Transcript Singularity Handling on PUMA in Operational Space Formulation Lim Ser Yong**
Singularity Handling on PUMA in Operational Space Formulation
Author: Denny Oetomo*, Marcelo Ang Jr*, Lim Ser Yong** * National University of Singapore, ** Gintic Institute of Manufacturing Technology ISER 2000, Honolulu, HI, Dec 12, 2000 1
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Problem Statement Singularity
• Motion across singularities – increased usable workspace for task execution • Traditional methods – motion-based – forcing Jacobian to be non-singular, etc • Task (operational) space methods – Motion/Forces at End-Effector are directly controlled via Joint Torques 3
Task-Based Control
Operational space • Force on hand/tool – virtual - to cause motion – real - actual forces exerted • Singularity handling needs to be in both motion and force control • Actuation signals to robot joints computed to effect forces 4
Operational Space vs Inverse Kinematics
• Differential Motion
dq
J dx
Ndq
• Operational Space Formulation 0
J T F
N T
0 [Chang and Khatib, 1994]
F
(
x
)
x
(
x
,
x
)
p
(
x
)
N
[
I
J J
]
J
A
1
J T
(
JA
1
J T
) 1 5
Work Done
• Analysis and resolution of singularities in operational space • Remove degenerate directions – lower order (in terms of task space) non-singular but redundant mechanism • Graceful escape algorithms using null space motion – stable and smooth motions from singular to non singular regions • Experimental verification 6
d2 Z 1 X 1 Y 1 a2
PUMA Singularities
Z 4 , Z 6 Z 2 X 2 Y 4 , Z 5 X 4 , X 5 d4 Z 3 Singularities in PUMA 500 series: •Wrist •Elbow •Head d3 X 3 Y 3
Det(J) = a 2 (d 4 C 3 - a 3 S 3 ) (d 4 S 23 + a 2 C 2 + a 3 C 23 ) S 5
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Jacobian in Frame 0, 0 J Head Singularity Z 1 0
R
1
T
.
0
J
1
F
1
R
0 .
0
F
; Remove 2nd row X 1 1 J T 1 F N T 0 0 at head singularity 1 J =
i a2 C2 d2 d3 + a3 C23 + d4 S23 k 0 0 0 1 d4 C23 a2 C2 a2 S2 a3 S23 0 a3 C23 0 1 0 d4 S23 d4 C23 0 a3 C23 0 1 0 a3 S23 d4 S23 0 0 0 0 0 0 0 0 S23 0 C23 S4 C5 S23 + C23 C4 S5 C4 0 S4 S5 C4 S23 S5
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{ C23 S23 S4 C23 C5 y
Elbow Singularity d4 a2 d3 Z B Wrist point Degenerate direction d2 b b Tan 1 a 2 d 2 d 3 a 3 C 3 d 4 S 3 X B 0 B J x x x x x 0 x x x x x 0 x x x x x 0 x x x x x 0 x x x x x 0 x x x x x 10
Wrist Singularity Z4 q 6 Y4 q 5 q 4 X4 x 4 J x x x x x x x x x x x x x x x x x 0 0 0 0 x x 0 0 0 0 x x 0 0 0 0 x x 11
Experimental Sets
Four sets of result were collected, consisting of the
position
and
orientation (tracking) error
in: 1. PUMA tracing a non-singular trajectory 2. PUMA going through wrist singularity, not in the singular direction.
3. PUMA escaping from wrist singularity into a path in singular direction 4. PUMA escaping from elbow singularity into a path in singular direction 13
Escape into Singular Direction
Utilising Null Space Motion: Type 1: Null Space Motion creates motion in singular direction Joint 3 Type 2: Null Space Motion creates internal motion which shifts the singular direction.
Desired path and non-feasible path The initially non-feasible direction Desired path 14
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0.0015
0.001
0.0005
-1 0 -0.0005
0 -0.001
-0.0015
-0.002
-0.0025
-0.003
Position error - wrist lock - feasible path
1 2 3 4 5 6 x error
time(s)
y error z error 7 8 16
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Polishing Application
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Conclusions
• The singularity handling algorithm implemented.
– By removing the singular component in operational space • Graceful escape algorithms using null space motion – stable and smooth motions from singular to non singular regions • Experimental Verification 19
Future Work
•This is one of the ‘infra structure’ of a larger project.
(further work would be done on the larger project).
•Extension of the work into inherently redundant robots.
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