模塊化機(jī)器人理論與應(yīng)用(公開(kāi))

模塊化機(jī)器人理論與應(yīng)用楊桂林中科院寧波工業(yè)技術(shù)研究院先進(jìn)制造技術(shù)研究所致謝：新加坡制造技術(shù)研究院 （SIMTech）OutlineIntroduction

to

Modular

Reconfigurable

Robot

System

(MRRS)Serial

MRRSModularRobot

AssemblyRepresentationConfiguration-independent

Kinematics

andDynamics

ModelingKinematicCalibrationTask-based

Designof

Serial

MRRS

ConfigurationsParallelMRRSKinematics

Modelling

andAnalysisSingularityandWorkspace

AnalysisKinematicCalibrationTask-based

Designof

Parallel

MRRS

ConfigurationsQ&AConventional

Industrial

RobotsAconventional

industrial

robot

consists

ofmonolithically

designed

joints

andlinks,

which

has

a

fixed

configurationonce

it

iscommissioned.6-DOF

PUMA

Type4-DOF

SCARA

Type6-DOF

Parallel

TypeModular

Reconfigurable

Robot

SystemModularFlexible

Re-usableEconomicalA Modular Reconfigurable Robot System (MRRS) consistsof a collection ofstandard

actuator

(joint)

and

end-effector

modules

and

customer

designed

links/connectors that can be assembled into different robotdiversity

of

tasks.configurationsfor aModular

Recon幣gurable

Robot

Systems先進(jìn)制造技術(shù)研究所@

中國(guó)科學(xué)問(wèn)材陽(yáng)與工程研究所思rJ

中國(guó)科學(xué)院寧波工業(yè)技術(shù)研究院

(籌〉

Embedded

MotionControllerNetworkCommunicationIntegrated

ServoAmplifierBuilt-inEncoderand

SensorsBrushlessDCServoMotorGear-boxMagneticBrakeBearingAnIMAis

aself-contained,

independent,and

intelligent

actuator

withitsownmotion

control

capabilities.AnIMAis

acompactintegration

of

aservo-motor,

an

amplifier,acontroller,anencoder,

abrake,agearbox,and

thecommunication

interface.Advantages:– Compact

-highlyintegrated

product––––––Intelligent

-

built-in

controller

andsensorSimplewiring

-

onecable

technologyEasy-to-use

-plug&play

capabilitiesReconfigurable-

self-containedRapidlyDeployableEconomicalIntelligent

Modular

Actuator(IMA)智能模塊化執(zhí)行機(jī)構(gòu)Network-based

Distributed

ControlSchemeConventional

Centralized

Control(Complicated

Wiring

Scheme)Decentralized/Distributed

ControlMRJModuleARM

ModuleSmart

MotorPowerCube

ModuleSilverMaxMotorCoolMuscle

MotorMAC

MotorCommercial

Available

IMAsCommercial

Available IMAs

(SHUNK)PRUniversol

rotory

module

with

integroted

con

trol

e

lectroni(sPan-Tilt-Unit

with

integroted

control

elec↑lonics(ompoct

Rotory

module

withintegrated

control

ele【tronicsRotory

module

with

integroted【ontrol

electronics

0

nd

hollow

sh

oftMRDHigh

Power

Rotory

module

with

integroted

con

trol

elec↑ron

icsRotary

module

with

torque

motor(ompoctwrist

module

with

in↑egrated

control

electron

icsMinioture

rotary

module

with

inte

grote

d

control

electronics先進(jìn)制造技術(shù)研究所中國(guó)科學(xué)院寧波材料技術(shù)與工程研究所中國(guó)科學(xué)院寧波工業(yè)技術(shù)研究院(籌〉

Commercial

Available IMAs

(SHUNK)PSM(ompoct

Servo

Drive

withintegroted

control

electroni

【SPlS5αew

driven

lineor

Dxis

with

servoarive

PSMPDU(ompoct

Servo

Drive

with

in愴

groted【ontrol

electronics

ond

pre【ision中國(guó)科學(xué)院寧波工業(yè)技術(shù)研究院(籌〉PLB8elt

driven

li

neor

oxis

with

servo

drive

PDU

MSMHigh

Power

Servo

Motor

with

integroted

(ontrol

electronicsHMulti缸is

light-Weight-Arm

F.斤Sensing

Pockoge先進(jìn)制造技術(shù)研究所中國(guó)科學(xué)院寧波材料技術(shù)與工程研究所Commercial

Available

IMAs

(SHUNK)'KHH&EE--wrshHVnv薩、，嚴(yán)LRUV

aMHLJH陽(yáng)uwf

.f，a、

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LE-2.fin

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Parolle

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Grippe

r

with

vorio

ble

strokePEHLong

-stroke

Gripper

with

integrated

contml

electmnicsEZNServ<rele

cfric

3-Finger

Gentrli.cGrippe

rGripper

for

smoll

(omponentsWSG2-Fillger

PO

ralle

I

Grippe

r

withintegratedl

，electronics

in

the

fingersEGNServo"，electric

2-Finger

Porollel

GripperSDH3-finger

dextrous

roboti

c

hondwith

7

DOF中國(guó)科學(xué)院寧波工業(yè)技術(shù)研究院(籌〉

先進(jìn)制造技術(shù)研究所中國(guó)科學(xué)院寧波材料技術(shù)與工程研究所Modular

Robot

Assembly

RepresentationA

MRRS

canhave

many

possible

robot

configurations.To

automaticallyformulate

the

kinematics

and

dynamicsmodels,

a

mathematic

representationmethod

needs

tobe

proposed

for

modularrobot

assemblies.Modular

Robot

Assembly

Representation-

AGraph

Based

ApproachKinematic

Graph:Kinematic

chains

oflinks

and

joints

can

berepresented

by

graphs.One

logical

way

to

convert

a

kinematic

chain

to

a

graph

is

to

replace

thejoints

by

edges

and

links

by

vertices

as

a

joint

can

be

connected

to

two

linksonly

and

alink

may

accept

more

than

one

joint.Such

a

graph

representation

is

called

a

kinematic

graph

and

has

been

widelyused

in

mechanism

synthesis

and

design.b dabcdfeffcfaeModular

Robot

Assembly

RepresentationJp3Jr2J

r1J

f1L

c2L

p2L

r1L

r1L

c1 Basezyx(a)

ModularRobot

Assembly(b)

Kinematic

GraphL

c1J

f1Base(p1,

p2),

wherep1,

p2

{

x,

y,

z}

andp1

p2.z yx zy xxyzzyxL

r1J

r1L

r1Jr2L

p2Jp3L

c2c

20

Lr1

Lc1

Lr1

J

f

1

0

(

z,

y)

(

z,

y)

0 0(

z,

x) 0(

x,

y) (

z,

y)000Jr1 Jr

2 J

p3Assembly

Incidence

Matrix

(AIM)Extended

Incidence

Matrix

(eIM)p

2

c

2

r1Definition

(Assembly

Incidence

Matrix):

LetG

bethespecialized

graph

of

amodular

robot

and

M(G)

be

its

eIM.

The

assembly

incidence

matrix

(AIM)

of

this

robotA(G)

is

defined

by

substituting

each

of

1s

in

M(G)

with

the

corresponding

port

vector

P

=Incidence

Matrix

0 0 (

z,

x) (

z,

x)Lp2

000(

y,

z)L

11010000Lc1

r1

00101101L

0001L

J

f

1J

r1J

r

2J

p30

Mathematical

Background(Group

Theory

andDifferential

Geometry)Rigidbody

motion

–

SpecialEuclidean

Group,

SE(3) (

Lie

Group)Instantaneous

rigid

body

motion

-se(3) (

Lie

Algebra

)Relationshipbetween

LieGroup

(T

SE(3)

)and

Lie

Algebra

( ?s

se(3))

SE(3),

with

R

SO(3)

and

p

3

14

4

0 1

R p

T

so(3)

and

v

(v

,

v

,

v

)

3

1

4

4

0

0

z

y

0with

ω?

se(3),v

0 0

s?

ω?

x

x y zy xzv

(v

,

v

,

v

)

3

1with

(

,

,

)

3

1

ands?

s

(v,

)

6

1,x y zx y zsuch

that s?

Log(T

)se(3)LogSE(3)

Exp

SE(3) such

that

T

es?

.se(3)Mathematical

BackgroundMatrix(Twist)

Exponential(1)(1)Mathematical

BackgroundMatrixLogarithm:Adjoint

RepresentationAn

element

ofaLie

group

can

be

identifiedwitha

linearmapping

betweenitsLie

algebra

viathe

AdjointRepresentation.Forevery

T

SE(3)

and s?

se(3),

theadjoint

map

AdT

:

se(3)

se(3)

is

defined

by:Mathematical

Background.s

R ?pR

0 R

AdT

(

s

)

AdT

(

?s

)

T

?s

T6

6

1

Elements

ofa

Liealgebra

can

also

be

identified

withalinear

mapping

betweenitsLie

algebra

viathe

Liebracket.

Given

an

element

x

se(3),

its

adjointrepresentation

isthe

linear

map

adx

:

se(3)

se(3),

defined

by

adx(y)

=[x,

y].Ifx

=(

1,

1)

and

y

=

(

2,

2)

are

elements

of

se(3),Kinematic

Modeling:

D-HParameters

v.s.

POEParametersKinematic

transformation

for

two

adjacent

links– Dyad

Kinematics

with

D-HRepresentation– Dyad

Kinematics

with

POE

RepresentationConversionFrom

POE

parameters

to

D-H

parameters:

DirectandUniqueFromD-Hparameters

toPOEparameters:

Non-unique

but

Flexible?si

qiTi

1,i

(qi

)

Ti

1,i

(0)eFor

a

revolute

joint,

qi

i

;

For

a

prismatic

joint,

qi

di

.dicos

i

1,i10sin

i

1,icos

i

1,i0

sin

icos

icos

i

1,icos

i

sin

i

1,i0

sin

i

sin

i

1,i

di

sin

i

1,i

(q

)

sin

icos

i

1,icos

i

i

1,i

,

ai

1,i

,

i,

and

diai

1,ii

1,i iT

D

-

Hparameters;

0a-1,ix-1yi-1zi-1zixiyiAxis

i-1Axis

iLink

i-1Link

id

i-1,i

iziyixizi

-1xi

-1yi

-1sii

-1iLinki-2linki

-1link

iProduct-of-Exponentials

(POE)

FormulaDyadKinematicsLocal

POE

Formulaziyixizi

-1xi

-1yi

-1sii

-1iLink

i-2link

i

-1link

iq

; e

s?i

qi

SE(3).s?

s

[

ω

,

v

]T

6

1;i i i iwith ω?

i

so(3)

and

vi

;

se(3)(0)

SO(3)

and

p (0)

3

1;with

R

SE(3)

Ri

1,1

(0) pi

1,i

(0)

Ti

1,i

(0)

0 0

ω? v

s?

0 1ii iii

1,ii

1,i

4

4

4

4

3

1?si

qiTi

1,i

(qi)

Ti

1,i

(0)eBasex0y0z0xi-1yi-1xiAxis

i-1xnxn+1zi-1

(si-1

)(s

n+1

)Toolynyizi

(si

)(sn)n+1yn+1znzn+1AxisiAxisnT0,n

1

(q1

,

q2

,

...

,

qn

)

T0,1

(q1

)

T1,2

(q2

)

...

Tn

1,n

(qn

)

Tn,n

1

T (0)e

s?1q1

T (0)e

s?2q2

...

T (0)e

s?nqn

T0,1 1,2 n

1,n n,n

10 01 0 0

0 0 1 0

0 0

0

0 1

0T

(0)

0

1

0,1Automatic

Generation

ofKinematics

Model

23

1 00 1 0

1 0 0 d

0 0 0 1

0

T

(0)

0

0

2,31

d34

0

0

1

0 0

1 0 0

0 0 10 0

0T

(0)

3,4

0 1

1 0 0 0

0 1 0

0 0 0 1

0

T4,5

0

d45Jp3Jr2J

r1J

f1L

c2L

p2L

r1L

r1L

c1yzxyxzxyzzyxzyx(a)

Modular

Robot

Assembly(b)

Kinematic

GraphBaseL

c1J

f1L

r1J

r1L

r1Jr24L

p2Jp35L

c21 (0)Base0

Lc

2

Lr1

Lp2

Lc1

r1

(

z,

y)

(

z,

y)0 0 0(

z,

x) 0 0(

x,

y) (

z,

y) 00 (

z,

x) (

z,

x)0 0 (

y,

z)000p3r

2r1f

1JJJJL23Based

onthe

AIM

and

LocalPOE

formula,

wehave:T (q)

T (0)es?1q1T (0)es?2q2

T (0)es?3q3

T (0)es?4q4

T0,1 1,2 2,3 3,4 4,50,5d12d23d34d45s1

(

1,

1

)

(0,

0,

0,

0,

0,

0)s2

(

2

,

2

)

(0,

0,

0,

0,

0,

1)s3

(

3

,

3

)

(0,

0,

0,

0,

0,

1)s

(

,

)

(0,

0,

1,

0,

0,

0)4 4 412

0 01 0 0

0 0 1 d

0 0 1

0

T(0)

0

1

1,2q

(

,

100mm,

,

0,

,

,

,

,

,

)4 4 4 4 4 4 4 4xyz1Base

0x11yzx8

yzBaseLr1Lp1Jr2Lr1Lr2

JLr2Branch

1

Lc2Jr3Lr2Jr1Jp1f2 r2JLr2Lr2Lr2Lc2Jr3 Jr3Jr3Jr3Branch

2234576109Forward

Kinematics

Computation

ExampleT10,11

T0,1

(0)e T1,2

(0)e T2,3

(0)e T3,4

(0)e T4,6

(0)e T6,8

(0)e T8,10

(0)e

T0,9

(q)

T0,11(q)

s?

q s?

qs?

qs?

qs?

qs?

q s?

qT0,1

(0)e T1,2

(0)e T2,3

(0)e T3,4

(0)e T4,5

(0)e T5,7

(0)e T7,9

(0)s?1q1 s?2q2 s?3q3 s?4q4 s?5q5 s?7

q710101

1 2

2 3

3 4

4 6

6 8

8dqindT0,n

q

T0,ni

1i0,i

i

i,n)T

T T (0)es?i

qi

s?

T0,i

1

i

1,i i

i,ni,n

qi

qies?i

qii

1,i

(0)0,i

10,n

T s?

T

(T

T

TndTo,n

T0,i

s?iTi,n

dqii

1nAdT si

dqi0

,in

i

1

T0,i

siT0,i

dqini

1 i

1dqi

dTo,nT0,n

T0,i

siTi,nT0

,n

1

1

1

??T1,2

(0)e ...Tn

1,n

(0)eT0,n

(q1,

q2

,

...

,

qn

)

T0,1

(0)e

log(T

d

T

1

)

log(T

d

T

1

)4

4dT T

1

(T

d

T )T

1

T

d

T

1

I0.n 0,n0.n 0,n0n 0,n 0.n 0,no,n 0,n 0.nAdT si

dqi0

,in

ilog(T nT n

)

d

1

0. 0,Inverse

Kinematics

Model

(A

numerical

solution

approach)The

purpose

ofinverse

kinematics

istodetermine

therequired

jointangles

withgivenend-effector

pose.s?1q1 s?2q2 s?n

qn

T

JdqInverse

Kinematics

Modeldq

(dq

,

dq

,...,

dq

)

n

1

1 2 ns

]

6

n

,

spatial

manipulator

Jacobian21

T

log(T

d

T

1

)

6

1,0.n

0,nAd s ... AdJ

[

Ad sT0,nT0,

2T0,1nThe

differential

kinematics

model

can

be

written

as:pose

difference

vector

viewed

at

thebase

framedifferential

changeof

the

joint

anglesRobot

Description

T(0),S

Given

Desired

Tool

Pose

TdIntial

GuessSolution

q0q

=

q +dqii-1iDerivation

of

dqdq

=J*.

log

(T T )i 0

,n

0

,n

1 di

=

1YesTerminateNo

<

6,7(0)es?6q6

T4,5 5,6(0)es?4q4

T (0)es?5q5

T3,4(0)es?3q3T2,3(0)es?2q2

T1,2(0)es?1q1T0,10,7T (q)

TInverse

Kinematics

Computation

ExamplezxyzBasexyJr1BaseLr1Lr1Lr1Lr1Lr1Lr2Lc2Jr1

Jr1Jr1Jr3

Jr223456701

Lc

2

0

Lr1

Lr

2

Lr1

Lr1

Lr1

r1

00

0

AIM

0 0(

z

,

y

) 0(

x

,

z

) (

z

,

y

)(

x

,

y

)

(

z

,

x

)

0 (

y,

x

)

0 0000L

1164.93

115.24

149.23

0.4571

0.4267

0.78030.8750

0.2286

0.426800.2286

0.58220.78030100,6T

dq=[0.7854,

0.7854,

0.7854,

0.7854,

0.7854,

0.7854

]07

(z

,

y)00000

(

x

,

y

)(

z

,

y

)00000000(

x

,

y

)(

z

,

y

)00000(

x

,

y

)J

r1J

r1J

r1J

r1J

r

2J

r

3Inverse

Kinematics

Computation

ExampleComparison

withthe

D-H

approachCasesContentsPOEMethodD-HMethod1Initial

Guess[0,0,

0,0,

0,0][0,0,

0,0,

0,0]Iterations617Times

(ms)2.25.98Solution[-.445,-.785,

2.356,

-.446,

.785,

.785][.525,

-.583,

2.375,

-.88,-.167,

.879]2Initial

Guess[0.2,

0.2,

0.2,

0.2.,

0.2,

0.2][0.2,

0.2,

0.2,

0.2.,

0.2,

0.2]Iterations1496Times

(ms)4.7236.85Solution[.525,

-.583,

2.375,

-.878,

-.167,

.879][.785,

-.446,

2.356,

-.785,

.-446,

.785]3Initial

Guess[0.4,

0.4,

0.4,

0.4.,

0.4,

0.4][0.4,

0.4,

0.4,

0.4.,

0.4,

0.4]Iterations540Times

(ms)1.8717.96Solution[.183,

.58,

.767,

.878,

1.416,

.878][-.446,-.785,

2.356,

-.446,

.785,

.785]4Initial

Guess[0.6,

0.6,

0.6,

0.6.,

0.6,

0.6][0.6,

0.6,

0.6,

0.6.,

0.6,

0.6]Iterations58Times

(ms)1.762.86Solution[.183,

.58,

.767,

.878,

1.416,

.878][.183,

.58,

.767,

.878,

1.416,

.878]5Initial

Guess[0.75,

0.75,

0.75,

0.75.,

0.75,

0.75][0.75,

0.75,

0.75,

0.75.,

0.75,

0.75]Iterations34Times

(ms)1.11.37Solution[.785,

.785,

.785,

.785,

.785,

.785][.785,

.785,

.785,

.785,

.785,

.785]6Initial

Guess[0.8,

0.8,

0.8,

0.8.,

0.8,

0.8][0.8,

0.8,

0.8,

0.8.,

0.8,

0.8]Iterations33Times

(ms)0.990.93Solution[.785,

.785,

.785,

.785,

.785,

.785][.785,

.785,

.785,

.785,

.785,

.785]Configuration-Independent

Dynamicsj*Link

jJoint

jNewton-Euler

Equation

for

aLink

AssemblyajFj

=(fj,

j)

6

6Configuration-Independent

DynamicsAfter

substitution,

wehavej*Joint

jjFj

=(fj,

j)Link

jNewton-Euler

Equation

expressed

in

Module

frame(j)Letthekinematictransformation

fromj*

toj

be

Tj*,J=(Rj*,j,

pj*,j),

then-1If,thenDynamics

of

a

Tree-structured

Modular

RobotRecursiveNewton-Euler

AlgorithmForward

Iteration:Initialization:Recursion:1234567zyxzyxzBranch

2Lc2Jp3Lp2Jr3Lr2Jf2Branch

1Lc2Jp3Lp2Jr3Lr2Jf2Lc1Jp1BaseyBasexLp1(a)

A

tree-structured

robot(b)the

kinematic

graphConfiguration-Independent

DynamicsBackward

Iteration:Initialization:

starting

fromall

pendent

linksRecursion:1234567zyxzyxzBranch

2Lc2Jp3Lp2Jr3Lr2Jf2Branch

1Lc2Jp3Lp2Jr3Lr2Jf2Lc1Jp1BaseyBasexLp1(a)

A

tree-structured

robot(b)the

kinematic

graphClosed

FormEquations

ofMotionExpanding

therecursiveequations,

matrix-form

equations

ofvelocities,

accelerations,

and

forces

canbe

given

by:generalizedClosed

Form

Equations

of

Motion.Lt\_

1

==

diag

[-ad咐1，一αd

s坤

f

-

v

一αd時(shí)機(jī)l

ER67zxh

;A

2

=

dmg

[-GdF1744γ.;

-adfn

] εWMn

;FE，==

COl1lmn[

P{

，P2，，F(xiàn)

] ζ

6nx

l

is

the

external

\\rrench

vectoI':HTo==EE，d

，dA

Aι9毛6nx6Ady礦16x

6 0TMAdTJ I6×60016x6H

==IT13Ad:ri"31r23AdT?31。。oIε

W6nx

6η..TIJdTFJThAdzuhnA

向L1

·咿

?

16x

6Not

e that 冗(G)=[r'i，]jε捉(n+取(葉1)is

the

aocessibility lnatrix oft

he

robot

's

kinenlatic

digraph

G.@

中國(guó)科學(xué)問(wèn)材料技術(shù)與工程研究所飛

中國(guó)科學(xué)院寧波工業(yè)技術(shù)研究院(籌〉Closed-form

Equations

of

MotionThe

closed

form

equations

of

motion

can

be

written

as:Computational

ComplexityImplementation

and

ExamplesInverse

DynamicsSolvingtheinversedynamics

problem

isnecessaryfor

robot

motionplanningandcontrolalgorithm

implementation.Once

arobot

trajectory

isspecified

interms

ofjointangles,velocities,

andaccelerations,

andtheexternal

forces

are

alsogiven,

the

inversedynamicscomputes

the

torques

to

be

applied

tothe

jointstoobtain

the

desired

motion.This

computationisalsouseful

for

verifying

thefeasibility

of

the

imposedtrajectory

andforcompensatingnonlinear

terms

inthe

dynamicmodel.The

computationofinversedynamics

becomes

quitestraightforward

once

thedynamicmodel

isderived.Implementation

and

ExamplesForward

DynamicsSolvingthe

forwarddynamics

problem

isnecessaryfor

the

robot

simulation.Forward

dynamicsallows

describing

the

motionof

therobot

system

interms

ofthe

joint

accelerations,

whenaset

ofassignedjoint

torques

isapplied

totherobot.The

jointvelocities

and

anglescanbe

obtained

byintegrating

the

system

ofnonlinear

differential

equations.The

closed-formequationsof

motionderived

form

the

proposed

algorithm

aremore

convenient

forthis

purpose

because

they

provide

the

explicit

relationshipbetween

thejointtorques

(aswell

as

the

external

forces),

angles,velocities,andaccelerations.Implementation

and

ExamplesAutomatic

dynamic

model

generalization(a)

A

tree-structuredrobot(b)

the

kinematic

graphBase1234567yxzyxzyxzJp1BaseBranch

2Lc2Jp3Lp2Jr3Lr2Jf2Branch

1Lc2Jp3Lp2Jr3Lr2Jf2Lc1Lp1Implementation

and

ExamplesAutomatic

dynamic

model

generalization(a)

A

tree-structuredrobot(b)

the

kinematic

graphBase1234567yxzyxzyxzJp1BaseBranch

2Lc2Jp3Lp2Jr3Lr2Jf2Branch

1Lc2Jp3Lp2Jr3Lr2Jf2Lc1Lp110,

thenAssumeForward

Dynamics

Example(a)

A

tree-structured

robot(b)the

kinematic

graphBase1234567yxzyxzyxzJp1Lp1

BaseBranch

2Lc2Jp3Lp2Jr3Lr2Jf2Branch

1Lc2Jp3Lp2Jr3Lr2Jf2Lc1Known

conditions:Implementation

and

ExamplesImplementation

and

ExamplesForward

Dynamics

Example0.80.60.40.20-0.2-0.4-0.6-0.80.20.40.60.811.21.4Joint

Acceleration

(1/s

2

or

m/s

2)Time

(second)Joint

5Joint

6,7Joint

1Joint

40.80.60.40.20-0.2-0.4-0.6-0.80.2 0.4 0.6 0.811.2 1.4Joi

nt

Veloci

ty

(rad/s

or

m/

s)Time

(second)Joint

5Joint

1Joint

6,7Joint

40.80.60.40.20-0.2-0.4-0.6-0.80.20.40.60.811.21.4Joint

Displacement

(

rad

orm)Time

(second)Joint

5Joint

4Joint

1Joint

6,7(a)

A

tree-structured

robot(b)the

kinematic

graphBase1234567yxzyxzyxzJp1BaseBranch

2Lc2Jp3Lp2Jr3Lr2Jf2Branch

1Lc2Jp3Lp2Jr3Lr2Jf2Lc1Lp1Implementation

and

ExamplesForward

Dynamics

Examplet

=

0t

=

0.3t

=

0.6t

=

0.9t

=

1.2t

=

1.5Modular

Robot

CalibrationThe

Need

forCalibrationKinematic

Calibration

--

an

effective

approachtoimprove

robot

accuracyAn

integrated

process

ofmodel

formulation,

pose

measurement,

parameter

identification,andcalibration

result

implementation.Modular

Robot

Calibration

--

Configuration-independent

ApproachMachining

toleranceAssemblyerrorsWear

anddriftDiscrepancyModular

RobotControllerActualParametersNominalParametersKinematic

Calibration

ModelForward

Kinematics

Tn,n

1

)(e

Ti

1,i

(0); eT0,n

1

e etn

1tn

1t1 ?s1q1 t2 ?s1q1titn snqne e ...e e e?????? ?Assumptions:–

Kinematicerrorsexist

only

in,,)

retain

theirnominal

values.Error

Model

—

A

linear

equation

with

respect

totn

1t2

...

Ad

t1

Ad

1T0

,n

1T0

,n

1?s2q2?t1 ?t2?s1q1e?t1

e?s1q1...e ee e e e?snqn?tn

?

?

?

?tiTi

,i

1

(0)

hence

inti

se

(3)

denoted

by

ti

;–

The

joint

axes

(

?si

)and

thejoint

angle

(qi??log(T0,n

1

T0,n

1

)Va

1=iT0

,in

i

1