + αnJ
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R 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 To Spatial 4R Spatial 7R Spatial Conclusions �
Optimally 2× 4 Fault Tolerant Jacobian (Two Failures)
J = [ j1 j2 j3 j4 ] =
[ 1√2
12 0 −1
2
0 12
1√2
12
]
16/45
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 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
-32
1
21
1
1
211
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
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 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
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
17/45
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R 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 To Spatial 4R Spatial 7R Spatial Conclusions �
Published 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 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 complicatedDenavit and Hartenberg (DH) parameters are used to describethe kinematic, parameters including link lengths
1 ai : Link length2 αi : Link twist3 di : Joint offset4 θi : Joint value
Any spatial Jacobian consisting all rotational joints isrepresented by a 6× n matrix, s.t.
ji =
[viωi
]=
[zi−1 × pi−1
zi−1
]Determining the DH parameters from a Jacobian becomesharder than the case for a planar manipulator
Computing a maximum reach and a total workspace volume isnot simple either
20/45
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Spatial Manipulators
The geometry becomes more complicatedDenavit and Hartenberg (DH) parameters are used to describethe kinematic, parameters including link lengths
1 ai : Link length2 αi : Link twist3 di : Joint offset4 θi : Joint value
Any spatial Jacobian consisting all rotational joints isrepresented by a 6× n matrix, s.t.
ji =
[viωi
]=
[zi−1 × pi−1
zi−1
]Determining the DH parameters from a Jacobian becomesharder than the case for a planar manipulator
Computing a maximum reach and a total workspace volume isnot simple either
20/45
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Spatial Manipulators
The geometry becomes more complicatedDenavit and Hartenberg (DH) parameters are used to describethe kinematic, parameters including link lengths
1 ai : Link length2 αi : Link twist3 di : Joint offset4 θi : Joint value
Any spatial Jacobian consisting all rotational joints isrepresented by a 6× n matrix, s.t.
ji =
[viωi
]=
[zi−1 × pi−1
zi−1
]Determining the DH parameters from a Jacobian becomesharder than the case for a planar manipulator
Computing a maximum reach and a total workspace volume isnot simple either
20/45
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Spatial Manipulators
The geometry becomes more complicatedDenavit and Hartenberg (DH) parameters are used to describethe kinematic, parameters including link lengths
1 ai : Link length2 αi : Link twist3 di : Joint offset4 θi : Joint value
Any spatial Jacobian consisting all rotational joints isrepresented by a 6× n matrix, s.t.
ji =
[viωi
]=
[zi−1 × pi−1
zi−1
]Determining the DH parameters from a Jacobian becomesharder than the case for a planar manipulator
Computing a maximum reach and a total workspace volume isnot simple either
20/45
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 ] =
−√
34
√112
√112
√112
0 −√
23
√16
√16
0 0 −√
12
√12
21/45
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 viωi is normalized=⇒ ωi = f (βi ) =⇒ DH = f (β)
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
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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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Monte Carlo Integration Algorithm
x2
x1
x3
one million random points
boundary sphere for Monte Carlo approximation
unknown true boundary
R is 110% maximum reach
nrn
10,000 uniformly distributed random samples within a sphereof radius of 110% of the maximum reach are used
WK = nf /nr
One million uniformly distributed random samples aregenerated in the joint space
The maximum reach is computed from the largest norm
24/45
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Monte Carlo Integration Algorithm
x2
x1
x3
one million random points
boundary sphere for Monte Carlo approximation
fault tolerant workspace boundary
unknown true boundary
R is 110% maximum reach
nf
nrn
10,000 uniformly distributed random samples within a sphereof radius of 110% of the maximum reach are used
WK = nf /nr
One million uniformly distributed random samples aregenerated in the joint space
The maximum reach is computed from the largest norm
24/45
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Monte Carlo Integration Algorithm
x2
x1
x3
one million random points
boundary sphere for Monte Carlo approximation
fault tolerant workspace boundary
unknown true boundary
R is 110% maximum reach
nf
nrn
10,000 uniformly distributed random samples within a sphereof radius of 110% of the maximum reach are used
WK = nf /nr
One million uniformly distributed random samples aregenerated in the joint space
The maximum reach is computed from the largest norm
24/45
Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Monte Carlo Integration Algorithm
x2
x1
x3
one 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 sphereof radius of 110% of the maximum reach are used
WK = nf /nr
One million uniformly distributed random samples aregenerated 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
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
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 To Spatial 4R Spatial 7R 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
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 To Spatial 4R Spatial 7R Spatial Conclusions �
Examples of Manipulators with Common Link TwistParameters
There is a design flexibility within this entire family ofmanipulators as DH = f (β)
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 To Spatial 4R Spatial 7R 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 To Spatial 4R Spatial 7R 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 To Spatial 4R Spatial 7R 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 To Spatial 4R Spatial 7R 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 To Spatial 4R Spatial 7R Spatial Conclusions �
Columns Permutation and/or Sign Change Effect
Sign change for very column is equivalent to reversing thedirection of the corresponding joint axis
A permutation is only equivalent to either a rotation or areflection of the original Jacobian
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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, “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 Transactions on Robotics, Vol. 29, No. 5, pp.
1300–1307, Oct. 2013.
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
6× 7 Optimally Fault Tolerant Jacobian
J =
−√
67
√142
√142
√142
√142
√142
√142
0 −√
56
√130
√130
√130
√130
√130
0 0 −√
45
√120
√120
√120
√120
0 0 0 −√
34
√112
√112
√112
0 0 0 0 −√
23
√16
√16
0 0 0 0 0 −√
12
√12
Unfortunately, it is not feasible to identify directly a rotationalmanipulator from this canonical J
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
6× 7 Optimally Fault Tolerant Jacobian
J =
−√
67
√142
√142
√142
√142
√142
√142
0 −√
56
√130
√130
√130
√130
√130
0 0 −√
45
√120
√120
√120
√120
0 0 0 −√
34
√112
√112
√112
0 0 0 0 −√
23
√16
√16
0 0 0 0 0 −√
12
√12
Unfortunately, it is not feasible to identify directly a rotationalmanipulator 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 )
0
cos(αi ) cos(βi )
sin(αi ) cos(βi )sin(βi )
α
γ ω
z
y
x
β
ii
i
i
vi
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Two ApproachesNo exact solution was found
Two approaches were used1 Solving for 21 nonlinear equations that correspond to the
constraints of the optimally fault tolerant Jacobian2 Solving directly for equal and maximum f σ6 and equal
magnitudes for the null vector elements
Each approach’s objective functions converges to the samevalues
Theoretical values 21 equations Direct optimization for KK 0.5774 0.5196 0.5714
σ1 1.5275 1.5829 1.6455
σ6 1.5275 1.4726 1.4169
|nJi |’s equal Not equal equal
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
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 withina 3D workspace has its own υoiBy integrating for the whole 6D volume we get:
V6d ≈Nr∑i=1
∆υr · υoi =Vr
Nr
Nr∑i=1
υoi
Δ }h
Ai
i
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
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 withina 3D workspace has its own υoiBy integrating for the whole 6D volume we get:
V6d ≈Nr∑i=1
∆υr · υoi =Vr
Nr
Nr∑i=1
υoi
Δ }h
Ai
i
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
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 withina 3D workspace has its own υoiBy integrating for the whole 6D volume we get:
V6d ≈Nr∑i=1
∆υr · υoi =Vr
Nr
Nr∑i=1
υoi
υ oΔυ r
i
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
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 withina 3D workspace has its own υoiBy integrating for the whole 6D volume we get:
V6d ≈Nr∑i=1
∆υr · υoi =Vr
Nr
Nr∑i=1
υoi
υ oΔυ r
i
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Orientation Volume
Unit quaternions q were used:
q = [s , v ] =[cos(θ/2) , sin(θ/2)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 uniquelyall possible orientationsBy uniformly sampling on the half of 4D sphere surface, weget by using Monte Carlo integration:
Voi ≈ Vomax
Noi
No= π2
Noi
No.
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Orientation Volume
Unit quaternions q were used:
q = [s , v ] =[cos(θ/2) , sin(θ/2)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 uniquelyall possible orientationsBy uniformly sampling on the half of 4D sphere surface, weget by using Monte Carlo integration:
Voi ≈ Vomax
Noi
No= π2
Noi
No.
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Orientation Volume
Unit quaternions q were used:
q = [s , v ] =[cos(θ/2) , sin(θ/2)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 uniquelyall possible orientationsBy uniformly sampling on the half of 4D sphere surface, weget by using Monte Carlo integration:
Voi ≈ Vomax
Noi
No= π2
Noi
No.
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
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
one 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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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
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
one 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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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
6D Reachable Volume
x2
x1
x3
total workspace boundary
φ
ψ
θ
orientation volume of Pi
calculate orientation
volume
position (Pi)
boundaryfor Monte Carlo approximation
(c)
(b)(a)
orientation (Qi)
V6d ≈Vr
Nr
Nr∑i=1
υoi
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
6D Reachable Volume
x2
x1
x3
total workspace boundary
φ
ψ
θ
orientation volume of Pi
calculate orientation
volume
position (Pi)
boundaryfor Monte Carlo approximation
(c)
(b)(a)
orientation (Qi)
V6d ≈Vr
Nr
Nr∑i=1
υoi
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
6D Fault Tolerant Volume
x2
x1
x3
total workspace boundary
φ
ψ
θ
orientation volume of Pi
calculate orientation
volume
position (Pi)
boundaryfor 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
orientation is fault tolerantfault tolerant
workspace boundary
(at least one orientation is fault tolerant)
(c)
(b)(a)
orientation (Qi)
VFT 3d≈(NFT 3d
N
)4
3πR3
VFT ≈VFT3d
NFT3d
NFT3d∑i=1
VFToi
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
6D Fault Tolerant Volume
x2
x1
x3
total workspace boundary
φ
ψ
θ
orientation volume of Pi
calculate orientation
volume
position (Pi)
boundaryfor 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
orientation is fault tolerantfault tolerant
workspace boundary
(at least one orientation is fault tolerant)
(c)
(b)(a)
orientation (Qi)
VFT 3d≈(NFT 3d
N
)4
3πR3
VFT ≈VFT3d
NFT3d
NFT3d∑i=1
VFToi
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Maximizing KSearching in all self-motion manifoldsOptimizing locally using ∇KLeast norm solution
θ = J+x+αnJ∇K
−6 −4 −2 0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
Distance from design point [m]
K
−6 −4 −2 0 2 4 6 80
1
2
3
4
5
6
Distance from design point [m]
|∆ θ
| [ra
d]
Self Motion
Self Motion
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Maximizing KSearching in all self-motion manifoldsOptimizing locally using ∇KLeast norm solution
θ = J+x+αnJ∇K
−6 −4 −2 0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
Distance from design point [m]
K
−6 −4 −2 0 2 4 6 80
1
2
3
4
5
6
Distance from design point [m]
|∆ θ
| [ra
d]
Self Motion∇K
Self Motion∇K
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Maximizing KSearching in all self-motion manifoldsOptimizing locally using ∇KLeast norm solution
θ = J+x+αnJ∇K
−6 −4 −2 0 2 4 6 80
0.1
0.2
0.3
0.4
0.5
0.6
Distance from design point [m]
K
−6 −4 −2 0 2 4 6 80
1
2
3
4
5
6
Distance from design point [m]
|∆ θ
| [ra
d]
Self Motion∇KLeast Norm
Self Motion∇KLeast Norm
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Columns Permutation
There are 7! Jacobians
All of different found feasiblesolutions were evaluated tobe almost equivalent
All of different significantdesigns may be found usingonly one feasible J with its 7!permutations
0 1000 2000 3000 4000 5000 60005
6
7
8
9
10
11
12
13
permutation index
max
rea
ch
J
1
J2
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Examplesmin Robot of J1 mid Robot of J1
Robots from J1 Robots from J2Max reach Rmax [m] Max reach Rmax [m]min mid max min mid max6.0 9.4 12.1 5.9 8.7 11.7
Volumes[%]
Vr 99 96 62 97 95 68VFT3d
81 81 51 81 73 59
V6d 49 46 26 49 41 28VFT 27 24 12 26 19 13
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Examplesmin Robot of J1 mid Robot of J1
Robots from J1 Robots from J2K ≥ 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 68VFT3d
81 81 51 81 73 59
V6d 49 46 26 49 41 28VFT 27 24 12 26 19 13
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Examplesmin Robot of J1 mid Robot of J1
Robots from J1 Robots from J2K ≥ 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 68VFT3d
81 81 51 81 73 59
V6d 49 46 26 49 41 28VFT 27 24 12 26 19 13
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
6D Volumes Plots
6D reachable volume of min Robot for J1 6D FT volume of min Robot for J1
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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 Example of aSeven Joint Manipulator Optimized for Kinematic Fault Tolerance,” acceptedto appear in IEEE International Conference on Systems, Man, and Cybernetics,pp. XXX–XXX, San Diego, CA, Oct. 5–8, 2014.
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R 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
Characterize and count self-motion manifolds
Explore all of optimal 7R manipulators
Study 8R optimally fault tolerant Jacobians
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Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions �
Thanks!
?
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