Post on 14-Dec-2015
Chin Pei Tang May 3, 2004Slide 1 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Chin Pei Tang (chintang@eng.buffalo.edu)
Advisor : Dr. Venkat Krovi
Mechanical and Aerospace Engineering
State University of New York at Buffalo
Manipulability-Based Analysis of Cooperative Payload Transport by
Robot Collectives
Chin Pei Tang May 3, 2004Slide 2 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Agenda
Motivation & Our System
Literature Survey & Research Issues
Kinematic Model
Twist-Distribution Analysis
Manipulability
Cooperative Systems
Conclusion & Future Work
Part I
Part II
Chin Pei Tang May 3, 2004Slide 3 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Motivation
Why Cooperation?
– Tasks are too complex
– Distinct benefits – “Two hands are better than one”
– Instead of building a single all-powerful system, build multiple simpler systems
– Motivated by the biological communities
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 4 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Our System
Flexible, scalable and modular framework for cooperative payload transport
Autonomous wheeled mobile manipulator
– Differentially-driven wheeled mobile robots (DD-WMR)
– Multi-link manipulator mounted on the top
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 5 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Features
Accommodate changes in the relative configuration
Detect relative configuration changes
Compensate for external disturbances
Using the compliant linkage
Using sensed articulation
Using redundant actuation of the bases
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 6 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Research Issues
Challenges
– Nonholonomic (wheel) / holonomic (closed-loop) constraints
– Mobility / workspace increased (but also increases redundancy)
– Mixture of active/passive components
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 7 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Literature Survey
Applications of Robot Collectives
– Collective foraging, map-building and reconnaissance
Coordination & Control
– Formation Paradigm• Leader-follower [Desai et. al., 2001]
• Virtual structures [Lewis and Tan, 1997]
• Mixture of approaches [Leonard and Fiorelli, 2001],
[Lawton, Beard and Young, 2003]
No physical interaction
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 8 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Literature Survey
Physical Interaction
– Teams of simple robots • box pushing [Stilwell and Bay, 1993], [Donald et. al., 1997]
• caging [Pereira et. al., 2002], [Wang & Kumar, 2002]
– Teams of mobile manipulators [Khatib et. al., 1996]
– Design modifications [Kosuge et. al., 1998],
[Humberstone & Smith, 2000]
Upenn MARS Univ. of Alberta CRIPNASA Cooperative Rovers
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 9 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Literature Survey
Performance Measures– Single agent
• Service angle [Vinogradov et. al, 1971], conditioning [Yang and Lai,
1985], manipulability [Yoshikawa, 1985], singularity [Gosselin and
Angeles, 1990], dexterity [Kumar and Waldron, 1981], etc.
– Multiple agents (Robot teams)• Social entropy – Measuring diversity of robots in a team
(Information-theoretic) [Balch, 2000]
• Kinetic energy – Left-invariant Riemannian metrics [Bhatt et. al., 2004]
Manipulability– Serial chain – Yoshikawa’s measure [Yoshikawa, 1985],
condition number [Craig and Salisbury, 1982], isotropy index [Zanganeh and Angeles, 1997]
– Closed chain [Bicchi and Prattichizza, 2000], [Wen and Wilfinger, 1999]
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 10 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Research Issues
Part I – Physical Cooperation
– System level constraints
– Motion planning strategy
Part II – Performance Evaluation & Optimization
– Performance measures
– Formulation that takes holonomic/nonholonomic constraints and active/passive joints into account
– Different actuation schemes
– Optimal configuration
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 11 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Mathematical Preliminaries
( )20 0 1
F FMF
M
R dA SE
é ùê ú= Îê úê úë û
r
1M F F FM M MT A A
-é ù é ù=ë û ë û&
1N FF N F NM F M FT A T A
-é ù é ù é ùé ù=ë û ë û ë ûë û
0
0
0 0 0
z x z
z y x
y
v
v v
v
w w
w
é ù é ù-ê ú ê úê ú ê úÛê ú ê úê ú ê úê ú ê úë ûë û
Twist Matrix Twist Vector
Similarity Transformation
Body-fixed Twist
Homogeneous Matrix Representation
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 12 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Kinematic Model
Mobile Platform
cos sin
si
0 0 1
n cosFMA
x
y
ff
ff
é ùê úê ú
= ê úê úê úê û
-
úë
( ) ( ), 2FF FM MA R Sd E= Î
%
1M F F FM M MT A A-é ù=ë û
&
Reaching any point
in the plane
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 13 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Kinematic Model
cos sin
si
0
0
0 0 0
n cosM F
M
x y
x yT
ff
ff
f
f
+é ù-ê úê úé ù ê ú=ë û ê úê úê úë
- +
û
&
&
& &
& &
sin cos 0x yff- + =& &
cos sin Mx y vff+ =& &Nonholonomic
Constraints
Mobile Platform
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 14 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Kinematic Model
0
0
0 0 0
0
M
M FM
v
T
f
f
é ù-ê úê úé ù ê ú=ë û ê úê úê úë û
&
&
sin cos 0x yff- + =& &
cos sin Mx y vff+ =& &Nonholonomic
Constraints
Mobile Platform
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 15 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Kinematic Model
0 1 0 0 0 1
1 0 0 0 0 0
0 0 0 0 0 0M M
vM
M M
M FM M
TT
vT
f
f
é ù é ùê úê ú ë ûë û
é ù é ù-ê ú ê úê ú ê úé ù= +ê ú ê úë û ê ú ê úê ú ê úê ú ê úë û ë û
&
1444442444443 14444 44443
&
2
Mobile Platform
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 16 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Kinematic Model
1 1 1 1
1 1 1 1
cos sin cos
sin cos sin
0 0 1
MA
L
A L
q q q
q q q
é ù-ê úê ú
= ê úê úê úê úë û
2 2 2 2
2 2 2 2
cos sin cos
sin cos sin
0 0 1
AB
L
A L
q q q
q q q
é ù-ê úê ú
= ê úê úê úê úë û
3 3 3 3
3 3 3 3
cos sin cos
sin cos sin
0 0 1
kB
E
L
A L
q q q
q q q
é ù-ê úê ú
= ê úê úê úê úë û
k kM M A B
A BE EA A A A=
Manipulator
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 17 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Kinematic Model
1 2 3
kk kk
k k
EE EEM M A BA BE E
T T T Tq q qé ùé ù é ùé ù= + +ë ûë û ë ûë û& & &
1E M M ME E ET A A-é ù=ë û
&
Manipulator
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 18 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Kinematic Model
kE tf&%
k
M
Evt% 1
kE tq&% 2
kE tq&% 3
kE tq&%
1 23 2 3
1 23 2 3 3
1
L S L S
LC L C L
é ùê úê ú
+ê úê úê ú+ +ê úë û
123
123
0
C
S
é ùê úê úê úê úê ú-ê úë û
1 23 2 3
1 23 2 3 3
1
L S L S
LC L C L
é ùê úê ú
+ê úê úê ú+ +ê úë û
2 3
2 3 3
1
L S
L C L
é ùê úê úê úê úê ú+ê úë û 3
1
0
L
é ùê úê úê úê úê úê úë û
Twist Vectors
Assembled
1 2 31
2
3
kk k k
k
kk
M
E
M
E E EE FE
Ev
v
t t t tt tq q qf
f
q
q
q
é ùê úê úê úê ú
é ùê úé ù= ê úê úë û ë ûê úê úê úê úê úë û
& & & &&
%%
&
% %&
&
% %
( )kE J h%
Jacobian
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 19 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Mobility Verification
- Verify that arbitrary end-effector motion is feasible.
- Partitioning of feasible motion distribution:
- Actively-realizable
(using wheeled bases)
- Passively-accommodating
(using articulations)
- Configuration dependent partitioning
- Steer the actively-realizable vector-fields
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 20 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Twist-Distribution Analysis
Partition the Jacobian
pa
E FE T T pa JJt h hé ùé ù= +ë
éë û úû
ùêë û
& &%%%
1 2 3
k k k
p
E E ETJ t t t
q q qé ù= ê úë û& & &% % %
1
2
3
p
q
h q
q
é ùê úê úê ú=ê úê úê úë û
&
&&%
&
Passive Distributions
k k
a M
E ET vJ t t
fé ù= ê úë û&% %a
Mv
fh
é ùê ú= ê úê úë û
&&%
Active Distributions
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 21 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Twist-Distribution Analysis
Can any arbitrary twist be realized using only the active distribution?
Feasibility check
a
ETG J té ùé ù= ê úë ûë û
M%
( ) ( )aT
rank G rank J=
Not constructive
ReciprocalWrench
Alternate constructive approach
1 23 2 3 123
1 23 2 3 3 123
1 0
aTJ L S L S C
LC L C L S
é ùê úê ú
= +ê úê úê ú+ + -ê úë û
1 1 2 12 3 123
123
123
a
LC L C L C
w S
C
é ù- - -ê úê ú
= ê úê úê úê úë û
%
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 22 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Condition:
Transform an arbitrary twist from {Ek} to {M}:
0The Motion Planning Strategy
Given arbitrary twistT
EEz Ex Eyt v vwé ù= ê úë û%
To understand this condition better:
Achieved by aligning the forward travel direction with the direction of the velocity
Twist-Distribution Analysis
[ ] [ ] [ ]
[ ] [ ] [ ]1 1 2 12 3 123 123 123
1 1 2 12 3 123 123 123
Ez
M FE Ez Ex Ey
Ez Ex Ey
t L S L S L S C v S v
LC L C L C S v C v
w
w
w
é ùê úê úé ù ê ú= + + + -ë û ê úê ú- - - + +ê úë û
%
[ ] [ ] [ ]1 1 2 12 3 123 123 123 0Ez Ex EyLC L C L C S v C vw- - - + + =
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 23 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Manipulability
Jacobian Matrix
( )1 1E
m T nm nt J q h´ ´´
é ù= ë û &% % %
{ }: , 1E EV Tt t Je h h= = =& &
% % %%Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
1n
nh ´ Î& ¡%
1E m
mt ´ Î ¡%
( )T m n
m n
J q´
´
é ùë ûÎ
%¡
Joint manipulation rates space Task velocity space
Manipulability is defined as the measure of the flexibility of the manipulator to transmit the end-effector motion in response to a unit norm motion of the rates of the active joints in the system
Chin Pei Tang May 3, 2004Slide 24 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Manipulability – SVD
Singular Value Decomposition
TTJ VU= S
T Tm mU U UU I ´= =
T Tn nV V VV I ´= =
1 2
1 2
, , , ,0, ,0m n kn kk
k
diag s s s
s s s
´-
æ ö÷ç ÷S = ç ÷ç ÷÷çè ø³ ³ ³
L L144244314444244443
L
( )1 1E
m T nm nt J q h´ ´´
é ù= ë û &% % %
{ }: , 1E EV Tt t Je h h= = =& &
% % %%Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
TT TJ J
TT TJ J
Chin Pei Tang May 3, 2004Slide 25 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
-3 -2 -1 0 1 2 3-3
-2
-1
0
1
2
3
Y
X
Y X
Y
X
Y
X
Y
X
Y
X
Y
X
Y
X
Y
X
Y
X
Y
X
Y
X
Manipulability ellipsoids of Two Link at F frame
x (m)
y (m
)
RR Manipulator Example
1 2L m= 2 1L m=
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
1
1 1 2 12 2 12
21 1 2 12 2 12
1 1E
E
E
X L S L S L S
LC L C L CY
q
q
é ù é ùQê ú ê úé ùê ú ê úê úê ú= - - -ê úê úê ú ê úê úê ú ë ûê ú+ê ú ê úë ûë û
&&
&&
&
Chin Pei Tang May 3, 2004Slide 26 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Manipulability Indices
Yoshikawa’s Measure (Volume of Ellipsoid)
Condition Number
Isotropy Index
( ) ( ) ( ) 1 2det detT TY T T T kJ J J s s sG = = SS = × ×L
( ) 1CN T
k
Jss
G =
( )1
kI TJ
ss
G =
Not able to distinguish the ratio of major/minor axes of ellipsoid
Value goes out of bound at singular
position
Better numerical behavior0 1I£ G £
1 CN£ G £ ¥
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 27 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Yoshikawa’s Measure
Condition Number
Isotropy Index
( ) ( )det TY T T TJ J JG =
( ) 1CN T
k
Jss
G =
( )1
kI TJ
ss
G =
Adopted measure
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
RR Manipulator Example
Chin Pei Tang May 3, 2004Slide 28 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Manipulability (Closed-Loop)
a p
T T Th h hé ù= ê úë û% % %
( ) ETJ th h=&
%%%
( ) 0CJ h h=&%%%
{ }: , 1, 0E EV T a Ct t J Je h h h= = = =& & &
% % %% %
Generalized Coordinates
Forward Kinematic
Closed-Loop Kinematic Constraints
Not easy to compute
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 29 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Manipulability (Closed-Loop)
a pT T TJ J Jé ù= ê úë û
a pC C CJ J Jé ù= ê úë û
a p
ET a T pJ J th h+ =& &
%% %0
a pC a C pJ Jh h+ =& &%% %
1
p ap C C aJ Jh h-=-& &% % p a pp C C a CJ J Jh h x+=- + %& &
%% %
p p
ET a T Ct J J Jh x= + %&
% %%
a p C apT T T CJ J J J J+= -
{ }: , 1E EV T a at t Je h h= = =& &
% % % %
Partition according to active/passive manipulation variable rates
Exact Actuation Redundant Actuation
Manipulability Jacobian
Solved explicitly
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
0p pC CJ J =%
%
{ }: , 1, 0E EV T a Ct t J Je h h h= = = =& & &
% % %% %
Chin Pei Tang May 3, 2004Slide 30 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Cooperative Model
Team up
End-effectors need to be re-aligned
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 31 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Kinematic Model (with end-effector offset angle)
cos sin 0
sin cos 0
0 0 1
k
k k
Ek kE
A
d d
d d
é ù-ê úê ú
= ê úê úê úê úë û
( ) ( )
( ) ( )1 2 3 2 3 3
1 2 3 2 3 3
1
sin sin sin
cos cos cos
k k k
k k k
L L L
L L L
d q q d q d
d q q d q d
é ùê úê úê ú= - - - - - -ê úê ú- - + - +ê úë û
( )
( )1 2 3
1 2 3
0
cos
sin
k
k
d q q q
d q q q
é ùê úê úê ú= - - -ê úê ú- - -ê úë û
( ) ( )
( ) ( )1 2 3 2 3 3
1 2 3 2 3 3
1
sin sin sin
cos cos cos
k k k
k k k
L L L
L L L
d q q d q d
d q q d q d
é ùê úê úê ú= - - - - - -ê úê ú- - + - +ê úë û
( )
( )2 3 3
2 3 3
1
sin sin
cos cos
k k
k k
L L
L L
d q d
d q d
é ùê úê úê ú= - - -ê úê ú- +ê úë û
3
3
1
sin
cosk
k
L
L
d
d
é ùê úê ú
= -ê úê úê úê úë û
Similarity Transformation
1
Etq&%
2
Etq&% 3
Etq&%
Etf&% M
Evt%
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 32 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Simulation
Step 1: Identify
Step 2: Construct manipulability Jacobian
Step 3: Compute isotropy index
Case I – MB static, R1 actuated
Case II – MB static, R2 actuated
Case III – MB moves, R1 & R2 passive
Case IV – MB moves, R1 locked
Case V – MB moves, R2 locked
TE E E
x yt v vé ù= ê úë û%
ah&%
ph&% aT
JpT
JaC
JpC
Ja p
ET a T pJ J th h+ =& &
%% %0
a pC a C pJ Jh h+ =& &%% %
a p C apT T T CJ J J J J+= -
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 33 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Simulation Parameters (3-RRR Nomenclature)
-1 0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
Y
X
Location of MB and geometry of platform
x (m)
y (m
)
( ) ( )1 1, 0,0I Ix y = ( ) ( )1 1, 3.4641,2I I I Ix y = ( ) ( )1 1, 0,4I I I I I Ix y =
330Id = ° 210I Id = ° 90I I Id = °
1Iel = 1I Iel = 1I I Iel =
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 34 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case I: MB static R1 actuated
1 1 1
TI II I I I
ah q q qé ù= ê úë û& & &&
%
2 3 2 3 2 3
TI I II I I I I I I I I
ph q q q q q qé ù= ê úë û& & & & & &&
%
10 0
a
E ITJ t
qé ù= ê úë û&% % %
2 30 0 0 0
p
E I E ITJ t t
q qé ù= ê úë û& &% % % % % %
1 1
1 1
0
0a
E I E II
C E I E III
t tJ
t t
q q
q q
é ù-ê ú= ê ú-ê úë û
& &
& &
% % %
% % %
2 3 2 3
2 3 2 3
0 0
0 0p
E I E I E II E II
C E I E I E III E III
t t t tJ
t t t t
q q q q
q q q q
é ù- -ê ú= ê ú- -ê úë û
& & & &
& & & &
% % % % % %
% % % % % %
Generalized Coordinates
Forward Kinematics
General Constraints
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 35 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case Study I-A
1 2k kL L¹
-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
x (m)
y (m
)
Contour plot
0.1
0.10.1
0.1
0.1
0.1
0.10.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.10.1
0.1
0.10.2
0.2
0.2 0.20.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.30.30.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.40.4
0.5
0.5
0.50.5
0.5
0.5
0.5
0.50.
5
0.5
0.5
0.5
0.60.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.80.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.8
0.90.9
0.9
0.9
0.9
0.9
0.90.9
1 2kL m= 2 1.5kL m= 3 1kL m=
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 36 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case Study I-B
1 2k kL L=
-0.5 0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
x (m)
y (m
)
Contour plot
0.1
0.1
0.1 0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.20.
2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.30.3
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.6
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.8
0.8
0.8
0.8
0.8
0.9
1 1.5kL m= 2 1.5kL m= 3 1kL m=
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 37 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case II: MB static, R2 actuated
20 0
a
E ITJ t
qé ù= ê úë û&% % %
1 30 0 0 0
p
E I E ITJ t t
q qé ù= ê úë û& &% % % % % %
2 2
2 2
0
0a
E I E II
C E I E III
t tJ
t t
q q
q q
é ù-ê ú= ê ú-ê úë û
& &
& &
% % %
% % %
1 3 1 3
1 3 1 3
0 0
0 0p
E I E I E II E II
C E I E I E III E III
t t t tJ
t t t t
q q q q
q q q q
é ù- -ê ú= ê ú- -ê úë û
& & & &
& & & &
% % % % % %
% % % % % %
2 2 2
TI II I I I
ah q q qé ù= ê úë û& & &&
%
1 3 1 3 1 3
TI I II I I I I I I I I
ph q q q q q qé ù= ê úë û& & & & & &&
%
Generalized Coordinates
Forward Kinematics
General Constraints
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 38 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case II: MB static, R2 actuated
-0.5 0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
x (m)
y (m
)
Contour plot
0.10.1
0.10.
1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.8
0.9
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 39 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case III: MB moves, R1 and R2 passive
TI I II I I I I I I I I
a M M Mv v vh ff fé ù= ê úë û& & &&
%
1 2 3 1 2 3 1 2 3
TI I I I I I I I I I I I I I I I I I
ph q q q q q q q q qé ù= ê úë û& & & & & & & & &&
%
0 0 0 0a M
E I E IT vJ t t
fé ù= ê úë û&% % % % % %
1 2 30 0 0 0 0 0
p
E I E I E ITJ t t t
q q qé ù= ê úë û& & &% % % % % % % % %
0 0
0 0
M M
a
M M
E I E I E II E IIv v
C E I E I E III E IIIv v
t t t tJ
t t t t
ff
ff
é ù- -ê ú= ê ú- -ê úë û
& &
& &
% % % % % %
% % % % % %
1 2 3 1 2 3
1 2 3 1 2 3
0 0 0
0 0 0p
E I E I E I E II E II E II
C E I E I E I E III E III E III
t t t t t tJ
t t t t t t
q q q q q q
q q q q q q
é ù- - -ê ú= ê ú- - -ê úë û
& & & & & &
& & & & & &
% % % % % % % % %
% % % % % % % % %
Generalized Coordinates
Forward Kinematics
General Constraints
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 40 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Self-Motion
0a pC a C pJ Jh h+ =& &
%% %p a pp C C a CJ J Jh h x+=- + %& &
%% %
Feasible motions of passive joints due to the actuations butnot violating constraints
Feasible self-motion when all the active
joints locked
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
0p pC CJ J =%
%
m n´
m n<
( )n n m´ -
n m-Underconstrained
Dimension of self-motion manifold
Lock this number of joints
Chin Pei Tang May 3, 2004Slide 41 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Self-Motion
1 2 3 1 2 3
1 2 3 1 2 3
0 0 0
0 0 0p
E I E I E I E II E II E II
C E I E I E I E III E III E III
t t t t t tJ
t t t t t t
q q q q q q
q q q q q q
é ù- - -ê ú= ê ú- - -ê úë û
& & & & & &
& & & & & &
% % % % % % % % %
% % % % % % % % %
6 9´ 6m= 9n =
9 6 3n m- = - =Lock this number
of joints
2 Cases:- Locking R1- Locking R2
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 42 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case IV: MB moves, R1 locked
TI I II I I I I I I I I
a M M Mv v vh ff fé ù= ê úë û& & &&
%
2 3 2 3 2 3
TI I II I I I I I I I I
ph q q q q q qé ù= ê úë û& & & & & &&
%
0 0 0 0a M
E I E IT vJ t t
fé ù= ê úë û&% % % % % %
2 30 0 0 0
p
E I E ITJ t t
q qé ù= ê úë û& &% % % % % %
0 0
0 0
M M
a
M M
E I E I E II E IIv v
C E I E I E III E IIIv v
t t t tJ
t t t t
ff
ff
é ù- -ê ú= ê ú- -ê úë û
& &
& &
% % % % % %
% % % % % %
2 3 2 3
2 3 2 3
0 0
0 0p
E I E I E II E II
C E I E I E III E III
t t t tJ
t t t t
q q q q
q q q q
é ù- -ê ú= ê ú- -ê úë û
& & & &
& & & &
% % % % % %
% % % % % %
Generalized Coordinates
Forward Kinematics
General Constraints
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 43 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case IV: MB moves, R1 locked
-0.5 0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
x (m)
y (m
)
Contour plot
0.1
0.1
0.1 0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.30.
3
0.3
0.3
0.3
0.3
0.3
0.40.4
0.4
0.4
0.4
0.4
0.4
0.40.4
0.4
0.4
0.50.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.60.6
0.6
0.6
0.6
0.6
0.6
0.6
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.8
0.8
0.80.8
0.8
0.90.9
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 44 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case V: MB moves, R2 locked
TI I II I I I I I I I I
a M M Mv v vh ff fé ù= ê úë û& & &&
%
1 3 1 3 1 3
TI I II I I I I I I I I
ph q q q q q qé ù= ê úë û& & & & & &&
%
0 0 0 0a M
E I E IT vJ t t
fé ù= ê úë û&% % % % % %
1 30 0 0 0
p
E I E ITJ t t
q qé ù= ê úë û& &% % % % % %
0 0
0 0
M M
a
M M
E I E I E II E IIv v
C E I E I E III E IIIv v
t t t tJ
t t t t
ff
ff
é ù- -ê ú= ê ú- -ê úë û
& &
& &
% % % % % %
% % % % % %
1 3 1 3
1 3 1 3
0 0
0 0p
E I E I E II E II
C E I E I E III E III
t t t tJ
t t t t
q q q q
q q q q
é ù- -ê ú= ê ú- -ê úë û
& & & &
& & & &
% % % % % %
% % % % % %
Generalized Coordinates
Forward Kinematics
General Constraints
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 45 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case V: MB moves, R2 locked
-0.5 0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
x (m)
y (m
)
Contour plot
0.1 0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.20.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3 0.3
0.3
0.3
0.3
0.30.4
0.4
0.4
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.50.5
0.5
0.6
0.6
0.6
0.6
0.6 0.7
0.7
0.7
0.8
0.8
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 46 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Case Study – Configuration Optimization
( )max IqqG
% %T
E Eq x yé ù= ê úë û%
( )max IhhG
% %T
I II I I Ih q q qé ù= ê úë û% % %%1 2 3
Tk k k kq q q qé ù= ê úë û% % % %
Subject to: Closed-Kinematic Loop Constraints
ConstrainedOptimization
Problem
UnconstrainedOptimization
Problem
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 47 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Configuration Optimization – Case IV
-1 0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
Y
X
Optimal Configuration (Case IV)
x (m)
y (m
)
( ) ( ), 1.4205,2.1885E Ex y =
* 1.0000IG =-0.5 0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
x (m)
y (m
)
Contour plot
0.1
0.1
0.1 0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.3
0.40.4
0.4
0.4
0.4
0.4
0.4
0.40.4
0.4
0.4
0.50.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.60.6
0.6
0.6
0.6
0.6
0.6
0.6
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.8
0.8
0.80.8
0.8
0.90.9
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 48 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
-1 0 1 2 3 4
0
0.5
1
1.5
2
2.5
3
3.5
4
Y
X
Optimal Configuration (Case IV)
x (m)
y (m
)
Configuration Optimization – Case V
( ) ( ), 0.8660,1.5000E Ex y =
* 0.8660IG =-0.5 0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
x (m)
y (m
)
Contour plot
0.1 0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.20.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3 0.3
0.3
0.3
0.3
0.30.4
0.4
0.4
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.5
0.5
0.6
0.6
0.6
0.6
0.6 0.7
0.7
0.7
0.8
0.8
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 49 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Conclusion
Modular Formulation
Motion-Distribution Analysis
Evaluation of Performance Measures
Manipulability Jacobian Matrix Formulation
Effect of Different Actuation Schemes
Optimal Configuration Determination
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 50 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Future Work
Global Manipulability
Force Manipulability
Singularity Analysis
Decentralized Control
Redundant Actuation
IW
W
dW
dWm
G=òò
2
,min2
,max
I
I
sæ öG ÷ç ÷=ç ÷ç ÷÷çGè ø
1 1T
n mm nJ Ft ´ ´´é ù= ë û% %
Introduction Model Distribution Analysis Manipulability Cooperative System Conclusion
Chin Pei Tang May 3, 2004Slide 51 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Thank You!
Acknowledgments:Dr. V. KroviDr. T. Singh
Dr. J. L. Crassidis& all the audience…
Chin Pei Tang May 3, 2004Slide 52 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo
Twist Matrix as Velocity Operator
1
0 0 0
E EF FE E EF F F
E E E
vdT A A
dt
-é ùé ù é ùWé ù ê úë û ë ûé ù é ù é ù= =ê ú ê úë û ë û ë ûê úë û ê úë û
%
0
0zE F
Ez
w
w
-é ùê úé ùW =ë û ê úê úë û
xE FE
y
vv v
é ùé ù ê ú=ë û ê úë û%
zE FEt v
wé ùé ù ê ú=ë û ê úë û% %
1 2
1 11 21 21 1 2 2
N
E EF E E E E NE E E E
T T T
T A T A A T A T- -é ù é ù é ù é ù é ù é ùé ù é ù= + + +ë û ë ûë û ë û ë û ë û ë û ë ûL
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