chin pei tang ([email protected]) advisor : dr. venkat krovi

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

http://mechatronics.eng.buffalo.edu/home_new.html


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

00

0 0 0

z x z

z y x

y

vv v

v

w ww

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 sinsi0 0 1n cosF

MAxy

ffff

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 sinsi

00

0 0 0n 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+ =& &

NonholonomicConstraints

Mobile Platform




Kinematic Model

00

0 0 00M

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+ =& &

NonholonomicConstraints

Mobile Platform




Kinematic Model

0 1 0 0 0 11 0 0 0 0 00 0 0 0 0 0

M MvM

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 cossin cos sin0 0 1

MA

LA L

q q qq q q

é ù-ê úê ú= ê úê úê úê úë û2 2 2 2

2 2 2 2


AB

LA L

q q qq q q

é ù-ê úê ú= ê úê úê úê úë û3 3 3 3

3 3 3 3


kB

E

LA L

q q qq 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




Kinematic Model

1 2 3kk kk

k k

EE EEM M A BA BE ET 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

1L S L S

LC L C L

é ùê úê ú+ê úê úê ú+ +ê úë û123

123

0CS

é ùê úê úê úê úê ú-ê úë û1 23 2 3

1 23 2 3 3

1L S L S

LC L C L

é ùê úê ú+ê úê úê ú+ +ê úë û2 3

2 3 3

1L S

L C L

é ùê úê úê úê úê ú+ê úë û 3

10L

é ùê úê úê úê úê úê úë û

Twist Vectors

Assembled

1 2 31

2

3

k k k k

k

kk

M

E

M

E E EE FE

Ev

v

t t t tt t q q qf

f

q

q

q

é ùê úê úê úê úé 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 tq q q

é 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 tf

é ù= ê úë û&% %aMvf

hé 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 ûM % ( ) ( )aTrank G rank J=

Not constructive

ReciprocalWrench

Alternate constructive approach

1 23 2 3 123

1 23 2 3 3 123

1 0aTJ 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 Cw S

C

é ù- - -ê úê ú= ê úê úê úê úë û%




Condition:

Transform an arbitrary twist from {Ek} to {M}:

0The Motion Planning Strategy

Given arbitrary twistTE

Ez 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

EzM F

E 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= = =& &% % %%


1n

nh ´ Î& ¡% 1

E mmt ´ Î ¡%

( )T m nm 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 L144244314444244443L

( )1 1E

m T nm nt J q h´ ´´

é ù= ë û &% % %

{ }: , 1E EV Tt t Je h h= = =& &% % %%


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=


11 1 2 12 2 12

21 1 2 12 2 12

1 1E

E

E

X L S L S L SLC 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

J ss

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

J ss

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+ =& & %% %

0a pC a C pJ Jh h+ =& & %% %

1p 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 0sin cos 00 0 1

k

k kE

k kEAd dd d

é ù-ê úê ú= ê úê úê úê úë û

( ) ( )( ) ( )

1 2 3 2 3 3

1 2 3 2 3 3

1sin sin sincos 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

0cossin

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

1sin sin sincos 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

1sin sincos 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 û3

3

1sincos

k

k

LL

dd

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 actuatedCase II – MB static, R2 actuatedCase III – MB moves, R1 & R2 passiveCase IV – MB moves, R1 lockedCase V – MB moves, R2 locked

TE E Ex yt v vé ù= ê úë û%

ah&% ph&% aTJ pTJ aCJ pCJa p

ET a T pJ J th h+ =& & %% %

0a 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, 3.4641,2II IIx y = ( ) ( )1 1, 0,4II I II Ix y =330Id = ° 210IId = ° 90I I Id = °1Iel = 1I Iel = 1II Iel =




Case I: MB static R1 actuated

1 1 1TI II I I I

ah q q qé ù= ê úë û& & &&

%2 3 2 3 2 3

TI I I I I I II I I I Iph q q q q q qé ù= ê úë û

& & & & & &&%

10 0

a

E ITJ tqé ù= ê úë û&% % %

2 30 0 0 0

p

E I E ITJ t tq q

é ù= ê úë û& &% % % % % %

1 1

1 1

00a

E I E II

C E I E III

t tJ

t tq q

q q

é ù-ê ú= ê ú-ê úë û

& &

& &

% % %% % %

2 3 2 3

2 3 2 3

0 00 0p

E I E I E II E II

C E I E I E III E II I

t t t tJ

t t t tq q q q

q q q q

é ù- -ê ú= ê ú- -ê úë û

& & & &

& & & &

% % % % % %% % % % % %


Forward Kinematics

General Constraints




Case Study I-A1 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-B1 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

0.50.5

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 tqé ù= ê úë û&% % %1 3

0 0 0 0p

E I E ITJ t tq q

é ù= ê úë û& &% % % % % %

2 2

2 2

00a

E I E II

C E I E III

t tJ

t tq q

q q

é ù-ê ú= ê ú-ê úë û

& &

& &

% % %% % %

1 3 1 3

1 3 1 3

0 00 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 tq q q q

q q q q


& & & &

& & & &

% % % % % %% % % % % %

2 2 2TI II I I I

ah q q qé ù= ê úë û& & &&

%1 3 1 3 1 3

TI I I I I I I I I I I Iph 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

TI I I I I I I I I II Ia 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 Iph q q q q q q q q qé ù= ê úë û

& & & & & & & & &&%

0 0 0 0a M

E I E IT vJ t tf

é ù= ê úë û&% % % % % %

1 2 30 0 0 0 0 0

p

E I E I E ITJ t t tq q q

é ù= ê úë û& & &% % % % % % % % %

0 00 0

M M

a

M M

E I E I E II E IIv v

C E I E I E III E II Iv v

t t t tJ

t t t tff

ff


& &

& &

% % % % % %% % % % % %

1 2 3 1 2 3

1 2 3 1 2 3

0 0 00 0 0p

E I E I E I E II E II E II

C E I E I E I E III E II I E II I

t t t t t tJ

t t t t t tq 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-Motion0

a 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 manifoldLock this number

of joints



Self-Motion

1 2 3 1 2 3

1 2 3 1 2 3

0 0 00 0 0p

E I E I E I E II E II E II

C E I E I E I E II I E II I E III

t t t t t tJ

t t t t t tq 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

TI I I I I I I II II Ia M M Mv v vh ff fé ù= ê úë û

& & &&%

2 3 2 3 2 3TI I I I I I II 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 tf

é ù= ê úë û&% % % % % %2 3

0 0 0 0p

E I E ITJ t tq q

é ù= ê úë û& &% % % % % %

0 00 0

M M

a

M M


C E I E I E III E IIIv v

t t t tJ

t t t tff

ff


& &

& &

% % % % % %% % % % % %

2 3 2 3

2 3 2 3

0 00 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 tq 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

0.70.7

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

TI I I I I I I I I II Ia M M Mv v vh ff fé ù= ê úë û

& & &&%

1 3 1 3 1 3TI I II I I II 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 tf

é ù= ê úë û&% % % % % %

1 30 0 0 0

p

E I E ITJ t tq q

é ù= ê úë û& &% % % % % %

0 00 0

M M

a

M M


C E I E I E III E IIIv v

t t t tJ

t t t tff

ff


& &

& &

% % % % % %% % % % % %

1 3 1 3

1 3 1 3

0 00 0p

E I E I E II E II

C E I E I E III E II I

t t t tJ

t t t tq 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

% %TI 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

0.70.7

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

0.50.5

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 Adt

-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

%

00zE F

Ez

ww

-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 ûL14444444244444443 14444444244444443 1442443



Single Module Payload Transport

T

a Mvh fé ù= ê úë û&

%a M

E ET vJ t tf

é ù= ê úë û&% %a

ET at J h= &% %


chin pei tang ([email protected]) advisor : dr. venkat krovi

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