Chin Pei Tang May 3, 2004 Slide 1 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo Chin Pei Tang ([email protected]) Advisor : Dr

Chin Pei Tang May 3, 2004Slide 1 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo

Chin Pei Tang ([email protected])

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


Agenda

Motivation & Our System

Literature Survey & Research Issues

Kinematic Model

Twist-Distribution Analysis

Manipulability

Cooperative Systems

Conclusion & Future Work

Part I

Part II


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


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



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



Research Issues

Challenges

– Nonholonomic (wheel) / holonomic (closed-loop) constraints

– Mobility / workspace increased (but also increases redundancy)

– Mixture of active/passive components



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



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



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]



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



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



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



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



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



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



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


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


k kM M A B

A BE EA A A A=

Manipulator



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



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



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




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




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


%



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


[ ] [ ] [ ]

[ ] [ ] [ ]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- - - + + =



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


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


-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=


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

&&

&&

&


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 £ ¥



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


RR Manipulator Example


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



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


0p pC CJ J =%

%

{ }: , 1, 0E EV T a Ct t J Je h h h= = = =& & &

% % %% %


Cooperative Model

Team up

End-effectors need to be re-aligned



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


( ) ( )

( ) ( )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%



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+= -



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 =



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

& & & &

& & & &

% % % % % %

% % % % % %


Forward Kinematics

General Constraints



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=



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=



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


t t t tJ

t t t t

q q q q

q q q q


& & & &

& & & &

% % % % % %

% % % % % %

2 2 2

TI II I I I

ah q q qé ù= ê úë û& & &&

%

1 3 1 3 1 3


ph q q q q q qé ù= ê úë û& & & & & &&

%


Forward Kinematics

General Constraints



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



Case III: MB moves, R1 and R2 passive


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


& &

& &

% % % % % %

% % % % % %

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

& & & & & &

& & & & & &

% % % % % % % % %

% % % % % % % % %


Forward Kinematics

General Constraints



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


0p pC CJ J =%

%

m n´

m n<

( )n n m´ -

n m-Underconstrained

Dimension of self-motion manifold

Lock this number of joints


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



Case IV: MB moves, R1 locked



%

2 3 2 3 2 3


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



t t t tJ

t t t t

ff

ff


& &

& &

% % % % % %

% % % % % %

2 3 2 3

2 3 2 3

0 0

0 0p

E I E I E II E II


t t t tJ

t t t t

q q q q

q q q q


& & & &

& & & &

% % % % % %

% % % % % %


Forward Kinematics

General Constraints



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



Case V: MB moves, R2 locked



%

1 3 1 3 1 3


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



t t t tJ

t t t t

ff

ff


& &

& &

% % % % % %

% % % % % %

1 3 1 3

1 3 1 3

0 0

0 0p

E I E I E II E II


t t t tJ

t t t t

q q q q

q q q q


& & & &

& & & &

% % % % % %

% % % % % %


Forward Kinematics

General Constraints



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



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



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



-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



Conclusion

Modular Formulation

Motion-Distribution Analysis

Evaluation of Performance Measures

Manipulability Jacobian Matrix Formulation

Effect of Different Actuation Schemes

Optimal Configuration Determination



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% %



Thank You!

Acknowledgments:Dr. V. KroviDr. T. Singh

Dr. J. L. Crassidis& all the audience…


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

14444444244444443 14444444244444443 1442443


Single Module Payload Transport

T

a Mvh fé ù= ê úë û&

%

a M

E ET vJ t t

fé ù= ê úë û&% %

a

ET at J h= &

% %

Documents

Chin Pei Tang May 3, 2004 Slide 1 of 51 Automation, Robotics and Mechatronics Lab, SUNY at Buffalo Chin Pei Tang ([email protected]) Advisor : Dr