CONSTRAINT TREATMENT # //_) TECHNIQUES AND PARALLEL ...

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Nov 7, 1990 - on bodies i and j is given by f(i)= + fala. ;. fU) = w fala ...... A. V., Utku, S., Wada, B. K., and Chen, G. S. "Effect of Im- perfection on Static. Control.
CU-CSSC-90-26

CENTER

FOR

SPACE

STRUCTURES

AND

CONTROLS

='? .,; _i // //

CONSTRAINT

TREATMENT

TECHNIQUES

AND

PARALLEL

ALGORITHMS

FOR

MULTIBODY

DYNAMIC

#

//_)

ANALYSIS

by Jin-Chern

Chiou

_ j K

Un c ] (;._/;O

O_

_s

i'_ k:1 7 m m

November

1990

COLLEGE

OF

UNIVERSITY CAMPUS BOULDER,

ENGINEERING OF

BOX

COLORADO

429

COLORADO

80309

CONSTRAINT

TREATMENT

AND

FOR

PARALLEL

TECHNIQUES

ALGORITHMS

MULTIBODY

DYNAMIC

ANALYSIS

by JINoCHERN

CHIOU

University,

Taiwan,

B. S., Feng-Chia

M. S., University

of Colorado

A dissertation Faculty University of the

of the

of Colorado requirement Doctor

Aerospace

at Boulder.

submitted Graduate

R. O. C.,

in partial for the

of the fulfillment

degree

of Philosophy Engineering 1990

ig86

to the School

Sciences

1981

of

Chiou,

Jin-Chern

Constraint

(Ph.D.,

Treatment

for Multibody Dissertation

by Professor

Computational

procedures

multibody

bust

constraint and

e_cient

stabilization

modularity

of the

Second, staggered

angular

are

then

obtained between

Third, mentation

algorithm,

parameters

of the present

the

and

and

solution

and

takes

angular procedures

constraint The

a two-stage advantage

of a

integration

a two-stage

staggered

coordinates

of bodies via

in MBD

the

velocities.

ro-

analysis.

translational

algorithm

two

programming

parallelizable uses

de-

developed.

of motion, that

orientations

to obviate

are

for MBD

equations

a robust

to minimize

f. e., penalty

addressed

of from

matrices,

stability,

an implicit

and

ill-conditioned

accuracy,

to integrate

by using

First,

process

this algorithm

The angular

Euler

the combination accurate

Mainly,

algorithm

velocities.

and

viewpoint.

scheme,

numerical

analysis

are developed

partitioning

governing

procedure

dynamic

systems

techniques,

are also

the

is developed.

the

involving

enhancing

to treat

difference

tionship

for

and

integration

treatment

techniques

solution

algorithm, central

solution

explicit-implicit

partitioned

(DAEs)

and natural

issues

kinematic (MBD)

the time

constraint

technique

computational

for

dynamic

during force

Algorithms

K. C. Park

equations

violations

Sciences)

Analysis

differential-algebraic

graded

Engineering and Parallel

directed

constraint

more

Techniques

Dynamic

three-dimensional the

Aerospace

and systems

kinematic

rela-

It is shown

that

yields

a computationally

procedures,

parallel

solution. to speed

of the

present

up the

computational

constraint

treatment

techniques,

two-stage

implestag-

PRECEDING

PAGE BLANK

NOT

FILMED

iv

gered

explicit-implicit

To this end,

the

transformed

into

been

derived.

niques, used

DAEs

systems

tial and parallel

racy the gate

and

of the speed

the

testbed

efficiency

up of the

to which

conjugate

gradient

written

This

testbed

of the constraint

staggered

and

explicit-implicit

Schur-complement-basecl

algorithm

on a parallel

implemented used

treatment

computer.

been

form

has tech-

algorithm

is

form.

in the present in both

sequen-

to demonstrate

techniques,

numerical parallel

have

complement

developed

has been

out.

analysis

numerical

in Schur

carried

complement

structural

procedures

designed

techniques

Schur

matrix

computational

effciently

treatment

the sparse

equations

computers.

two-stage

gradient

matrices

has been

has been

constraint

preconditioned

the

a software

robustness

the

algorithm

By fully exploiting

To evaluate work,

and

arrowhead

a parallel to solve

numerical

the

algorithm,

preconditioned

the accuand conju-

This dissertation for the

Doctor

Jin-Chern has

been

of Philosophy Chiou

approved

Department Aerospace

Engineering

for the of Sciences

by

/, -/'_

/ i



. Park

C. A. Felippa

_-.

"

degree

by

V

To my mother

and

to my wife, and

in memory

of my father.

Min-Hui,

vi

ACKNOWLEDGEMENTS

I would ance,

like to thank

patience,

Without

and encouragement

his excellent

dynamics,

the

suggestions

puters.

To

the

Adjoint reading

also

Professors

J. Housner my thesis

Finally,

the

of Scientific

acknowledged.

financial

Research Research

for the

knowledge

making support

Center

valuable provided

through

through

grant

grant

studies.

of computational possible.

for his invaluable of running long

com-

he has

as a result

spent

of their

people.

members; for

support

parallel

hours

gained

to all these

E. Schmitz;

for his guid-

of my thesis

been

many

my committee

and

and

have

aspects

I am indebted

like to thank

course

C. Farhat

The

K. C. Park,

knowledge

not

practical

writing.

through

NASA/Langley Office

the

is immeasurable. I would

would

C. A. Felippa

thesis

the

and outstanding

Professor

concerning

Professor

during

and thesis

to thank

Professor

in critiquing assistance

research

research

I also wish and

my advisor,

Professor

devoting

R. Su,

their

time

to

suggestions. by

Dr.

J. Housner

NAG-I-756

F49620-87-C-0074

and

Air

of the Force

is gratefully

CONTENTS CHAPTER I.

INTRODUCTION 1.1

Overview Systematic

1.1.2

Evaluations

Study

of MBD

of Existing

1.2.1

Robust

1.2.2

Explicit-Implicit

1.2.3

Parallel Implementation Procedures Analyzer ..........................................

2

Procedures

....

6

.............................................. and Efficient

Dissertation

2.1

Introduction

2.2

Reference

2.3

Angular

2.4

Euler

2.5

Rodrigues

2.6

Euler

2.7

Angular

2.8

Time

2.9

Velocity

Treatment

Integration

Outline

KINEMATICS

8 of Constraints

of MBD

Equations

Frames

Angles

Derivatives and and

......

10

...................................

13

...................................

14 14

..................................... of a Particle

Parameters

Velocity

9

12

14 .....................

..........................................

Parameters

........

for MBD

..........................................

Orientations

Velocity

...............

Computational

Objectives

2.10

1

Formulations

1.2

SPATIAL

1

...............................................

1.1.1

1.3 II.

...........................................

17 19

.................................

21

......................................

22

......................................

25

of Euler

Acceleration Acceleration

Parameters

.................

26

of a Particle

.................

28

of a Rigid

Body

............

28

PRECEDING

PAGE

BLANK

NOT

FILMED

.o°

Vlll

2.11 Ill.

EQUATIONS Introduction

3.2

The

3.3

D'Alembert's

3.4

Governing

3.5

Interactions

3.5.2

3.6

The

.................................

MOTION

FOR

MBD

30

SYSTEMS

.........

..........................................

Principle

Work

Principle

................................

Equations

.........................

of Motion

for Rigid

and

Flexible

Bodies

Bodies

Equations Interaction

of Motion: of Flexible

and

Concluding

Remarks JOINTS

33 37

........................

of Motion: of Flexible

38 .............

.....................

Rigid

Bodies

FORCE

43

44

..........

.................................. AND

32 32

of Virtual

Equations Interaction

KINEMATIC

46 48

ELEMENTS

..........................................

..........

50

4.1

Introduction

4.2

Spherical

Joint

........................................

51

4.3

Universal

Joint

. .......................................

54

4.4

Revolute

........................................

56

4.5

Cylindrical

4.6

Rigid

Joint

4.7

Force

Elements

Joint Joint

50

......................................

59

............................................ in MBD

Systems

......................

4.7.2

External

Forces

4.7.3

Actuator

Forces

.....................................

63

4.7.4

Damping

Forces

.....................................

64

4.7.5

Spring

Forces

Concluding

..................................

61

Gravitational

SOLUTION

Force

61

4.7.1

4.8 V.

The

Remarks

OF

3.1

3.5.1

IV.

Concluding

and

Moments

62

.......................

........................................

Remarks

PROCEDURES

65

.................................. FOR

DAEs

62

66 ...................

67

ix

5.1

Introduction

5.2

Reviewing

of Existing

5.2.1

Stiffly-Stable

5.2.2

Direct

5.2.3

Generalized

5.2.4

Baumgarte's Method

5.2.5

Null

Gear

Integration

Space

Solution

Procedures

Method Method

Coordinate

67

68

...........................

71

Constraint Violation ..........................................

Method

..................................

78

Stabilization

Technique

5.4

Natural

Partitioning

Scheme

..........................

5.4.2

A Mutiple

5.4.3

A Closed-Loop

Open

Chain

Open

Chain

Partitioning

5.5.2

Two-Stage

5.5.3

Update

of Translational

5.5.4

Update

of Euler

5.5.5

Update

of Constraint

5.5.6

Penalty Constraint Implementation

Concluding

the

System

MBD

System

System

Solution

5.5.1

Natural

MBD

MBD

Explict-Implicit

72

75

Constraint

A Single

.........

Stabilization

Penalty

5.4.1

67

..........................

Partitioning

Method

.............

5.3

5.5

VI.

..........................................

Staggered

Equations

Explicit-Implicit and

Parameters Forces

81 84

..................

84

................

88

........................

Procedures

Governing

............

Angular

90

...................

92

of Motion

.....

Procedure Velocities

.... .....

Scheme

Remarks

PARALLEL IMPLEMENTATION GOVERNING EQUATIONS 6.1

Introduction

6.2

Parallel Implementation Stabilization Technique

94 101

........................

102

........................

103

Stabilization Technique .................................

Partitioning

92

104

Implementation

.......

................................. OF THE OF MOTION

......................................... of Penalty Constraint ............................

104 105

............

107 107

108

X

6.3

6.4

A Parallel

6.5

Preconditioners

6.5.1

Diagonal

6.5.2

Scaled

6.6 VII.

Parallel Implementation of Natural Scheme ............................................

Concluding

NUMERICAL

Gradient

Solution

Preconditioner Preconditioner Remarks

7.3

Deployment

7.4

Dynamics

7.5

Dynamic

Simulation

of a Closed

7.6

Dynamic

Simulation

of a Space

7.7

Dynamic

Simulation

of Automobile

Summary

8.2

Directions

REFERENCE

117 118 119

.........................................

Crank-Slider

8.1

116

................................

The

Mechanism

of Three-Bar of a Bowling

119

......................... Manipulator

Ball

119

...............

Four-Bar Crane

134 Linkage

................

for Further

147

165 166

....................................

.....................................................

143

157

.................................

Research

....

Suspension

........................................... of Work

128

..........................

........................................... Remarks

112 115

.................................

7.2

CONCLUSIONS

......

..............................

Introduction

Concluding

Method

............................

7.1

7.8

111

......................................

EXAMPLES

Systems

VIII.

Conjugate

Partitioning

......................

166 170

171

Figures

Figure 1.1

MBD

Equations

of Motion

Computational 2.1

Position

Vector

2.2

System

Rotational

2.3

Translation

Corresponding

................................

Coordinate Rotation

Space

3 .................

...............................

of a Body

Interaction

of Flexible

bodies

3.2

Interaction

of Flexible

and

in Three-Dimensional

4.1

A Spherical

4.2

An Universal

4.3

A Revolute

Joint

4.4

A Modified

Revolute

4.5

A Cylindrical

Joint

4.6

A Point

Acting

4.7

An

4.8

A Damper

4.9

A Spring

Acting

5.1

Example

of a Open

Joint

Actuator

Joint

52 55 57

...................................

59

.......................................... on a Body on Two Between

Between

Two Two

Chain

MBD

Triple Systems

with

5.3

Example of a Closed-Loop MBD The Crank-Slider Mechanism Mechanism

63

.........................

Bodies Bodies

64

......................

65

.......................

66

System:

Pendulum

Example

60

.............................

Bodies

5.2

Crank-Slider

47

............................................

Acting

of MBD

.....................

..........................................

Acting

Three-Dimensional

Bodies

44

............................................

Joint

Force

29

................................

Rigid

15 18

.....................................................

3.1

7.2.1 The

Their

in Three-Dimensional

and

Space

and

Procedures

Multiple

...................... Branches

System: ............................. .............................

85 ...........

89

90 120

xii 7.2.2

Histories of the Generalized Coordinates: Penalty Constraint Stabilization Technique

of the 7.2.3 Histories Baumgarte's 7.2.4

Errors

Generalized Technique

in Constraint

Techniques

............

122

Coordinates: ................................

Conditions,

124

Performance

of Two

.............................................

125

7°2.5

Position

Error

of Implicit

Algorithm

Via

GCP

............

126

7.2.6

Velocity

Error

of Implicit

Algorithm

Via

GCP

............

126

7.2.7

Position

Error

of 2-Stage

Staggered

Algorithm 7.2.8

Velocity

Error

Algorithm

of 2-Stage

Configuration

7.3.2

Deployment

7.3.3

Performance

of Two

Solution

Accuracy

of Three-Link

Spherical

Angular

Explicit-Implicit

Manipulator

Ball

.................

Manipulator

Stabilization -

127

10 -6

Rolling

Velocities

7.4.5

Angular

Techniques,

on a Flat

7.4.6

Time

7.4.7

Convergence Procedure

Histories

Surface

with

138

with

Sphere

with

Constraint

Studies on Present and .............................................

Initial

7.5.2

Motion

7.5.3

Angular

Configuration and

of a Four-Bar

Trajectories

Velocities

of the

of the

...........

139

139

Offset

Forces

.............. with

Offset

140 ....

140

Conventional 142 wl for Three 142

Linkage

Four-Bar

Four-Bar

135

Sphere

Accuracy Comparison on Angular Velocity Different Time Steps ..................................

7.5.1

.............

of Three Constraint Forces ..............................................

of Three

Sphere

133

Offset

of the

132

.............................

No

Velocities

of the

129

...................

Projected on Three-Dimensional ................................................

Time Histories No Offset

7.4.8

Staggered

of Three-Link

Ball Track Surface

127

..............................................

7.3.1

Solid

Explicit-Implicit

..............................................

Linkage

...............

Linkage

..........

...............

144 144 145

moo

XUl

7.5.4

Constraint

Forces

7.6.1

Crane

Trajectory

7.6.2

Specified Crane Tip Velocity of Rigid Members ..............................................

Tip

of the

Four-Bar

of Rigid

Linkage

and

Flexible

7.6.3

Crane

Joint

Torques

(Rigid

7.6.4

Crane

Joint

Torques

(Flexible

7.6.5

Crane

Configuration

7.6.6

Specified Crane Tip Velocities of Rigid Members ..............................................

7.6.7

Crane

7.6.8

Joint

Angular

7.6.9

Time

History

7.6.10

7.6.11

Trajectory:

and

Velocities

(Rigid

of Joint

Torques

Time .vs.

and

..............

149 151

Flexible

Z - Y Plane

Members)

Joint

(Flexible

...........

.................

Members)

Velocities (Flexible Members: .......................................... Members:

........

149

152

Joint Angular 12 Elements) Torques

..........

Time

Motion

(Rigid

148

148

Members)

and

......

Flexible

.vs.

Subsequent

146

Members

and

Members)

X - Y Plane

................

..........

153 154 154

156

12 Elements)

..........

156

7.7.1

Automobile

7.7.2

Four Partitioned Suspension

Subsystems of the Automobile Systems ...................................

159

7.7.3

Force

in Front

160

7.7.4

Displacement

History

7.7.5

Displacement

Histories

of Body

2

.........................

161

7.7.6

Displacement

Histories

of Body

3

.........................

161

7.7.7

Displacement

Histories

of Body

4

.........................

162

7.7.8

Displacement

Histories

of Body

5

.........................

162

7.7_9

Total Computer P.C.S.T..vs. Speed

Suspension

Storage

..........................

Springs

............................

of Body

1

..........................

Run Time on Alliant FX/8: N.P.S ....................................

Up on Alliant

Efficiency

Systems

on Alliant

158

160

163

FX/8:

P.C.S.T..vs.

N.P.S ..........

163

FX/8:

P.C.S.T..vs.

N.P.S ..........

164

CHAPTER

I

INTRODUCTION 1.1

OVERVIEW The

kinematic

dynamic

(MBD)

over

past

the

systems

and

systems

two

simulation systems

of MBD

on improving system

either and

These

dynamic

based

possess

merically

integrate

the

robot

arm

maneuvers,

However, ciency may groups order

becomes require

to make to speed

realistic

if the

research and

the

these

analysis

effective the

activities verification

analysis

approachc.s

of the

control

use of parallel

bce[I

analysis

or the prob-

progra_._

developed.

'l'hcsc allld Iltl-

problems vehicle

complex,

such

design

mo_ivaa_d

several

technology systems,

thus

as

dynamics.

comput.ational

preliminary

of MBD

system,

gellorale

ground

computational

concentrated

computer

and

has

in tht,

engineering

dynamics,

This

computer

of bodies

primarily

of multibody

the

systems hardware'.

the

have

to automatically

during

mechanical

by using

mamber

of motion

iterations.

dynamics

many

in making

have

become

researchers

of computer

general-purpose

systems issue

that

of multidisciplinary

capability

equations

a dominant

up

more

spacecraft

when

many

role

multibody

of many

modeled

important

on different

programs

fact

development

stand-alone

computer

to the

be effectively

rapid

dimensional

attention

an

design

several

are

the

of three

the

is due

can

played

the

As a result,

[1-111 which

attracted

systems

small.

design

lems.

also

analysis

This

Moreover

has

remain

has

in industry

bodies.

software

dynamic

decades.

of interest

of linked

and

etti-

stage

that

a,searcl_ 12-1.1' ultimately

in

PR_.CE['dNG

achieving

real-time

exploiting

the

structure erate

for the

parallelism

a robust

time

Study

In general, expressed

chosen

be

of MBD

the

the

depending

upon

of the

coordinates

of the

is chosen,

basic

approach

gives

a set

of differential-algebraic

approach

gives

a set

of second-order

In order

tained growing

approach

to understand

coordinates

to derive

will be discussed. which

number

equations,

elimination

of two

the

and

ble

a versatile

an easy

in

data to gen-

treatment

of system interpretation

which in which

the

constraint in the

as shown

conditions derivation

of bodies set

as the

systems.

Thus

if

equations

augmentation

in Fig.

1.1.

whereas

The the

disadvanlag('s tour

is to use

are

are used

first

second

equations. and

of system

that

the

of motion,

position

kinematic

dynamic

equations

A minimal

in terms

in the

approaches:

choice

be derived

of coordinates

joints

final system

advantages

first

type

the

bodies

equations

the

variables.

is given

complexity

The

determine

of state

the

can

An important

treat

differential

the

systems the

bodies.

is how they

relationships

as results

the

for MBD

approach

coordinates,

issues

procedure

and

of motion

forms

set of coordinates

ordinates

an automatic

algorithm,

configuration

kinematic

interpreted

different

include

key

FILM_o

Formulations

equations

of these

an arbitrary can

the

to describe

to describe

systems

NOT

The

a streamlined

integration

in various

characteristic

problems.

topology,

of motion,

_LANK

results.

Systematic

and

large-scale in MBD

system

equations

simulation

1.1.1

for

inherent

describing

system

constraints, of the

simulation

i"AGE

choi¢_,s

a set with

absent. number

or co-

of iizde,lwtlde,r_r the

of second-order independent

ot" usil_x

least

possi-

differ¢'ntial

variables,

is ob-

Howcvc, r, the' rapidly of variablt's

im:reas_,s.

3

M B D SYSTEMS Augmentation

Elimination

Approach

Approach

Differential

Ordinary Differential

Algebraic Equations (D A Es)

Equations (O D Es)

Solution Procedures O DEs Numerical

Constraint Stabilization

N + ODEs

Algorithms

M

Constraint Stabilization Numerical Algorithms

Fig.

1.1

MBD Equations Computational

of Motion Procedures

and

Their

Corresponding

4 and the high degreeof nonlinearity of the equationsof motion make these coordinates difficult to implement in a general-purposecomputer program. The secondchoice is that of relative define

the

moving the

orientation

body

of each

or another

number

For

via

multipliers,

that

body

in that

in the

in which

do

thus

Furthermore,

when

preprocessing

is needed

case

not

the

system

directly

to either

a non-

For an open

tree

to the

number

structure,

of independent

equations

number

are

of relative

imposed

coordinates

coordinates

determine of the

consists

to identify

respect

Relative

postprocessing

the

which

constraint

coordinates.

they

system;

body.

system,

[1-4,15,16],

with

is equal

a closed-loop

of independent

disadvantage

body

moving

coordinates

coordinates.

exceeds

adjacent

of relative

Lagrange

moving

coordinates

the

simulation

of several

set

position

the

of each

results

closed

an appropriate

have

is needed.

loops,

extensive

of independent

vari-

ables. The

third

a body

using

moving

coordinates

and

can

two

is that

formulation However,

by

located adjacent

they

lead

presence

is that

be reached

dependencies matrix

which

in parallel during

to become need

the

causes

preferably

the

at the main

mass

computation. process

singular.

to be constructed

that

Should

is used

in order

of aatural

co-

and

easy

happen,

equations.

renders

Another

this

mechanism,

constraint

matrix

of numerical

matrix

of tile

advantage

linear

define

to il. 'Fhes(_

implementation

or

populated

which

attached

joints

computer

quadratic

r17,18],

rigidly

The

to a simple with

coordinates

coordinates

bodies.

of a fully

less attractive

coordinates

nation

are

of natural

moving

in conjunction the

dinates

is that

or more

be shared

ordinates

can

choice

these

drawback

integration

coor-

o1 tht,st, a t)ositiot_

to identify

the

variable

a new

linear

combi-

to continue

the

irlt¢_gration

process. The last choice is that of the

position

spect

of each

to an inertial

defined

by the

gles.

The

easy

to derive,

puter

programs.

redundant the

reference

body-fixed

main

advantages which

and

is paid

form.

of motion highly

but

When multipliers

with

applied

to the

to the

independent

ber

of equations

the

simplicity

amenable

used,

to the

leads

appearance

of the

of motion

are com-

of equations, as part

inefficiency

of

unless

equations

of motion

expressed

in differential

to a minimal

of virtual

systems

based

set

solution

equations.

are

of equations matrices

preserve kinematic

the

sparsity

relationships. in modular,

together

an(t

equations

for MBD augmented

system

of cqtlati(ms

(DAEs).

DAEs

make

of the

solution

Furthermore, general-purpose

natu-

equations

combined equations

wiLh l,agrang(+

coordinates,

resulting

differential

These

approach,

work

on relative

the

of differential-algebraic

to implementation

an-

to be solved

of dense

coordinates,

coordinates yet

the

of freedom

by a set of second-order

class

is

issues.

approach

principle

constraint

body

descriptions.

or Cartesian

are given

have

re-

or Euler

set

to computational

are

the

d'Alembert's

algebraic

belong

from

equations

a maximal

multipliers lead

with

of general-purpose

yield

degrees

this

kinematic

ral coordinates systems

of system

suffers

are

may

the

define

of each

parameters

development

Lagrange

system

orientation

is that

coordinates

[5-8], which in the

via Euler

to computational

Obviously

nonlinear

the

body

angular

frame

coordinates

in terms

equation

The

which

If independent generated

frame.

of this choice

these

process,

attention

individual

facilitates

Since

coordinates

in each

reference

coordinates

simulation

special

particle

Cartesian

In contrast

use of a larger matrix the MBI)

num-

as well as approach programs.

is

6 1.1.2

Evaluations A closed-form

solution

for highly

simplified

ble except must

be used

tions. tion

For

to obtain

the

both

ferential

may

of numerical Orlandea sparse

et al. matrix

discrete

[6] have

Gear

but

and

solution

encountered

to be solved

governing

the

As for difa special

restricted

DAE

class

problems.

together

problems

become

and

Newton-Raphson

technique

often

elimina-

formula

solutions.

numerical

equa-

by the

investigated

solution

algorithms

difference

accurate

of some

this

system

with

has

impossi-

integration

produced

central

[19,20]

applied

equations

oi' the

explicit

is in general

time

equations

stable

for the

formulation

system

Thus

in conjunction

reasonably

algorithms

equations

solution

modified

equations,

Procedures

MBD

problems.

formulas

yield

algebraic

of the

differential

the

as stiffly-stable

algorithm

Computational

the numerical

second-order

approach,

as well

of Existinl_

with

a

because

the

stiff

and

numerically

ill-conditioned. An

alternative

relies

on augmenting

twice

time-differentiated

ferential

equations

algorithms errors

approach, the second-order

solvers

provide

only

stabilize

the

modified ferential

the

original

constraint

some

MBD

problems

from

ill-conditioning

constraint equations.

solution.

the

was

Park

proposed

Baumgarte's in the solution

and

One

a set

numerical

multipliers,

con-

approach [23,241

to who

of relaxation

have

stabilization

for Lagrange

integration

the constraint

accuracy.

[25,26]

with

ord in ary dif-

numerical

by Baumgarte

Chiou

i21,22),

equations

As a result,

to form

constraint

Petzold

numerical

eventually

desired

equations

and

equilibrium

However,

so that

within

violations

Gear

so that

be applied.

accumulate

satisfied

constraint

equations

an approximate

and

are no longer

can

by

governing

constraint

will propagate

ditions

advocated

shown te(:hniquc

dif-

that

for

suffers

l,'urthermore,

7 for complicated MBD systems, the process of ation

parameters

specific

that

problems

are

may

the

second-order

differential

fying

dependent

system

Jacobian can

matrix

be formed

In order eral

and

to find

numerical matrix.

titioning

scheme

Since

computational can

coordinates

specified

accuracy

choose

a new

set

null

used

are

expensive,

during

the When

of independent

coordinates

methods

on O(n)

multipliers.

the

this

sev-

constraint

coordinate [28,29],

set

the

chosen

the

by repcatii_g

r_atural [30-32].

of independent set

simulation

occurs,

par-

scheme

the

numerical

matrix

variables,

null space

for determining

violated.

Lagrange

generalized

the

constraint

Jacobian

to decompose the

and

computationally

criteria

the

of

by identi-

the given

decomposition

[17,18],

is maintained

from

con-

a set

be done

independent

employed

schemes

can

so that

constraint

superior

value

to each

violations

DAEs

This

of the

include:

singular scheme

from

to eliminate

been

relax-

violations

constraint

variables

space

algorithms

become

pendent

independent

have

[27], the

partitioning

coordinates

is obtained.

of numerically

These

coordinates these

equations

consequently a set

constraint

to avoid

multipliers

the

algorithms

Jacobian

approach

optimal

difficulties.

Lagrange

and

so that

the

computational

different

of eliminating

to tailor

encounter

An ultimately sists

used

determining

of indeuntil

the

it is rtcccssary

to

the

idcillificalioll

process. Recently, of generalized

coordinates,

39].

These

tree

topologies

in O(n) significant

based

algorithms

modifications

several

are primarily

in which

operations.

and

their

If the

are required

where

variations

applicable

equations

system

algorithms,

to MBD

of motion

topology in order

have

may

embodies to obtain

n is the number

been

proposed

systems

consisting

be recursivelv multiple numerical

[33of

solved

closed

loops,

solutions.

8 Moreover, the presenceof closedloops may cause O(n) the

simplicity

1.2

of open

tree

topology

in parallel

algorithms

to loose

computations.

OBJECTIVES Since

backs

in the

alternative

the

computer

during

solution

involving

computing

the

the

rithms

treatments

mentation

of this

DAEs

of linked

the

configuration

are

adopted

coordinates

efficiently

objectives

bodies and

and

orientations

angular

representations

inertial

coordinates

or constraint constraint

constraint

matrix;

difficulties

force

expense

in

inefticiency

in

in extending

the

attached

are for

the

bodies

to the in the

determined. translational

and

existing

in the

center system Note

simulation

review

the

system. of the

and algo-

a parallel

of coordinates

of each

robust

integration

third,

dynamics

two sets

of mass

first,

explicit-implicit

we will first

of bodies

center

of the

are to develop:

multibody

by employing

rigidly

angular

Alternative

computational

accurately;

in mind,

velocity

to locate are

and

second,

for a general

these

degraded

Jacobian

dissertation

of constraints;

procedure With

algorithms;

difficulties.

unacceptable

large

constraint

us to look for

stabilization

integration;

draw-

computations.

objectives

to solve

of the

various

motivated

disadvantages:

matrix;

suffer

those

constraint

of time

iterative

to parallel The

either

ill-conditioned

null space

have

overcome

following

process

procedures

these

that

involve

the

the

implicit

algorithms

efficient

that

overcome

violation

matics

procedures

procedures

elimination

solution

implementation,

solution

solution

using

aforementioned

implecapacity.

spatial that

kine-

describe

Inertial

coordinatos

bodies.

Body-tixcd

of mass

of each

can

be obtained

as soon

the

of chosing

that

motions

and

body

purpose bodv-tixed

so that

coordi-

as

9 nates for the angular motions is to decouplethe inertia matrix to obtain the translational and rotational equations separately. After the kinematics of a body in spacehas been determined, a formulation basedon d'Alembert's principle of virtual work is adopted to derive the system governing equations by treating the bodies in the system as originally independent of each other. Nonlinear kinematic constraint conditions, which are imposed to describe the interconnectivity between the various bodies, are appended to the formulation using Lagrange multipliers. The final system equations of motion, which are characterizedas DAEs, not only enhanceprogramming modularity but can also be generatedautomatically. As mentioned previously, the useof existing numerical time integration solution proceduresmay encounter computational difficulties. In this regard, two newly developed schemesbased on constraint stabilization (penalty constraint stabilization scheme)and constraint elimination (natural partitioning scheme),are introduced to correct for the constraint violations accurately and el[iciently. 1.2.1

Robust The

vation

that

nique. tem

is given the

equations and

Thus,

stabilization

as time time

by the

difficulties this

that

generalized

scheme of penalty

progresses

constants.

time

rate

have

scheme

can be processed the

of Constraints

equations

to intrinsic

Furthermore,

module

constraint

in time.

which

overcome

Treatment

time-differentiated

according

form,

Efficient

penalty

characteristic cay

and

been offers

constraint

attractive

in two modules: constraint

penalty

encountered the

forces

form

constraint

This

of the

is based

the module.

on the

retain

obser-

a parabolic

violations

will de-

time-differentiated forces,

enables

us to

in Baumgarte's feature generalized This

that

techthe

sys-

coordinates se[)aratiorl

[i_s

10 nicely in the framework of time

partitioned

procedure

[40-42]

adopted

for the

integration. The

penalty nates

natural

partitioning

constraint

relying

tioned

previously.

heavily The

Applying

erning

DAEs

leads

of second-order

system

independent

to the

tripetal

that

integration rations, Hence,

rotational the

be the

best

quantities.

of the

have

consists

been

menmatrix

of several matrix

multipliers

that

expressed

are

coordi-

Jacobian

Jacobian

the

coordinates

Lagrange

equations

velocities,

forces,

open

to the govand

yields

in terms

of

central

angular

motions

systems is used

angular

[40-,t2],

difference

and

time

formula

while

cenequations

Direct

time

kinematic

configu-

of a body

in space.

which

has

the

treating

capability interaction

integrate

translational

and

parallelizable

integration

algo-

integration

partitioned

in the

bodies.

simple

of equations

a robust

the

appear

orientation

to separately

difference

velocity-dependent

of linked

for some

procedure

to treat

central

the

Equations

accelerations

except

yield

To obtain

candidate

both

the rotational

coupled

explicit

of MBD

angular

solution

solve

If the

the

directly

equations.

rithm,

that

topology

of the constraint

formulation,

represent

as external

system

from

physical

independent

of the constraint

elimination

and

a partitioned

to separately

and

null space

Integration

present

not

existing

algorithms

if the

different

variables.

of angular does

the

dependent

differential

accelerations

of motion

uses

is quite

numerical

resulting

Explicit-Implicit In the

terms

on the

the null space

a set

1.2.2

their

in parallel

which

scheme,

identifying

be generated

chains.

scheme,

stabilization

by explicitly

without

can

the

formula

translational is adopted,

is thought and

the

rotational

approximation

to

11 of the angular velocity for the evaluation of angular acceleration leads to numerical instability for the governing equationsof motion. This has motivated us to exploit a two-stageexplicit procedurewhich stabilizes the central difference algorithm and delivers an accurate solution. The Euler parameters are used in the present solution procedure to represent the angular orientations of bodies. An implicit mid-point time integration algorithm is employedto integrate the Euler parameters by exploiting the kinematic relationship with the associated angular velocities. The specific implicit algorithm presentedhas been chosen becauseit is unconditionally numerically stable while it can be analytically inverted during actual computer implementation becauseof its special four by four matrix form. Combining thesesolution algorithms a two-stagestaggeredexplicitimplicit solution procedure [43] has been developed. This solution procedure, which invokes either the penalty constraint stabilization schenmor the natural partitioning schemeto stabilize the constraint violations, has been implemented in a computer program to validate and demonstrate its robustnessand accuracy. The presentsolution procedurebasedon the penalty constraint stabilization schemeconsistsof two modularized solvers: the generalizedcoordinate solver and the constraint force solver. The solution procedurebased on the natural partitioning schemeincludesthe generalizedcoordim_lesolver and the independentcoordinate solver. The generalizedcoor'dinalc solver combines an improved version of the explicit central difference algorithm for integration of the translational coordinatesand angular velocity with an implicit algorithm to update the Euler parameter representation of angular orientations by exploiting the uncoupled inertia expression. "Fhe

procedure

12 has successfully been interfaced with the penalty staggered stabilization technique which solvesthe constraint forces as independent variables by implicitly integrating a stabilized companion differential equation for the constraint forces in time. The combination of the two algorithms can be invoked in a sequential manner on the rigid and flexible componentsof the multibody system resulting in an attractive, modular solution procedure. As for the independentvariable solver, a procedurebased on body-by-body constraint Jacobianmatrices is developedto explicitly form the null spaceof the constraint Jacobianmatrix and consequentlyobtain systemindependent variables. 1.2.3

Parallel

Implementation

Since bodies, time.

the

an MBD numerical

To reduce

efficient

parallel

general,

issues

eration

of the

constraint

the

system

forces,

present

constraint

thrust

of this

tioned

conjugate

time

consume

A method

uses

gradient

has

the

been

vector

and

or the

generalized

algorithm

constraint

forces and

include

gen-

and

parallel

of the

independent

of the

two-stage

developed.

to decide

of

interpretation

parallelism

a Schur-complement-based numerical

In

or elimination and

elimination

of CPU

computers.

systems

incorporation

of

to develop

parallel

of MBD

for exploiting

procedure

thousands

amount

coordinates,

constraint

solution

or even

it is advantageous existing

of generalized

Analyzer

a prohibitive

dramatically,

of motion,

stabilization,

scheme

of hundreds

computations

equations

results.

acceleration

may consist

by incorporating parallel

method

for MBD

may

integration

explicit-implicit

stabilization

CPU

involving

simulation

eralized

solution

algorithms

of the

gered

system

Procedures

The

main

precondi-

either penalty,

stag-

tile

gen-

constraint

acceleration

vec-

13 tors of the natural partitioning scheme. The present algorithm has been implemented and tested on existing parallel computers and has provided encouragingresults in practical MBD problems. 1.3

DISSERTATION The

dissertation

tial kinematics ter

coupled

MBD

of motion

the

the

a set of algebraic

veloped the

difficulties

associated

technique.

Chapter procedures.

tion

the

procedures.

investigation multibody

numerical

systems.

penalty are

and

Chapter

7 gives

robustness

and

discusses

8 summarizes directions

Chapter

the

the for

to the

the

the

Two

newly

based

accomplishments research

DAEs deand

computational

on

parallelism

of utilizing

of

scheme

staggered

coordinates,

example

further

using

equations

A two-stage

translational

4

in the

with

to overcome

numerical efficiency

Chapter

stabilization

is developed

exploits

equations

formulation

5 deals

procedures.

governing

bodies

procedures.

employed

forces

connect

spa-

Chap-

the

of motion.

adjoined

constraint

solution

6 analyzes

are

the

constraint

dynamic

solution

for updating

constraint

to derive

that

in the

existing

those

formulation.

work

joints

form.

Chapter and

of the

matrix

scheme

MBD

equations

Jacobian

with

and

to demonstrate

differential

on the

2 reviews

of a set of algebraic

are introduced

algorithm

lar orientations,

of virtual

that

based

Chapter

present

equations

partitioning

explicit-implicit

solution

joints

several

natural

consist

constraint

schemes

in the

properties

in constraint

by reviewing

used

second-order

These

as follows.

principle

that

mechanical

system.

motion

bodies

d'Alembert's

with

derives

is organized

of rigid

3 employs

equations

OUTLINE

the

anguproposed

inherent problems the

in the in order

present of the in the

solupresent field

of

CHAPTER SPATIAL 2.1

without

the

begin

of the

Finally,

a rigid

body

2.2

and

thus

the

position

has

been

the

and

velocity,

ultimately

defined,

the

determine

to analyze the

systems.

In

rotational

representations

and

this

acceleration

particle

complete

used

as if it has the

bodies

design

chapter,

we and

of a particle been

derivation

in

attached

to

of kinematics

for

from we can

fundamental

technique

is using

vectorial

quan-

of a particle

in a given

reference

frame.

\Vhen

frame

to the

particle

by one or more

systems

the

origin

resolve

for a particular component

useful.

such

in terms

this use.

of the

To derive

2.1) expressed

of a Cartesian

a most position

vector

of coordinates

(Fig.

of bodies

been

and

Frames

to locate

tremely

has

finite

the

of particles

motions,

mechanical

position,

motion

body.

In mechanics,

between

these

of different

we will consider

Reference

tities

in the

the

of the

acceleration

derivation

obtain

space.

with

and

bodies

the

subsequently

is the study

associated

velocity,

with

rigid

which

forces

position,

geometry

the

KINEMATICS

Introduction Kinematics,

the

II

of that position In many

vector relations,

of three

reference vector

dynamics

in various

reference

let us consider components

parallel

problems, frames a position to the

relations prove vector three

exr_, axes

frame: r v = r__e I + r_e 2 + r__e3

(2.'2.1)

15 where (rl,

r2, r3) e are the coordinates

frame,

and

(_el,e2,_e3)

frame.

Similarly,

are

rp can

the

basis

be expressed

of the

particle

vectors

fixed

in another

in the

inertial

reference

in the

inertial

reference

reference

frame

as

rp = r_b_1 + rb_b2 + rbb_3 where

(rl,r2,r3)

frame,

and

b are

the

coordinates

(b_l, b2 ,_ha) are the

basis

(2.2.2)

of rp expressed

vectors

in the

of an arbitrary

b reference

moving

reference

frame.

Y(b2)Y(e2)

/

P

_- X(el)

Z (e$)__ z(b 3 )

Fig. Since

(2.2.1)

e reference relation projection

2.1 and

frame

between

Position (2.2.2) must

the

of rp with

r1

Vector

describe evidently

two bases respect

rp._e I

in Three-Dimensional the

same

be related

vector, to the

can be established to the

e basis

(e 1 b_l)r _ +(e_ l

vector

b.,)r_

Space

the

components

b reference by writing

frame. the

of the The

orthogonal

so that

v(e

1 b3)r:_{

( ..... l)

16 (2.2.4)

(2.2.5) These relations can be written in the following matrix form:

(2.2.6)

r e = RTr b

where R T

r e = [r_,r_,r_] rewritten

T, and

as the

e 1.b 1 el._b 2 e_l-b_ 3 e__ 2.b x e_2"b2 e_2"b3 e_a'b, e_3"b_2 _-bz

=

r b = [r_,rb,rb]

relation

of the

two

T.

Equation

basis

vectors:

(2.2.7)

(2.2.6)

can

be explicitly

e = RTb

where

matrix

R is called

cosine

matrix

between

Since basis

vectors

the

the

rp preserves

coordinate

two sets the

(2.2.8)

transformation

implies

or direction

of axes.

property

of constant

length

regardless

of the

selected,

rpb T rpb = r_TRTRr_,

which

matrix

= r_Tr_)

(2.'2.9)

that RTR

= I

(2.2.10)

or

R -_ =R r A matrix multiplying

satisfying (2.2.8)

relation by

R

(2.2.11) and

recalling

is called (2.2.10)

(2.':. 1l) an orthogonal yields

the

matrix. relations

Preof an

17 arbitrary moving referenceframe that is expressedin terms of the inertial referenceframe as b = Re

2.3

Angular

of a Particle

the

derivation,

From

Orientations

the

position

of a particle,

of reference passes,

frames,

the position

formation

matrix.

provides

us with

tations

so that

Euler's

theorem

and

previous

e, with

which

can

be specified

vector

orientation

This

leads

a foundation the

origin

can

of them

through

through

that

origin.

In short,

The

a proper

rotational

Note

that

if the

unit

transformation represented

+ cos¢(I

full coordinate

straints

three

are imposed.

transformation

R by nine the

lead

independent choice

to the

of these

be det_(+'rmine(]. basis one

axis

the

property

which

consequence if the

by

passes

by rotating

a

in Fig.

2.2.

by

(2.3.1)

is used, nine

b

am_ther

x I

of the

parameters

vectors

be represented

matrix

parameters

oricu-

¢ as shown

+ sin_n

parameters,

orthonormal

R will

The

- nn)

can

trans-

of angular

with an

As time

which

be described

an angle

vector

sets

theorem

dextral

about

can

n through

= nn

matrix by

axis

angle

of suitable

Euler

may

to coincide

rotation

R(n,¢)

However,

matrix

time

coordinate

types

oriented

a certain

on the

themselves.

of the

specific

matrix.

so does

various

be made

acting

to parametrize

and

development

any

at any

a transformation

changes

operator

we choose

in terms

two arbitrarily

one

about

be expressed

transformation

that

common

that

to develop

coordinate

states

we conclude

by

to the

rotating

vector

may

(2.2.12)

it means

direction

that

cosiues

3 \ :l coordiuate

that

it cau

six orthouormal presents

oul\"

be

con-

an hnportam

18 aspect in computing the angular orientation of bodies in MBD systems. Which kind of parameters one should adopt are a matter of judgment by individual researcher.The fewer the parameters,the fewer constraint equations need to be satisfied. However,sometimesit is necessaryto compute the coordinate transformation matrix at cancels

some

Moreover,

of the

the

transformation various with

use of three matrix

parameters, some

parameters

advantages

important and

Y ( b2)Y

used

so that

integration

a small

angles

commonly

corresponding

time

always

certain

properties

their

using

parameters

when several

by

every

the

rotational

lead

are

number

reached.

will be listed

kinematic

relationships

operator

are

(e2)

n

I )

3 ) z(b

Fig.

2.2

3)

System

Rotational

coordinate

To specify

parameters

Coordinate

which

of parameters.

to a singular

X(e

Z(e

step,

defined.

R with along of the

19 2.4

Euler

Angles

The angular

most

orientation

a set

of coordinates

angles

formalism

the

k-axis

rotate axes.

the

without

involving

To express

of these

three

of three

angle are

another

k"-axis

are Euler any

labeled

successive

Next,

"/ to produce rotational the

basis the

desired

rotational

operator

can

R(k,a)

k].

/'-axis

an angle

Euler

rotating

[i,j,

system

n through

= R(k",q)-R(i',13).

by

k"] are obtained.

the

provide

The

started

rotate

the

axis

angles

equations.

orthogonal

[i",j",

to describe

These

rotations,

[i',j',k'].

rotations,

R(n,¢)

angles.

successive

set of coordinates

effective

used

constraint

a of a specified

by an angle the

set of parameters

in space

coordinates

_ so that

minimal

of a body

consists

through

resulting angle

common

The by an

Finally, rotational ¢ in terms

be writlen

(2.4.1)

or

[ c_caR(n,

¢) =

sqca

s3c_sa

-c_sa-

s_c_ca

successive

rotations

c_c£ca

s_sa s_ca

-c_s_

sqs_ The

+ cqc£sa

-s_sa+

in (2.4.1)

b"

= R(k",'_)b'

R(i',

13)b

b = R(k,a)e

(c -= cos, s = sin),

R(k",

c_

are

b'=

with

(2A.2)

q) =

s i] c"/ 0

(213)

2O

R(i',/3)

=

0]

c/3

s/3

-_/3

c/3

(2.4.4)

COt

R(k, Ot)= Multiplying matrix they

in terms involve

compute; when

these

together

yields

of Euler

angles

trigonometric

second,

the

collinear.

represents

This

In mechanical angles

by using

which

-s(_ + _)

rotation

the

matrix

case

rotations

!]

a + _ about

k-axis.

it is necessary

transformation r12

r13

r21

r22

r23

r31

r32

r33

Euler

angles,

to

singular

along

k and

= 0 so that

to calculate

matrix

rll

first,

expensive becomes

by setting/3

sometimes

corresponding

both

s(a+_) 0

coordinate

difficulties:

numerically

c(,_ + _)

c(a+',/) 0

transformation

numerical

are

be illustrated

R(Ot,/3,_) =

To determine

some

in which

can

coordinate

transformation

analysis,

a given

The

(2.4.2).

coordinate

a single

0

functions

R =

which

COt

0

presents

/3 = nrr, n = 0,-t-1,-t-2,...,

k" become

-sOt

E_l]er

where

(2.4.5)

we calculate

first,

a = tan-l(r13)

(2. l.(3) r23

by recalling

(2.4.2)

so that/3

and

"_ can

s3 = r13sa+

c'_

=

rllca

--

r21s"

t

be evaluated

without

reaca

;

c3 = r33

;

=

rl.2ca

,s_

-

any

ambiguity

(2..t.7)

r.e.2,q_ ,

(2.-t.S)

21 2.5

Rodrigues The

Parameters Rodrigues

parameters

are defined

_/=

where angle.

n and

¢ are the

unit

vectors

¢

n tan

along

as

2

(2.5.1)

the

rotational

axes

and

the rotation

Obviously

_r_ = tan 2 ¢

(2.5.2)

2

so that ¢ 2 -

c°s2

1 1 + "/r-I

(2.5.3)

Since

cos¢ = 2cos _ _ - 1

(2.5..,)

2 substituting

(2.5.3)

into

(2.5.4)

yields

cos ¢ Again,

nsin¢

can

be written

n sin ¢

Replacement

=

of nsin¢

R-

i

cos¢

1 1 +'IT"/

(2.5.5)

,._T,.,/

+

in terms

2n sin --COS¢ ¢ 2 2 and

1 - "/T 3,

of "y as

=

, :3"_COS"

in (2.3.1)

., 0-'2

=

by (2.5.6)

[(1-

_/T'/)I

3'3 0 -_2

0 -3`3 Yl

+ 2(3'"/T

2"_ 1 _- _/T,_

--

and

(2.5.5)

-'_)]

(2.5.6)

leads

to

(:].5.7)

where =

As

in the

set

of three

case

of Euler

variables.

I

angles, Unlike

the the

-_l 3`2 0

Rodrigues

Euler

angles,

(2.5.S)

parameters the

Rodrigues

use

a minin:at parameters

22 expressiondo not involve trigonometric functions. This can be an advantage for actual numerical computation. However, they have the disadvantage of becoming infinite if the rotation angle ¢ becomes ±kzr,k = 1,2,. ..... Again it is sometimesdesirable to compute the Rodrigues coordinate

transformation

matrix

subtracting

symmetrically

from

is given. the

These

parameters

relations

off-diagonal

terms

are

if the

obtained

of (2.5.7)

which

by yield

¢ 4"_i = [1 + tan 2 _](Rjk

-

(2.5.9)

Rkj)

Since 4 1 + tr(R)

-

(2.5.10)

1 + tan 2 -¢ 2

therefore

by

computing

substituting

(2.5.9)

into

(2.5.8),

we obtain

the

expression

of

qi that 1 "_i -

Note

that

if 1 + tr(R)

= 0, then

¢ = =t=kzr as is concluded

2.6

Euler

values

minimal the

set

angular

following

from

Rk))

"/i approach

previous

degeneration

of the of three

of the

parametrizing parameters.

orientations

(2.5.11)

infinity

which

occurs

when

definition.

coordinate

variables, The

of bodies

choice in MBD

transformation one

has

of the systems

to use

more

parameters needs

matrix

for

than

the

to rot)resent to satisfy

the

requirements:

(1)

Singularity

should

(2)

To prevent

expensive

braic

-

Parameters

To avoid certain

1 + tr(R)(Rjk

description

not occur

for any

calculations

of finite

rotations

chosen

parameters.

of trigonometric is preferred.

functions,

an

alge-

23 (3) To avoid redundanciesin descriptions of parameters, a minimal set of parameters is preferred. In the presentresearch,Euler angular (1)

orientations

Euler

parameters

rotational (2)

Euler

The

for the

the

singularity

satisfy

parameters

such

preserve

in the

have

bodies

parametrizations

of bodies (3)

of the

parameters

the

been

following

chosen

angles

algebraic

the

reasons:

free property

as Euler

to represent

can

description

that

other

not

provide.

of finite

sets

of

rotations

systems.

use of Euler

parameters

may

drastically

simplify

the

mathematical

formulation. Euler

parameters

are

defined

as

¢ q0 = cos-

2

(2.6.1)

q,_ = n sin -¢ 2 with

the

constraint

equation

q_ +q_+q_

where

q_

= [q_,q2,q3]

T. The

time

derivative r

of (2.6.2)

the

standard

is given

by



qo0o + qnq,,

Introducing

(2.6.2)

+q3 2 = 1

trigonometric

(2.6.3)

= 0

relationships

¢

cos¢=2cos

2--1 2

¢

(2.6.4)

¢

sin ¢ = 2 sin - cos 2 2 and

substituting

(2.6.1)

into

(2.3.1),

the

rotational

operator

dyadic

R be-

comes R = (2q 2 -

1)I + 2(qnq

T - qo(tn)

(2.6.5)

24 or 1

qo2 + q_ R=2

qlq2 + qoq3

q_ q2 - qoq3 ql q3 + qoq2

qlq3 -- qoq2

1

q_ + q_ -

q2q3 + qoql

q2q3 - qoql

(2.6.6) 1

q_) + q_ -

where Cl,_ =

In contrast

to Euler

formation

matrix

corresponding R from

angles

(2.6.6)

Euler

(2.6.6)

q3 0 --q2

and can

Rodrigues

not

parameters

can

the coordinate

singular.

Again,

be determined

trans-

if R is given,

by taking

the

l+tr(R) 4

(2.6.8)

into the

diagonal

q_ = 1 + 2Rii-

Equations

(2.6.8)

rameters.

The

off-diagonal

(2.6.7)

the

trace

of

so that

Substituting

Euler

--ql q2 0 parameters,

become

q2_

of the

0 --q3 ql

and

(2.6.9)

off-diagonal

terms

terms

tr(R)

determine

the

of (2.6.6) and

4qo

qo =

;

qo-

magnitudes can

be used

adding

of the to decide

symmetrically

Euler

pa-

the

sign

from

the

4qo

;

qo --

4qo

R23

4qt R13 -- R31

R21 --RI2 q3 =

(2.6.9)

'

R32 ;

R13 -- R31 q2 =

in

yields

R32 - R23 ql =

results

1 2,3

i= '

Subtracting

of (2.6.6)

of (2.6.6)

4

terms

parameters.

(2.6.S)

4q2

(2.6.10)

R21 - RI2 "iq3

and R12 + R2_ qlq2

-

4 R23 + R32

q2q3 =

4

(2.6.11)

25 R3:

+ R13

q3ql --

4

According to previous derivations,the following algorithm is used to determined

the

Euler

det

= max

if(

det

parameters (tr(R),

when Rll,R22,R33

= tr(R)

) then

use

(2.6.8)

to find

use

(2.6.10a)

elseif

( det use

(2.6.9)

[44]:

)

q0

to find ql,

= Rt,

R is given

q2, and

q3

) then

to find

qi

Compute

q0 = (R_-Rjk) 4q{

Compute

qe

(RI,,+R,p) 4q,

from ,

(2.6.10b)

p _: i from

(2.6.11)

endif

2.7

Angular

Velocity

Consider as given

the

by (2.2.12).

orientation The

of the

time

derivative

1_ = Re

Since

e is a fixed

basis

vector,

which

1)=

To relate

R and

b basis

R, we differentiate

Re

with

respect

to the

e basis

of b is

+ Re

implies

(2.7.1)

6 = 0, therefore

= I_RTb

(2.7.2)

the identity

matrix

('2.'2.10)

with

rcs[)cc:

to time: R• By

assuming

that

R = SR

T

and

RTsTR

R + R

TR

substituting

+ RTsR

= 0 into

=- 0

(2.7.J) (2.7.3)

we get

(2.7.-t)

26

Premultiplying (2.7.4) by R and postmultiplying by R T Sff-S

which

implies

matic

equation

that

T _-0

(2.7.5)

S is a skew-symmetric

for the

rotation

can

yields

matrix.

be defined

1_ = SR

Hence

the

matrix

kine-

as

= _TR

(2.7.6)

or

if+T=

I_R T

(2.7.7)

where =

0

-ws

w3

0

--w2

The

three

moving

components b basis

following

form

in (2.7.8)

relative

to the

are the

(2.7.6)

l_ = l_RTb

where

the

angular

velocity

vector,

the

function

2.8

present

derivation,

of the coordinate

Time

Derivatives

In this are

derivative

of (2.6.5):

derived.

:

e basis into

4qoqoI

the

w, can

that

can

components be written

of" the into

the

= _Tb

be written

+ a;2b2

as

+ wab3 that matrix

(2.7.10)

the

angular

R and

veh)citv

its time

a_ is a

derivatives.

Parameters

relations

These

velocity

(2.7.2)

we conclude

of Euler

section,

velocities

angular

transformation

(2.7.8)

0

= QTRRTb

w = _Zlbl From

-COl

wl

inertial

by substituting

w2

relations

between are

Euler

parameters

estabtished

+ 2clnq T + 2q,_cl T - 20o(t,_

and

by taking

- 2qocl,t

angular the

time

(2.8.1)

27 Substituting (2.6.5) and (2.8.1) into (2.7.7), the angular velocity, ca,can be expressedin terms of

Euler

parameters

and

w = 2q0/l,_ - 2qoqn

their

time

derivatives

as

(2.8.2)

- 2Cln/t,_ ----2Tq

where T=

Differentiating

(2.8.2)

with

-ql

qo

q3

-q2

-q2

-q3

qo

ql

-q3

q2

respect

of the product

¢b = 2T_I and

Tq shows

its inverse

relation

that

is used.

the

scalar

Appending

to (2.8.2),

the

wTw the

as

form

can

{0} inverse

it vanishes,

of the

be written

as 4/tTq

in terms

q2

q3

-ql

qo

q3

-q2

032

-q2

-q3

qo

ql

033

-q3

q2

of (2.8.6)

so does

constraint

ql

=2

and

"£t:l. Hence

(2.8.5)

also be written

qo

0.11

The

(2.8.4)

T Tr.;j _ 41(wTw)q

= w 2 can

velocity

+ 2"i'_1

that

differential

angular

yields

is

q=21 Note

qo

to time

& = 2T_

Expansion

-ql

(2.8.3)

-ql

= _2

if (2.8.2)

equation

(2.6.3)

o[ Euler

ql 02 q3

qo

parameters

(2.8.6)

is

r/qo//0-_1-_2-w3 ,,,

Iq°} qt

q2

q3

where

UY3

q = [qo,ql,q2,q3]

formulate of bodies

the

computational

in the

system.

(422

T.

--U21

In chapter sequences

0

(2.s.;)

q3

5, these and

-- A(_,)q

derivations

obtain

the

will

angular

be used orientations

to

28 2.9

Velocity

and

In the fixed

in the

expression

previous

moving

frame

section,

time

are

of two

of a Particle we have

reference

for the

moving terms

Acceleration

frame.

derivative

varying

different

In the

the

time.

vectors

Such

as shown

rate

present

of a vector

with

basis

studied

change

section,

whose

of a vector

we derive

the

along

the

be expressed

in

components

a vector

can

in (2.2.1)

and

(2.2.2)

where

rpe = rpb = r_b,• + rbb2 + rbb3

Differentiating

(2.9.1)

with

respect

to time

and

(2.9.1)

making

the

use

of (2.7.9)

yields

• =rp+wxrp .b rp where time

rp • b denotes of rpb due

rate

(2.9.2)

the

represents

time

to the the

frame

whereas

reference

frame

that

point

with

respect

relative

rotational

time

reference

(2.9.2)

rate

vector

is valid

for any

to time,

(2.9.2)

to b basis

motion

derivative the

b

of the

of the

moving

position

rpb iS expressed vector

we obtain

w x rpb denotes

and

frame.

vector

an expression

Thus, for the

of a moving differentiating acceleration

•.e

-.b

.b

where

•.e is the rp

acceleration

in the

b basis,

•b is the 2w × rp

acceleration

in the

Velocity Having

priate

of

p: b

b

rp = rp + 2w x rp + & x rp + w x w x rr,

2.10

that

r_, in an inertial

in terms

in space.

Note

the

to study

and

b basis,

Coriolis

and

Acceleration

derived the

of p in the

the

velocity

e basis,

of a Rigid

and

centripetal

of p angular

acceleration.

Body

of a particle

acceleration

acceleration

& x rr,_' is the

acceleration,

¢z x w x rpb is the

kinematics

•.b is the rp

(2.9.3)

in space,

of a rigid

body.

it is approlh'cause

an

29

unconstrained

rigid

convenient point

2.3)

to choose

within

motion

body

the

of the

on the

rigid

of six degrees

six coordinates

body

body

possesses

and

three

in space.

body

which

that

consist

rotations

about

To this end, can

of freedom, of three that

consider

be decomposed

it is generally translations

point,

a position

of a

to describe vector

the

rp (Fig.

to

Y(b 2 )

x(b I )

Sp

z(b 3)

Y(e 2 )

X(e

Z(e

1)

3)

Fig.

2.3

Translation and Rotation of a Body in Three-Dimensional Space

rp = ro + sp = rTe

where

ro is the

to the

origin

position

vector

of the body-fixed

from

the

reference

origin frame,

+ 1Tb

of the

(2.10.1)

inertial

reference

sp is the position

vector

frame from

30 the origin of the body-fixed referenceframe to a point the

basis

of the the

vector

of the

body-fixed

position

position (2.9.2)

and

expressed

reference

vector

vector

inertial

frame

of point

of point

(2.9.3),

the

reference that

o in the

p in the velocity

frame,

and

b is the

describes

the

orientation.

inertial

reference

body-fixed and

p of the

acceleration

basis

frame.

vectors

e is

vectors Also

frame,

reference

body,

and By

r is

1 is the adopting

of point

p can

be

as i'v = i" + §p + w × sp

(2.10.2)

and _p =i;+g

Since

there rigid

body,

can

be expressed

that

equations so that

2.11

been

motion

between

§p = 0 and

Concluding

position, reviewed.

the final

(e l0

sv

particle

at

velocity

and

point

p and

acceleration

o for of rp

rp = r T e -b IT}3 = r T e q- 1T _ TD

(2.10.,I)

_p = _T e + IT_Tb

(2.10.5)

o and

of motion different

§p = 0. The

x s v +wxwx

as

if points

Spatial the

§v+w

is no relative

the

Note

v+2wx

p coincide,

by separating

numerical

+ IT_T(zTb

which the

algorithms

implies

1 = 0, we can

translational may

and

be applied

derive

rotational

the

motion

accordillgty.

Remarks kinematics

velocity

and

In discussing

relations

needed

acceleration the

motion

vectors

to calculate

quantities

of particles

of a particle

and

as being

such

bodies attached

as

have to

31 a rigid body, a frame of referencemust be specified so that the dynamical equations of the body can be derived. In the present derivation, an inertial referenceframe is usedto locate the position of the center of massof a body. A body-fixed referenceframe is then employed to locate the position of a particle in the body. Such hybrid referenceframes are chosen to decouple the translational and rotational equations so that an efficient numerical algorithm can be formulated as discussedin chapter 5. Three representations of rotation have been studied. The advantagesand disadvantagesof these angular representationsare discussed.Eulet parameters to simple

have

algebraic

metric

functions

Other

angular

suffer

from

been

chosen

equations and

they

representations

singularity

in the

that

present

do not

require

give a singularity-free may

for certain

require parameter

derivation the

because

evaluation

representation trigonometric values.

they

lead

of trigonoof rotations.

functions

and/or

CHAPTER EQUATIONS

3.1

for MBD

chapter

systems

of employing and

rigid

deals using

bodies

of kinetic

energy

constraint

forces the

velocities

FOR

MBD

and

SYSTEMS

the

translational

the

rotational

virtual

motion

mass This

work

their

equations

2, an

whereas

of motion.

systems

When of virtual that

coordinates

reference

to describe

Third.

the

frames

and

configuration

is used

to described

a body-fixed

frame

is used

to described

in the

representation

constraint

the

in terms

quantities.

of generalized

into

frame

kinematic

kinematic

separated

inertial

to be decoupled

dynamic

being

of particles

systems.

of bodies

to obtain

advantages

are formulated

regard,

2 will be adopted

several

scalar

use

of motion

the system

than

are

the

In this

matrix

the principle

to dynamic

of which

equations

are

First,

problems

Fourth,

in chapter

motion

the

both

in MBD

indicated

There

rather

dynamic

versatile.

in chapter

of the

methods.

as a whole

no work.

of bodies

As

equations.

work,

formulation

motion

causes

do

formulation

in dynamics.

Second,

and

reviewed

the

methods

is considered

components.

makes

with

variational

variational

individual

with

MOTION

Introduction This

and

OF

III

system. into

d'Alembert's work,

between and

are composed

thus

decomposition

the

and

into the

constraint

is used principle

of an arbitrary

rotational

the, t_rincil)h_

produce

principle

we extend

frame

translational

is introduced

relationships conditions

This

number

the

of

forces governing

in conjunction of virtual of rigid

work hod-

33 ies with an arbitrary number of constraints that are used to restrict the motion of the bodies. The presentformalism considersthe motions of individual bodies as initially independent,and then applies restriction on those motion by the introduction of kinematic constraints. Such constraints are incorporated through the method of Lagrange multipliers. The resulting system of equations, which consists of second-orderdifferential equations that introduce Lagrange multipliers as constraint forces as well as the algebraic constraint equationsasconstraint conditions, are known asdifferentialalgebraic equations (DAEs). To cover further developmentin flexible MBD systems, the equations of motion that include elastic deformations, which havebeen derived by Downer [52], are given in time discreteform by taking the advantageof the previously chosenreferenceframes and formulation. 3.2

The

ies, the the

Principle Since

an

study

of its dynamics

system

Newton's basing

the

equilibrium

virtual to zero.

The

than This

This

scalar work

on

compatible

mathematical

in many

respects

as a collection

principle

acting

of interconnected

will be used may

a system with

expression

the

as noted

quantity:

in static constraints

obeying

previously,

generalizett to establish

be stated

bod-

by considering

of components

is accomplished,

of virtual

forces

a number

is simplified

on an overall

conditions.

displacement

involves

rather

principle

all the

Work

system

of motion.

derivation the

by

MBD

as a whole laws

sequently,

done

of Virtual

work.

by Con-

the system's

as follows:

The

equilibrium,

work

during

of the system

is equal

is

Tt

/_W

= _-_P_

• _r,

= 0

of bodies,

6W

(3.2.1)

i=1

where

n denotes

the

total

number

denotes

the

virtual

work

a

34 of the system, Pi denotesthe resultant forcesacting on eachbody, and /_ri denotes

virtual

To

interconnect

system,

the

displacement

kinematic

and

of each restrict

constraints

are

body.

the

motion

imposed.

Two

of bodies types

in the

overall

of constraints

must

be distinguished: (1)

Holonomic:

constraints

restrictions

may

that

depend

be expressed

only

as algebraic

• h(rp,

The

variation

of the

holonomic

6_h

(2)

constraints

-- O_h_rv ar v

depend

Such

restrictions

are

expressed

of the

to holonomic

when

systems

are subjected

to constraints,

forces

Pi

applied

F_ and

Pi Substituting

(3.2.6)

into

(3.2.1)

and

and

as in differential

(3.2.-t)

by

(3.2.5) constraints.

may

separate

forces

the

Hence resultant

F_, so that

= F_ + F_

(3.2.6)

yields rt

=0 i=1

velocities.

relations:

= 0

is given

one

+ i=l

on position

nonholonomic

constraint

rt

=

(3.2.3)

= B,_h_Sr v = 0

h and

forces

both

constraints

where

into

refer

by

= 0

= B,_hrv

nonholonomic

6_,_h nh

(3.2.2)

= Bh6rp

that

The variation

kinematic

relations:

is given

constraints

_,_h(rv,rv,t)

Such

t)=0

Nonholonomic: kinematic

on position.

35

Since

the

work

performed

to zero,

variations

5ri

do not

violate

by the constraint

therefore

we conclude

the

forces

prescribed

constraint

in any virtual

forces,

displacement

the

is equal

that

EFt.

5r,

= 0

(3.2.8)

i=1

Note

that

not

for

all independent.

unconstrained the

systems

with Thus

system,

equilibrium

5W

principle

of the

=

the

we cannot

the

position

constraints,

interpret of virtual

system

F_-_r,

V is the

ments

5ri

tained

as expected:

are

displacements

(3.2.8) work

as F_

can

5ri

-

be used

0.

are

For

an

to calculate

as

OV.sri)=O = )2_ (_Z_.

= 5V

i:1

where

virtual

(3.2.9)

i:1

potential

energy.

all independent

Since

by hypothesis

static

equilibrium

the

the

virtual

conditions

displacecan

bc ob-

OV F_ If a system

is subjected

-

Ori

to holonomic

method

According an

of Lagrange to this

undetermined

potential

energy

multipliers

method, multiplier

(3.2.10)

:0

(3.2.11)

is used

we multiply Aj,

0

constraints

@(r)

the

-

to augment

each

and

add

of the

all

the

potent

constraints

resulting

ia

energy.

(3.2.tl)

expressions

to

by the

V to get m

v°=v +

(3.2.12) j=l

where

V a is the augmented

potential

of the

constraint

The

equations.

energy

variation

and of the

rn denotes

the total

number

augmented

potential

energy

36 subjected to the constraint condition (3.2.11) can be written rn

OV-6ri)+_-_(Aj.6_3.)

6v° = i_--I

Substituting

(3.2.3)

into

=0

(3.2.13)

3"=1

(3.2.13)

yields

trt

0V (_r/+

Zjb

0_j. •-_Tri ). 6ri = 0 ,

i=l,...,n

(3.2.14)

3"=1

Note

that

but

the

the

virtual

rn values

OV Ori

--

whereas

+

the

be treated

displacements

of )lj can

_

OCj Ori

°

_J 3"=1 remaining

6ri in (3.2.14)

be chosen

_

O,

as independent

+ _ Or iOV

hi

rn + l,n-

independent,

m + 2,...,n

displacements

variables

not

so that

i = n-

n - rn virtual

are still

6ri(i

(3.2.15)

= 1, ..., n - m) can

so that

O,

O¢ iJor-

i=

1,...,n-rn

(3.'2.16)

3"=1

From

(3.2.15)

and

(3.2.16),

we obtain

the

following

equilibrium

conditions

m

OV _--_ A3. 0@3. Or--_ + " Ori -

O,

i = 1,...,n

(3.2.17)

3"=1

This

procedure

dent

variables

n values the

us to treat

by expanding

of ri and

expression

system

enables

m values

of (3.2.7)

equilibrium

from

all the n-

of )_j.

and

(3.2.17),

conditions

virtual

rn unknowns Now,

by the

as indepcn-

to n * m unknowns

recalling

we arrive

are enforced

displacements

(3.2.10) at the

and

conclusion

presence

with

comparing that

the

of the constraint

forces 71

F: =

0¢ Or_ )=1

i = 1.... ,_

(3.'_,.is)

37

This

important

and

the

kinematic

of virtual

work

Nevertheless, recourse

provides

was

the

the

inertial

forces

of the

Fi :

we apply work,

the

dynamic

system

with

constraints.

principle

equilibrium.

systems

gives

states

if the

are considered density

p_p are

the

inertia principle

principle

that

inertial

as applied

p, the

d'Alembert's

the

to dynamic

forces

The

in static

which

dynamic

F i-F

where

system.

a system

be applied

principle system

forces with

for

in the

the constraint

by

simple

equilibrium

by

Principle

to a dynamic

for a body

can

between

bodies

stated

principle,

D'Alembert's

constraint

of the

originally

principle

D'Alembert's

plies

the relationship

conditions

to d'Alembert's

including

3.3

result

of virtual

law of state

forces

as well

forces

acting

equilibrium

_-F

forces,

the

and

as the

ap-

external

and

on the system. condition

Thus,

is given

_ =0

by

(3.3.1)

i:p are

in conjunction work

equilibrium

the with

is extended

acceleration the

by

vectors.

principle

writting

If

of virtual

the

following

equation: fv Where the

F _ may

body:

position

be

viscous

considered

forces

equilibrium;

and

to (3.3.2),

which

principle,

as d'Alembert's

use

this formulation

6rP • (p_p - F _ - F_)dV

includes

which

to resist

independent both

the

many

velocities; defined

the principle

principle

to derive

include

(3.3.2)

types spring

external

of virtual

of virtual equations

= 0

work. of motion

of force forces forces.

work

and

acting

which

on

restore

We shall

refer

d'Alembert's

In present for MBD

chapter, systems.

we

38 3.4

Governing

Equations

To derive an

the

unconstrained

many

employed. vectors 6rp

equations

rigid

possibilities

of point

p derived

be obtained

the

body

by

virtual

for MBD

using

6rp

(3.3.2)

depending

derivations, in (2.10.4),

and

= 5rTe

rotational

these

/v (_rTe

6rT[M(i

+

_ + RTrT_3

the

we start

F c =

the

The

0.

and

virtual

with

There

coordinates

velocity

(2.10.5).

+ 1T6b

tensor

= _rTe

are

one

has

acceleration displacement

two equations

_aT[b)

+

into

+

• [p(_Te

RT_T03

+ IT_Tb

(3.4.1)

6& is

_& = -_RR

Substituting

where upon

we adopt

systems,

as

5rp

and

of motion

of choosing

In present

can

of Motion

T

(3.3.2)

IT_Tb

(3.4.2)

yields

+

l T(a T_TD)

F] + _o_T[M_TRi:

) __

fT[,]di"

+ Job + &.|w - Mo]

= 0

(3.,t.3) where

M = fv

J =

pdV,

fV

pll

--T

dV,

Mr c :

fv

pidV (3..1.-I)

F =/y Performing tion

the

Mo

RTfdV' variation

for a unconstrained

= [ if dV .Iv

of 6r and rigid

body

6a can

independently, be written

M(i _ + RT_ T& + RT&_Tw)

M_TRi

_ + Jd, + &Jw

the in the

- F = 0

- Mo

= 0

equations following

of tooforms

(3.,t.5)

(3.-t .6)

39 A considerablesimplification can be made in the equations of motion if the body-fixed coordinatesarechosensuchthat the principle axescoincide with the center of mass. With this choice,all products of inertia vanish since re =6 and (3.4.5) and (3.4.6) reduce

Note

that

the

translational

(3.4.7)

to

Mi:' = F

(3.4.8)

J& + &Jw = Mo

(3.4.9)

and

rotational

mass

matrices

can

be expressed

as follows:

o°°

M=

J=

where

M and

tions

(3.4.8)

Euler

equations

of a rigid may

body.

rotational body.

the are

(3.4.9)

are

Note

that,

the

So if one

that

must orient

J1,3

J2,2 J3,2

J2,3 J3,3

and

moment

used

however,

bodies

in solving they

are

of co does can

to find

be chosen

of inertia

in order and

their

for the

of motion. motion

nonlinear

and

to any the

orienLation of a body,

the

between

relation

ot"

phys-

orientation

corresponding

it

_.' as a ['unction

to describe

to find

Equa-

rotational

not correspond

angular

body.

equations

velocity

be used the

of the

in general

for angular

integral that

(3.4.11)

as Newton-Euler's

analytically

the

m

J1,2

widely

wishes

0

(3.4.10)

|J2,1 LJ3,1

known

time

0

FJ1,1

mass

representation

of parameters

parameters

(3.4.9)

to solve

Furthermore,

of the set

and

be difficult

time. ical

J denote

m

angular

a the

velocity.

4O For MBD systems, the constraint conditions are introduced into d'Alembert's principle of virtual work via Lagrange multipliers to restrict the motion of the bodies in the system. From a formulation point of view, there are severalwaysto imposethe kinematic relationships betweenbodies during the motion. In the present derivation, we use the description of the unconstrained motion to describeeachof the bodies separately. Therefore, the virtual displacementof (3.4.1) is not a kinematically admissible one for the constrained systems. The method of Lagrange multipliers must then be introduced to incorporate the constraint conditions into d'Alembert's principle of virtual work as has been indicated in the previous sections. To apply this method the constraints are multiplied by undetermined 1,agrange multipliers

)_ and

added

to the

_rp

where

8rp, p, i:p, f, and

Lagrange The

multipliers

augmented

ing the

dV

terms

reaction

g_

defined are

represent

forces

work

of the

unconstrained

. (p_p - f) + 6_-,_]dV

are

and

virtual

which

in the variations

the

work

are

exerted

of the

of the on

(3.4.12)

= 0

previous

the

system:

derivations,

,_ arc

constraint

equations.

constraint

account

forces,

of given

the

provid-

kinematical

constraints. Substituting

(3.2.3)

D

fv[Srp

and

(3.2.5)

into

(3.4.12)

yields

(P_v - f) +_h'Ah+5*_h'_,_h]dV

,.t:;) _rp

V

Performing tion

the

for MBD

'

(p_p - f + BT_h

variation systems

of _r and can

be derived

+ B_h_,_ h)dV T

_a

independently, from

(3.4.13)

=

0

the

equations

o[ too.

in the

following

matrix

41 form

j

Augmenting

&

(3.4.14)

--k

+ BnhAnh

with

differential-algebraic

the

equations

constraint

are subjected

to satisfy

Mo

- gaJw

equations

(3.2.2)

and

(3.2.4),

+ BTA

holonomic

= F

(3.4.15)

constraints

,x,h(u,t) =0 and

nonholonomic

{i

constraints forces,

(or constraint

F is the

centrifugal mass

[i;,_] T, B is the

=

forces

for j-th

and

the

force

vector

of the

holonomic

and

matrix),

A is its corresponding

include

external

forces

and body

M j =

gradient

(3.4.17)

= B,_hfl = 0

Jacobian that

acceleration,

matrix

(3.4.16)

constraints

(_,_h(fl, u,t)

where

the

result:

Mfi

that

{ F } ,3414,

=

u is the

is given

generalized

and

displacement

by combining

(3.4.10)

"m

0

0

0

0

0

0

rn

0

0

0

0

0

0

rn

0

0

0

0

0

0

JI,1

Ji,2

']1,3

0

0

0

J2,1

J..,2

,]2,3

0

0

0

J3, I

J3,2

J3,3

body

is

for j-th

Mo - wJ_

inertia

and

nonholonomic constraint forces

due

vector. (3.,t.11)

to The

as

(3.4.18)

(3.4.19)

42 In the present derivation we havereplaceda systemhaving n by

one

with

additional

n + m

variables

fore imposing

of the

in (3.4.18)

can

which

and

Second,

method

equations

dependent

independent

that

defines

can

be generated

presence ment

so that

given

acceleration

Regardless constraint

matrix

independent

acceleration

equations

_h

+ _t

algorithm

treats

different

proce-

the

symmetry

distinguishing constraint

between

Jacobian

interconnected

joint

topologies,

as shown of equa-

with

between

The

modules.

require

bodies Fourth,

no special

variables for

ma-

the treat-

can be avoided.

holonomic

constraints

by

equation

_nh

the

the

be-

sets

preserves

without

of stand-alone

to identify

and

rotational

equations

Third,

system

of freedom

matrix

computational

relationships

in the

preprocessing

• _h The

and

A as

equations.

mass

multipliers

variables.

a set

the

translational

of Lagrange

using

of degrees

first,

for all coordinates

loops

velocity

are:

multipliers

of constraint

the translational

kinematic

by

of closed

The are

the

Lagrange

number

to a convenient

and

of the resulting and

into

the

number

derivation

will lead

the

total

m is the

be partitioned

equations

trix

considering

n is the

present

later

the rotational dures.

where

constraints

advantages

tions,

unknowns,

- m unknowns

form:

of the

nature

acceleration

= Bhfi

= Bhu + I_hti

+ 2¢_,t/_

for nonholonomic

= Bnh/i of the

+ I3nhfi

can

+ q_tt

constraints

+ 2_utfi

constraints,

equation

(3..t.20)

the be

(3.-t.21) is given

+ _tt equations

augmented

by

(3.-t.22) of motion into

the

with

f'ollowing

43 where c = -(13fl + 2_,,tli + _tt).

Since the left hand side of (3.4.23)

is symmetric and sparse, several researchgroups have developedsolution procedurestailored to solve these constraint-augmented equations. These solution procedureswill be discussedin chapter 5. 3.5

Interaction Up

rigid.

to now,

This

system The

Equations

for Rigid

the

bodies

assumption

are

subject

formulation

not

and

rigid

gular

rotations

as well as deformation.

bodies

(rigid

ies

other. are

that

specified

tions

into

e.g.,

that

rigid

the

in Downer

outlined. ered

purpose

of motion,

Since

There

in flexible

a specific kinematic

position

joint;

of this

or flexible which

are MBD

second,

relationship.

motivated

may

further

that

consist

undergo

large

rigid

of an-

sections,

initially

algebraic

account.

by

in previous

adjacent

been

mechanical into

systems

is to impose of multibody in time

of flexible

[521, only

the two

systems: a flexible These

been

have

the

independent or flexible

constraint

of bod-

equations

time.

section

essentially

be taken

are treated

between

can be written

et al.

in the

all of which

of nonlinear

bodies

formulation

bodies

must

has

systems

and

system

As discussed

relationships a set

MBD

of large-scale

bodies,

in MBD

through

on the

The ships

or flexible) Kinematic

depend

flexible

the

that

section

Bodies

the

when

design

interconnected

each

and

comprise

hold

in this

in analysis

Flexible

deformation

presented

developments

that

does

to elastic

and

body body

different

first,

two

body approaches

these

kinematic

systems discrete

so that

form,

dynamics interfacing

their

equa-

can be obtained.

is well

docuinentx?d,

requirements

connection flexible

relation-

cases

bodies

are

connects

a rigid

body

are

illustrated

will

bc

to be considconnected with

in the

by

a given following

44 sections as initial developmentfor the two-stagestaggeredexplicit-implicit algorithm, discussedin chapter 5, which is used to numerically integrate thesesetsof nonlinear equations. 3.5.1

The

Equations

The (Downer,

discrete

Park,

and

of Motion: equations Chiou

Interaction of motion

of Flexible

for this

[52]) as illustrated

approach

in Fig.

Bodies can

be expressed

3.1 where

(j) (hi) (1) ( i)

(I)

(k) (nk)

Y(e

2 )

) Z(e

X(e

I )

3)

Fig.

3.1

Interaction

of Flexible

Bodies

45

Mfi

+ D(fi)

+ S(u)

T

+ BT)ih

+ B,_h)_,_h

= F

(3.5.11)

or T

Mfi + BT,_h + B._,_h subject

to the

constraint

= Q

(3.5.12)

equations,

¢h(u,t) =0 ; ,i,_ (u, u, t) = n_hu--0

(3.5.13)

where

M_ Mj

and

Mnb

diag[M(,_b,1),...,

=

1_1 _-

0

o0

M

(3.5.14)

o Mk

M(nb,nd)]

[/_(i,l),'",(t(i,ni),/_(j,1),.'.,/_(j,n.i),_(k,1),...,(t(k,nk)]

Bh

=

[ B(i,ni) 0

(3.5.15)

T

B(j,1) 0 F(i,i)

0 B(j,nj)

- S(i,1)

-

0 B(k,1 )

(3.5.16)

]

D(i,i)

,..

F(i,ni)

-

F(j,1) q

S(i,ni)

-

- S(j,1)

__

-

D(i,ni)

D(j,1) (3.5.17)

...

F(j,,_) F(k,x)

-

S(j,,q)

- D(j,,q)

-

S(k,1)

- D(k,1)

**°

F(k,,_k) In the

above

points, M

subscript

is the

block

equations

mass

matrices

dependent

force,

hi, nj and

(nb, nd) matrix

of i-th,

of each S(-)

denotes j-th,

individual

is the

internal

- S(k,_k) nk are the

- D(k,,_k)

the

nd-th and

total node

k-th

number

of tile _rh-th

bodies,

body,

D(-)

force

operator

of discrete

diag

nodal

flexible

are

the

l)ody,

diagonal

is the

generalized

velocity-

due

to member

flexibility,

46 Bh is the gradient of the holonomic constraints that connectthe nodeswith prescribedjoints, Bnh is the nonholonomic constraint Jacobian matrix, Ah are the holonomic constraint forces, A,,h are the nonholonomic constraint forces,F are external forces, and u is the generalizeddisplacement vector. In the present time discrete form, the flexible bodies are initially treated as independent of each other. Their kinematic relationships are then imposedby given specificconstraint conditions at certain nodal points. Thus, the Lagrangemultipliers will only be computed via the quantities of these constraint nodal points. We further address this issuein chapter 5 where the two-stagestaggeredexplicit-implicit algorithm is developed. 3.5.2

The

Equations

The tains

to the

major

of Motion: difference

construction

cantly

affects

tions

of motion

the

introduced

of the

computation

for this

Mii

Interaction

case

by the presence

constraint of the

can

of Flexible

Jacobian

constraint

+ D(u)

+ S(u)

Mfi

+ BT2h

matrix,

as shown

T

+ BTAh

+ B_hA,_h

Rigid

of rigid

forces.

be expressed

and

The

= F

bodies

which discrete

in Fig.

Bodies per-

signifiequa-

3.2 where

(3.5.21)

or

subject

to the

constraint

• h(U,t)

T

+ B,,h._,,h

= Q

(3.5.22)

equations,

:0

;

_,_h(fi,u,t)=B,_hfi:O

(3.5.23)

where

M

_I_

0

0

o

Mj

o

0

0

Mk

/3.5.24)

47

Y(e

2)

) Z(e

X(e

I )

3)

Fig.

3.2

Interaction

of Flexible

and

Rigid

Bodies

or

"M(i,1) 0

M

0 ...

0 0

0 0

0 0

0 0

0 0

0

0

M(i,,_i)

0

0

0

0

0

0

0

Mj

0

0

0

0 0

0 0

0 0

0 0

0 ...

0 0

0

0

0

0

M(k,1) 0 0

0

M(k,_)

(3.5.25)

48 _-I =

[U(i,I),...,U(f,nf),Uj,U(k,I),...,U(k,nk)]

B=

[ B(i,,_O 0

BU,r_O B(j,_r)

F(i,1)

- S(i,1)

(3.5.26)

T

0 B(k,1)

(3.5.27)

]

- D(i,x)

°**

F(i,ni)

-

S(i,ni)

Q =

F.7 F(k,x)

- S(k,1)

F(k,,_k) In these

equations,

subscript

(a,b)

j denotes

the

flexible force,

body S(.)

gradient

ni and denotes

j-th

is the of the

flexible

body

of k-th

flexible

holonomic

to the body

nonholonomic A,_h are

forces,

Fk includes

loads,

3.6

and

the

Concluding

distinguished. so that eliminated

thus

hand

-

number

M

consists

forces

generalized

flexible of the

is the generalized due

the body

side of the j-th

matrix,

Ah are

constraint

forces,

due to centrifugal displacement

subscript

mass

matrix

ni-th and

rigid

the

points,

body,

flexibility,

connects rigid

nodal

of

velocity-dependent

to member

side of the j-th

Jacobian

of discrete

a-th

that

hand

D(k,,_k)

of the

operator

constraints

D(k,1)

Bh node

is the of i-th

the

nk-th

node

body,

B_a

is the

holonomic F(a,b ) are

acceleration

constraint the

external

and

external

vector.

Remarks

of bodies The

system

force

total

body,

j, D(-)

(3.5.28) -

S(k,_k)

node

rigid

nonholonomic

methodologies

a number

b-th

to the right

inertia

u is the

Two with

left

constraint

forces,

the

rigid body

internal

-

nk are the

connected

i, k and

D(i,ni)

-

for deriving subject

first

method

dependent

and

reducing

the

the

equations

to kinematic makes

system

constraint

explicit

independent equations

of motion

use

conditions

of constraint

variables to the

for systems

can

may conditions

be identified

number

be

and

of in(lepen-

49 dent variables. The secondmethod introduces Lagrange multipliers in the equationsof motion so that DAEs are obtained. The presentresearchmakes useof the latter method to generatethe system dynamic equations in DAE form. Advantagesgained by this choice are as follows. First as shown in (3.4.21), the symmetry of the DAEs for all coordinates is preservedwhich avoids having to distinguish dependent and independentsystem variables. Second,the constraint Jacobianmatrix that establishesthe kinematic relationships of contiguousbodiescan be generatedby using a set of stand-alone joint modules as presentedin the next chapter. Third, the system topology whether open or closed-loop,require no special treatment, viz., preprocessing for a-priori identification of independentvariables can be avoided. The incorporation of flexible bodies, such as beams, in multibody systemsis alsodiscussedin this chapter. The computational issuesregarding these discrete forms will be addressedduring the development of tile twostage staggeredexplicit-implicit algorithm.

CHAPTER KINEMATIC

4.1

AND

FORCE

ELEMENTS

Introduction The

equations

formations the

have

individual

of motion

been

derived

bodies

Kinematic

relations

equations.

In the

are

Jacobian

the

derivation

of the

pertaining itate

to specific

the

development A common

is the systems. motion of the

two

bodies,

constraints, translational provides

used

joints

from

equations

rigid

joint types

to describe

of nonholonomic

pertaining

interpretation

defined.

To complete

be derived

in detail

to facil-

constraint

of contiguous

the

may

allows

involve

constraints,

of holonomic constraint. joint,

from

zero

to

(noIlholoIlo_Ilic}

constraints.

universal

separation

contiguratit)ii-

universal,

In this

in .klBD no relative

relative

velocity-dependent Spherical,

conditions

bodies

which

allows

descriptions

to a spherical

of

matrix

of these

that

joints.

constraint

Jacobian

connectors,

and

examples

physical

other.

constraint

of kinematic

constraints

that

of each using

de-

program.

interaction

to devices

Two

provide

an example

must

computer

the

Consequently,

are

the

in the construction

between

(holonomic)

clearly

been

elastic

It is emphasized

established

not

systems

range

dependent

are

the

mechanical

may

of freedom.

chapter.

however,

of motion,

to describe

incorporating

as independent

bodies

has

of a modular

six degrees

previous

equations

Joints

bodies.

those

matrix

systems

treated

chapter,

feature

use of joints

in the

link

previous

constraint

of MBD

originally

that

the

straint

JOINTS

IV

revolutc,

aH(t

A rigid join_"

chapter, joint,

tile conrevolute

51 joint, cylindrical joint, and a rigid joint are derived. As for other joints such as translational and skewjoints, the kinematic principles discussedin this chapter can still be applied accordingly. After the kinematic joints are derived, the force elements such as gravity, external forces, moments, actuators, dampers, and springs will be incorporated into DAEs in order to complete the assemblyof a generalpurpose computer program. Force elementsare discussedafter the treatment of mechanicaljoints becausesomeof the constraint conditions usedin kinematic joints can be applied to the force elements thus avoiding unnecessaryderivations. 4.2

Spherical

Joint

A spherical positions allows

is characterized

of two connected three

relative

Fig.

4.1 shows

The

center

fixed

joint

coordinates

relative

adjacent spherical

of bodies

relation

_

at a specific

common

location.

i and called

and i and

of freedom

j connected

p, can

(b_,bJ,b33) 3,

during

equality

'

3-s3p

by the To

constraint

joint

motion.

by a spherical

respectively.

algebraic

This

dynamic

be represented

the

of the

joint. body-

restrict

the,

equatiotls

ar_

as

can

also

be obtained

eT_s

Since

the

bodies

¢,=ri+sp-r

This

imposing

degrees

joint,

(b_,b_,bi3),

motion

expressed

rotational

two

of the

bodies

by

is not

function

_-- rTe

of time,

• =0

by applying

+ lpibi T

"_,1)

(2.10.1

as

(-t.2.2)

- rT e _ lpjb T 3

if we differentiate

_

once

with

respect

to

52 time, the following relationship is established: _s = Bsfl

(4.2.3)

or

eT_s

where

Bs

containing

is the the

constraint

= eTBs6

Jacobian

translational

and

(4.2.4)

matrix

rotational

and

fi is the

components

velocity

of bodies

vector i and

j,

namely

= [i-,, _,

i-,, _,,.]_

(4.2.s)

( z(b 3) Y(b 2 )

x(b I )

3

Y(e2

)

rj

X(e

Z(e

I)

3)

Equation the

)

velocity

(4.2.3) vector

shows

Fig.

4.1

that

if one

fi from

the

A Spherical

time

wishes

Joint

to obtain

derivative

B_,

of the

we need constraint

to extract equations

53 and treat the remaining terms as the constraint Jacobian matrix. This can be demonstrated by the following algebraic calculations eT_s By substituting

(2.7.2)

_"

-

and

using

= /'Te + lpibi into

(4.2.6)

_Te- lpj_b, the

(4.2.6)

expression

of (2.7.7),

we

obtain T

~T



=

T

-

-

~T

lpjwj

Rj)e (4.2.7)

= :(_, + RTCo,lp,-_, - Ry_:pj) The

cross

product

of two vectors

a and

a x b = _b

Making

the

use

of (4.2.8), eT_s

=

= -l_a

(4.2.7) eT(ri

b is given

can

-T

x a

(4.2.8)

be transformed

T~T R i lpiwi

÷

= -b

by

-/'j - Rj

T

~T

= ,: [I,(1,,,Rd ,-I,(1,.R,)

to T-T lpjwj)

T]

Wi

(4.2.9)

i-j

035

Comparing constraint

the

results

Jacobian

of (4.2.4) matrix

Bs

and

I denotes

tive

of _

the

is given

= [I,

and

(4.2.9),

we obtain

B_ for a spherical

(ipiRi)

(3 x 3) identity

joint

T , -I,

matrix.

expression

of the

where

-(lpjRs)

Similarly,

the

T ]

the second

(4.2.10)

time

deriva-

by _ = B_fi

+ I3_£1 • T-T

• T-T

= B_fi

+ R i lviwi

= Bsfi

+ R i _ilpi03i

= Bsfi

+ [0, Ri

T

- Rj

-T

T

lvj03 J T -

-T

- R 3 wjlvj03 ?

~T

&ilpi,0,

T

-T

R ) _ajlp/]

(4.2.11)

03i

i'j /-, 033

54

as

138 = [0,

-(ip,(_iRi)

T , o,

(4.2.12)

(ipi(_sR¢) r ]

and 11 =

4.3

Universal The

and

allows

Fig.

4.2

joint

universal

shows

two

for the

between

dicular.

joint

, r]

, (dj

bodies

universal

connected

This

fixes two bodies

rotational

If two vectors

vanishes.

, (di

(4.2.13)

]T

Joint

two relative

equations

[ ri

connected joint

bodies remain

kinematic

degrees by

at an arbitrary of freedom a universal

during

The

can be expressed

as if there

were

with

si and

two vectors

perpendicular

motion. constraint a spherical

sj that

are

perpen-

their

dot

product

at all times

can be expressed

in space

relative

joint.

relationship

as

= { _Pu =_sSi " Sj } =0

_unv

position

(4.3.1)

Since s = sTe

we conclude

= lTb

(-t.3.2)

s_ = RT/_

(-t.3.3)

that s_T = /TR

Replacement

= lTRe

of s_ and

8T in (4.3.1.b)

;

by

(4.3.3)

lead

= t_ R,Ryt, Time

differentiation

of (4.3.4)

to

(la )

yields

t mRyt,,-

= -

t, R'-Rs/-s -

(-t.3.5)

55

X(e I )

Z(e 3 )

Fig.

If (2.7.6)

is applied,

(4.3.5)

_,

magnitude

is a scalar,

Universal

the

Joint

becomes

transpose

+ (TR,RT@/j

of the

first

term

(,t.3.6)

of (4.3.6)

gives

the

same

as

: ;fR, Applying

An

= l_iT~Tw i RiRj/_;T

_u

Since

4.2

(4.2.8),

_u

+t:rR,Ry 2 ;

(4.3.7)

we obtain

=/2

T

RjRi

T ~T

l_i wi + l_Ti RiR;

T ~T

l_;.w: (4.3.8)

=[0,

I_;.RjR

i l_i , O,

/TR/R

]

/-;.

56 and the constraint Jacobian matrix Bu can be expressedas

B_ = [ 0, t_jRjR_ t_ , 0 _

Following

the

=

same

procedure

l_jwjRjR

i/_i

as in (4.2.11),

_ ]

the

+/jR/Riwil-i,0,1-

]3,

(4.3.9)

is found

iw iRiRjlj

to be

+ (4.3.10)

Hence

the

gradient

can

be written

4.4

Revolute

matrix

degree

connected joint

by

can

ies with other. equations

joint

attaches

two

of freedom

during

actual

a revolute

joint.

The

as if there

vectors

si and

Mathematically,

their

for the

revolute

(4.3.2)

sj

----

has concluded

that

bodies

for the

motion.

universal

that

s_ = RT/,

= I_T/_ = RT&/

for

joint

one two

the

connected

remained

be expressed

allow

4.3 shows

joint

is equal

=o_ Si × Sj

and

equations

always

product

_rv

Fig.

a spherical are

can

in space

constraint

were

cross

joint

_rev

Equation

equations

Joint

be expressed two

constraint

as

A revolute tional

of the

to zero.

parallel The

rotabodies

revolute two

bod-

to each cotlstrait_I

as

} =o time

= -RTlw

differentiation

(4.-,.1) of ,q yields

(4.4.2)

57

b2 )

,,

.

Sip

Y ( b2 )

sJ

(bl

z(b

3)

si

x (bI )

rj Y(e

2)

X(e

Z(e

Fig.

To obtain

Br,

the

time

¢_rv

Replacement

of s and

=

3)

4.3

A Revolute

differentiation

Si

X Sj +

s in (4.4.3)

1)

$i

of @r.

X Sj :

--Sj

by (4.3.2)

Joint

is taken

X Si

and

+

and

Si

given

by

X Sj

(4.4.2)

lead

(4.4.3)

to

¢_ = _j_y__,o,, - _,RT_,o, ,. oai

(4.4.4)

= [0, _,.R,_,, o, _,RTi,] _. % From

(4.4.4),

the

constraint

Jacobian

matrix

for @_

can

be easily

verified

58 to be Br The

time

derivative

f3r = [ O,

Hence

= [0,

§jRT_i,

of (4.4.5)

is found

§'iRT-l_i + §jRT_il_i,

the

constraint

Notice

that

if two

straint

equations

Jacobian

0,

-§iRTlj

]

(4.4.5)

to be

O,

matrix

-siRT_

for the

j -

§iRT_zj_3

revolute

joint

]

(4.4.6)

can

be written

&S

tions, can

of which

of the

largest

to remain

parallel

Equation

(4.4.5)

needed.

as linear the

the

proper

set

values

equations,

at all times, yields

are independent,

combination

absolute

constraint

of the

remaining from

of each

equation

row

select

the

two con-

algebraic

i.e., one of the

of equations

and

three

only

equa-

equations

equations. the

A tech-

overdetermined of the

two equations

set

gradient that

ma-

havc

the

terms. An

two

are

only two equations

for selecting

is to compare trix

are

be derived

nique

vectors

proper

lationships

alternative vectors as

a

that revolute

b i = [bl, b2, b3] i and attached for this

to bodies revolute

approach are

to modeling

capable

joint.

a revolute

of representing

This

approach

b j = [b_, b_, b3] j be two triad i and j respectively

joint

can

(Fig.

be expressed

4.4).

joint

their

kinematical

is followed of orthogonal The

is to set

kinematic

re-

below. unit

up

Let vectors

constraints

as

(I)_

}

(4.,t.8)

59 For any configuration, the constraint joint

is derived

and

given

Jacobian

B,

of the

revolute

by

c2(R

)

R

cI

i iT j j} ci3(Ri)TRJc_

B,=

where _, c_, and c_ are the components

x(b

matrix

(4.4.9)

=0

of b i and b j.

I

x(b I)

si

Y(b2)

a

j) !

!

Y(b 2 ) z(b

3)

q

Y(e 2 )

X(e I )

Z(e 3)

Fig.

4.5

Cylindrical

tions

A Modified

Revolute

Joint

Joint

A cylindrical of freedom

4.4

to two

for a cylindrical

joint connected joint.

provides

one

bodies. The

translational

Fig.

constraint

and

rotational

degree

4.5 shows

the

constraint

condi-

equations

are

derived

from

the

6O

condition that vector ui

must

_I_cl --" _c2:uJ

_c_l:

where

u_ and

frames

so that

obtained

from

u i are given the

remain

two

parallel

Ul

directional bodies

X xd

Uj

unit

will

slide

to vectors

= fiiUj :fild

vectors

uj and

d:

_. :0 S

based

according

(4.5.1)

on their

to that

body-fixed

axis,

and

d is

d = ri - rj. y(b, z(b

3) x(b I )

Y(b 2 )

,(b_)

(j)

Y(e 2)

X (e I )

Z(e 3 )

Fig.

Since

(4.5.1.a)

in the same

has

the

4.5

same

way as (4.4.5)

and

form

A Cylindrical

Joint

as in (4.4.1.b),

(4.4.6).

If the

Be1 and

first and second

B_I

can

time

be found

derivatives

61 of (4.5.1.b) are taken, Bc2 and I3c2 can be obtained as Be2 = [ fii,

dRTl_i , -fi,,

0]

(4.5.2)

i3_ = [6,, _RT__,+ aRY_,i, , -u, , o ] The

final

constraint

equations

for the

cylindrical

Bcyz :

4.6

Rigid

Thus

ical joint

joint

by definition

us a nonholonomic

with

statement

can

no relative

constraint

the gradient

be written

as

4.7

Elements

Force

In section discussed. include

Forces gravitational

that

among

of _rj

=[0,

of the

3.3, different are forces,

motion

I,

0,

constraint

two

as a spher-

bodies.

as following

is given

between

be imposed

the connected

= Wi -- Wj = Brjd

matrix

in MBD

can

°.

_rj

matrix

that

constraint

{

:

Jacobian

no relative

mathematically

Brj

Hence

(4.5.4)

Bc2

allows

velocities

be expressed

_rigid

The

are:

Joint

A rigid bodies.

joint

(4.5.3)

The

above

equations

}

= 0

(4.6.1)

by

-I]

equations

(4.6.2)

for the

rigid joint

can

Systems types commonly actuator

of forces

acting

encountered forces,

damping

on the bodies

have

in mechanical forces,

spring

been

systems forces,

62 and external forces. In the present section, these forces will be formulated and incorporated into DAEs as F = Fw -t- Fg -I- Fa + Fd -t- Fk -F Ff

where

F_,

Fg,

Fa,

damping,

spring

analyzed

in further

4.7.1

led,

and

inertial with

detail

Ff

denote

forces,

centripetal,

respectively.

gravity,

These

force

actuator,

elements

below.

the

acceleration frame

rng can

of gravity

fixed

in the

be calculated

is measured

earth,

by the

the

with

gravitational

respect force

ity.

fg is the

force

If we choose

inertial

frame,

the

field acting force

Fg that

(4.7.1.1)

and

g is the

on the

External

The

reference

z direction

by this

of the

gravitational

moment

Forces

and

T

Moments

a force

f(i)

acting

of f(i)

about

the

M(i)

where

of grav-

i is

Consider 4.6.

acceleration

negative

contributes

= 4.7.2

an

equation

by the gravity

a gravitational

reference

field on body

created

to

of a body

f9 = rngg

where

are

Force

reference mass

and

external

Gravitational Since

Fk

(4.71)

s (i) = lTb (i) is the frame.

The

position

contribution

on body origin

= s(i)

vector of

f(i)

i at point

of the

body

p as shown is

× f(i)

(4.7.2.1)

of point and

in Fig.

p in the

Mo (i) to the

i-th force

body-fixed vector

Ff

63 on body

i is

T If a pure

moment

m (i) acts

on body

i, then

(4.7.2.2)

the

force

vector

F/

on body

i

becomes F(f/) = [0, m(/)] T

(4.7.2.3)

Z(e 3 )

Fig.

4.7.3

Actuator

4.6

Force

Acting

on a Body

Forces

An actuator dependent

A Point

pair

is a force

of forces

tion

of these

forces

Fig.

4.7)

and

P3 is defined

where

acting

is defined

the

actuator

element

that

on two bodies by the connecting is installed.

provides

a constant

in MBD points

A vector

or a time-

systems.

The

direc-

of bodies

i and

j (see

li_ connecting

points

Pi

as

(4.7.3.1) The

actuator

force

f_ acting

on bodies

f(i)= + fala

i and

;

j is given

fU) = w fala

by

(4.7.3.2)

64

where

la =

given

by the

Fa

_.

The

i

actuators.

on bodies

i and

sign The

constitutes contribution

a pull

and

a push

of f(0

and

f(i)

forces

that

to the force

are

vector

j are

(4.7.3.3)

i_j 2

Z(e

3)

rj

Fig.

4.7.4

Damping

i and

4.7

An Actuator

Acting

on Two

Bodies

forces

Dampers energy

(j)

i)

dissipate

to dissipated j at point

relative

forms

Q/and

Qi

such (Fig.

body

motions

as heat.

The

4.8)

is found

by converting

damping

force

mechanical

between

bodies

to be

IT io (4.7.4.1) where

d is the

damping

coefficient

and

i,,, : ÷[e + rib, The

damping

forces

acting

on bodies

- ÷re -if'b, i and

(4.7.4.2)

j are

; /J")-- :Fi t

(4.7.4.3)

65 in

which

ld = _.

on bodies

i and

The contribution j can

be found

of f(0

from

and

fU)to

the

force

F d

vector

(4.7.3.3).

i) ( (j)

Z(e 3 ) b

Fig.

4.7.5

Spring

4.8

A Damper

and

Between

Two

Bodies

used

to restore

Forces

In mechanical equilibrium

Acting

systems,

of two bodies.

Sj of bodies

i and

springs

In Fig.

j.

The

are

often

4.9, a spring

spring

force

is attached

is calculated

position

to two

points

by

f. = k(l. - lo) where

k is the

l0 is the The

spring

undeformed

spring

forces

coefficient, length

acting

on bodies

f(O

where

Is = _.

The

along

= +fsls

contribution

(4.7.5.1)

ld = ISj - Sil the

vector

i and

;

S_

is the

between

deformed two points

length

and

Si and

S_.

j are

:U)

of f(s i) and

= q:fsls

f(J)

to the

(4.7.5.2)

force

vector

Fs on

66

bodies i

and

j can

be found

from

(4.7.3.3).

(i)

rj

Z_e3 )

Fig.

4.8

Concluding This

joints

and

straint

chapter

forces

matrix

differentiation

each

joint,

can

dures. and

Bodies

enables

Under

software

for multidisciplinary

vectors

such

constraint

of confusion. MBD

that

velocity

equations.

risk

for mechanical

It is emphasized

by a different

of motion

remaining

issues

5 reviews

with

systems.

expression

From

the confrom

the

circumstances,

Jacobian

matrix,

a programming

to possess

modularity

engineering

problems

generated.

disadvantages,

a numerically

Two

by extracting

without

development

equations

Chapter

conjunction

Between

mathematical

constraint

is represented

be automatically The

explicit

is obtained

separately

this the

Acting

to MBD

of the

which

be written

so that

gives

pertaining

Jacobian

standpoint,

A Spring

Remarks

time

can

4.9

and the

robust

regarding

existing

proposes

two-stage solution

DAEs

solution two

staggered procedure.

new

emphasize procedures,

constraint explicit-implicit

solution their

proce-

advantages

treatment algorithm

schemes

in

to form

CHAPTER SOLUTION

5.1

equations

chapters

3 and

to DAEs

cannot

methods

must

Several

existing

5.2.

of the

be applied

DAEs

regard,

stabilization 5.3 and

lack

of broadly

be derived algorithm

of the

Reviewing

only

and

5.5,

the

prevents

The Euler the

DAEs

are

focused

in the

are

criteria

also

stabilization a varying

with

both

degree

systems. constraint

developed called

or

algorithms

field of MBD

in sections

two-stage

stag-

translational

and

of this

It is shown but

in sectreatment

numerical

to integrate

equations.

instability

on the

offer

deal

algorithm

stability

discussed

constraint

that

is developed

separately.

equations

of DAEs

a numerical

procedure

nonlinear

robust

topic

solution

highly

methods

methods

elimination

by linearizing not

these

form

in

numerical

either

applicable

numerical

constraint

motions

while

formulated

problems,

primarily

involved

are

a closed

simplified these

as a challenging

two robust

In section

have

that

Since

for solving

which

However,

explicit-implicit

rotational

to solve

methods

remains

and 5.4.

in order

equations,

for solving

5.2

DAEs

systems

as DAEs.

methods

elimination.

MBD

for highly

numerical

the

nature

FOR

except

numerical

of success,

for

characterized

be found

constraint

In this

of motion

4 are

These

constraint

gered

PROCEDURES

Introduction The

tion

V

algorithm

that

maintains

will

the

present

the

explicit

algorithm.

of Existinl_

In reviewing

existing

Solution DAEs

Procedures solution

procedures,

a numerical

method

68 that is basedon Gear's backward differencealgorithm has been first developed to solve DAEs [20]. With little successby using this algorithm, Gear and Petzold [21,22]havesolved DAEs by differentiating the constraint equations twice in time and augmentingtheseequationswith the governing equations of motion to form a combinedset of second-orderdifferential equations. If the augmentedconstraint equationsare numerically integrated, however, constraint violations will generallyoccur becauseof accumulatedintegration errors. Severalsolution proceduresthat are basedon dinate

partitioning

technique

[23,24],

to overcome solution

this

key

so-called

the null space drawback.

Gear

earliest

of augmented

straint

equations

derivatives

Gear

system.

equations

following

have

sections

stabilizatior_ been

developed

we address

method

starts

with

that

to solve

been

these

by considering

equations

formula

and

represent

transforming

was the

the DAEs

re-

to form

algebraic

con-

in which

time

of motion solved

is the

to some used

equations

The

DAEs

applied

algorithm

of differential

difference

equations

has

This

of motion form

was used

[20] that

do not appear.

backward

constraint The

constraint

[30-32,54]

that

equations.

as a special

into the

algebraic

In the

algorithm

equations

of the variables

substituted

method

algorithm

differential-algebraic

a set

Baumgarte's

coor-

Method

numerical

stiffly-stable

stricted

[27,53],

generalized

in detail.

Stiffly-Stable The

the

and

procedures

5.2.1

with

technique

the

are

then

simul_alwouslv

kinematic into

the

joints

of

following

as M_" + BT_

:v

= F

(5.2.1.l)

(5.2.1.2)

69

(5.2.1.3)

O(q, t) =0

which

can

f(y,

be expressed

into

the

following

matrix

form

_', t) =

(5.2.1.4)

=G_,+g=0

where

[ oo

G=

g ----

y = [q,v,

Solutions

of (5.2.1.4)

depending

upon

may

the

driving

system,

the

present

systems

are

characterized

components. the

time

small.

with

means

to the

explicit that

a large

high-frequency

components

stiffly-stable for solving

solution

is adopted

DAEs procedure

to compute

f_k)Ay(k

high

and

low frequency

presence

the

schemes

stiff.

that

desirable.

parabolic

starts y, applied

with

be kept

damp

out

The

Gear

relatively to obtain

errors

associated

algorithms,

ranges,

arc

din'

thus

w_'ll

characteristics. the

Newtoh-Raphson

to (5.2.1.4)

) + f_k)A_r(k

components,

is required

for high-frequency

with

stiff

by low-frequency

must steps

of the

Numerically

of high-frequency

of time

components

eigenvalues

dominated

algorithms

number

are

and

numerically

solutions

characteristics stiff

}

= [dl,_',,_] T

numerical

Consequently,

The which

of the

0

become

by having due

0

g in (5.2.1.4)

may

solutions.

to their suited

step

This

accurate

system

However,

both

term

0

_%, F - BTA

A]T,y

contain

I

) = _f(k)

algorithm,

as

(5.2.1.5)

7O where k and

is the

iteration

cycle

A) (k), the pth order

ten

number.

Gear

To obtain

algorithm

the

relation

for (5.2.1.5)

between

is employed

Ay (k)

and

writ-

as p--I

y,+l

= _(ajyi-j)

+ b h f(yi+l,yi+l)

(5.2.1.6)

i=1

where

h is the

determined For

the

time

step,

depending k th and

and

upon

the

aj and

b are

order

the

coefficients

of the algorithms

(k + 1) th iterations,

(5.2.1.6)

can

that

need

one wishes

be rewritten

to be

to apply.

as

p--1

(yi+l)k

= (y_'(ajyi-j))k

+ b h f(yi+l,y,+l)k

(5.2.1.7)

i=1 p--1

(y,+,lk+, =

y,-Jllk+, + b h

(5.2.1.8/

i=1

Since the

the

summation

f th and

Subtracting

term

previous

time

(5.2.1.7)

of (5.2.1.7) steps,

from

they

(5.2.1.8)

A_,(k)

which into

gives (5.2.1.5),

the

relation

solving

remain

constant

are

only

a function

during

the

of

iteration.

yields

= --_-1Ay(k) bh

of Ay (k) and

A!}'(k)

(5.2.1.9)

Substituting

1 = (gy + _---_G) (k)Ay(k)

(k)Ay(k)

(5.2.1.10),

the

update

y(k+l)

value

= y(k)

= y(k) The

(5.2.1.8)

(5.2.1.9)

back

we obtain

1 (fy + _-_fy) Upon

and

drawbacks

2n + rn equations,

of this thus

procedure

are:

for a large-scale

of y and

= - f(k) y can

(5.2.1.10)

be obtained

+ Ay(k)

+ _h first,

(5.2.1.I1)

Ay(k )

it has expanded

system,

by

it presents

(5.2.1.12) rt

÷

some

r_l DA [_s iHto ineflh:iency

71 solving y and _, by using Newton-Raphson algorithm. Second,at starting time, there is no information availableto determine the Lagrangemultipliers )_. An poor

estimated

requirement effects

)_ may

of many

on the

time

constraint

cause

steps,

the

the

system

time

equations,

to diverge.

discretization

and

Third, may

eventually

lead

due

have

to the

compound

to useless

drifted

solutions.

5.2.2

Direct

Intel_ration

In the

previous

Lagrange

multipliers.

purposed

to convert

tions

state

equations

the

DAE

the

second

equations.

in the

where

¢ =

the

following

-(l_li

we have

system

+ 2_util

calculated

by solving/i

expression

substituted

(3.4.19)

m x rn matrix

+ _tt). first

the

can be computed (5.2.2.1)

BM-1B

by substituting

and

and

Petzold

[21,22]

differential

equa-

of the

constraint

(3.4.20)

with

constraint-augmented

The row

second

BM-1BTA

If the

Gear

of choosing

cquations the

governing

equation

can

be

,_ call

be

form:

of the into

difficulty

a set of ordinary

derivative

resulting

the

difficulty,

into

time

matrix

observed

this

Combining

of motion,

written

section,

To overcome

by appending

to the

Method

Lagrange

of (5.2.2.1) row

the

singular,

result

and

of (5.2.2.1)

= BM-1F

T is not

multipliers using

of (5.2.2.2)

resulting

so that

(5.2.2.2)

+ c

the

the

acceleration

vector

in to tile first

row o['

to yield

ii = M-x[F

- B T(BM-1BT)-I(BM-xF

+ e)]

ii

(5.2.2.3)

72 Numerical algorithms are then employed to compute the generalized velocity and position vectors at the next time step. Note that, upon using this approaches,several difficulties may occur: First, during the process of simulation, BM-1B numerical

accuracy

gorithms

are

numerical tion

of the

may

start

anism

to correct solution.

Third,

the

essarily the

accuracies.

this

defect,

the

numerical

time

where

nonlinear

proposed.

ing,

Baumgarte are

5.2.3

Generalized

developed the system

Coordinate

equations

straint

equations.

alized

coordinates

[53]. and

Their

al-

reason

that solu-

numerical

errors

conditions

are

is no numerical

gradually

diverge

mechthe

violations.

equations equations

no

from

as constraint

constraint

do with

not

nec-

fidelity

in

involved. several

corrective

the generalized the

Partitioning

Wehage

determine approaches

of DAEs

may

integration

methods

coordinate

penalty

have

partition-

constraint,

and

null

was

first

sequel.

coordinate

Calahan

are

include

in the

there

the

approximate

constraint

constraint

stabilization,

reviewed

the

difficulties,

methods

generalized by

algebraic

an

degrades

the

the

is known

of the

expressions

constraint

which

difficulty

aforementioned These

space,

The

original

for

only

Since

solution

derivatives

numerical

integration,

that

desired

the

been

point

to the

second

To avoid

time

which

will occur

provide

the

to the

represent

case

/i, difficulty

During

This

because

algorithms

to accumulate

satisfied

exact

integration

ill-conditioned,

Second,

to compute

equations.

longer

become

of (5.2.2.2).

used

time

T may

are not

Method

partitioning and

method

Haug

[27} use

independent are based

(GCPM) this

coordinates on the

all independent.

fact

If the

idea froru

that

the

to reduce the

con-

7z gener-

n coordinates

can

73 be partitioned into the

velocity

and

acceleration

fi_ accordingly, i denotes (3.4.14) the

rn dependent

where

the

d denotes

its time

following

n -

vectors

independent

and

and

dependent

generalized

(3.4.18)

and

Bfi+I3fl B d has

m

full

row

acceleration

rank,

vector -d

tioned

equations

of motion

d :

(3.4.19)

rid, fi,,

fig, and

coordinates, constraint

can

and

equations

be rewritten

= 0 --Bill

= {Bd[B

u

The

The

then

into

forms:

Sdfl

dependent

into

generalized

coordinates.

q,(u d, u

Since

coordinates,

can be partitioned

the

derivatives

m independent

i]

(5.2.z.3) O, therefore

indicates

the

by

= -Ba-I(B'fi

(3.4.13)

(5.2.3.2)

i

fii

which

is given

(5.2.3.1)

_ + Bfl)

can

(5.2.3.4)

be rewritten

into

the

following

patti-

form

M

Premultiplying

d

0

rid

by B d-r

on the

Ba-T

Substituting

,_ of (5.2.3.6)

Mifi i

Replacement

(M i

+

BiT

of fig in (5.2.3.7)

+

B*TMdB*)fi

i =

first

row

of (5.2.3.5)

Maii a + ._ = Bd-TF

into

(Bd-TF

(5.2.3.5)

Fd

the

second

d _

row

Bd-rMdii

by (5.2.3.4)

F i _

B'TF

d _

a

(5.2.3.6)

of (5.2.3.5)

d)

leads

yields

=

F i

yields

(5.2.3.7)

to

B-TMdBa-IBfl

(5.2.3.8)

74 where B* DAEs

= Bd-IB

where

cally,

these

a set

constraint

is used

to integrate

and

(5.2.3.3)

equations. the

by satisfying

vector,

factorizing

the

From

one

left

algorithm

(1)

Given

initial

(2)

Solve

(5.2.2.1)

(3)

Specify

(4)

Integrate

(5)

Solve

step.

may

hand

variables,

under

as follows:

conditions

u °, and and

(5.2.3.1)

process

fully

(5.2.3.8)

dependent

(5.2.3.1),

vari-

(5.2.3.2),

independent

computational

populated

matrix

cost

of

(5.2.3.8).

A

fi0.

variable

u ''_+1,

for u d'_+l

violat-

of view,

the

of

Numeri-

without

of solving

the

form

An.

(Compute)independent u in to obtain

point

estimate

of the

is stated

for fin,

be solved

equations

the

reduced is given.

whereas

constraint

During

not

side

can

a computational

the

the

equations

equations

independent

time

represents

differential

differential

at each

acceleration

(5.2.3.8)

of independent

are obtained

GCPM

Equation

independent

ing the

ables

i.

and

by using

u'

u i_+1 Newton-Raphson

method

; u "+1

]s

obtained. (6)

Solve

(5.2.3.2)

(7)

Go to step

(2) until

In step Raphson

(5),

iteration

the

a good may

proximation

of u d can

to the

terms

third

u d"+l and fl.+J

for

required estimate

converge

run

time

is reached.

of u d is needed

within

be obtained

is found.

a few

by taking

iterations.

the

Taylor

2

pendent

and

the

Newton-

A reasonable series

ap-

expansion

up

as

u dn+l

Wehage

so that

and

dependent

Haug

= u dn + hu dn + --u 2

[27] have

generalized

developed coordinates

..

d

an algorithm by using

(5.2.3.9)

to identify L-U

inde-

factorization

75 to decomposethe constraint Jacobianmatrix B. This algorithm occasionally leadsto poorly conditioned matrices. When this occurs, it is necessary to choosea newset of independentgeneralizedcoordinates by repeating the L-U factorization process. This strategy not only increasesthe computing time but also propagatesintegration errors. In recent years, many research groups havedevelopednumerical techniquessuch as the singular values decomposition [28,29] (SVD) and the QR decomposition [55] to factorize the constraint Jacobian matrix. These techniquesprovide some marginal advantages over L-U factorization, but the main idea remains basically the same. 5.2.4

Baumgarte's

Constraint

To stabilize Baumgarte This

[23,24]

method

differential

of error

and

second

constraint

proposed

modifies

laxation

the

the

Violation

the

violations

original

a stable

has the

response. by the

constraints,

stants used

to stabilize

tions

and

their

where

stability. the

time

error

(5.2.2.1),

stabilization

method.

to form

a set

to suppress

method,

the

Baumgarte

constraint

of regrowth

replaces

equations:

+/32(I ' = 0

(5.2.4.1)

and

constraints

for numerical

capability

In this

+ for nonholonomic

in solving

equations

following

(_ + 2a(_

Method

occur

violation

constraint

that

row of (5.2.2.1)

for holonomic

that

a constraint

equations

achieve

Stabilization

= 0 a, _32 and

Obviously, committed

derivatives.

by

To study

"_ are

2a(D,

f12_,

the

violation

the

arbitrary and

behav.ior

positive

-y(I) are

the

of constraint of the

method,

couterms equathe

76 generalsolution of the first order ordinary differential equation for constant "I (5.2.4.2) is expressedas _i = where ¢i

the

constant,

is decaying

ferential

ki,

as time

equation

kxi

conditions,

determined

(5.2.4.1),

and

k2i

the

from

= kli e_'t

are

the

q- k2ie_'_'t,i

conditions.

second-order

solution

integration

(5.2.4.3)

initial

For

general

Note

that

ordinary

for constant

dif-

a, _ is

= 1,...,m

constants

(5.2.4.4)

that

depend

on

the

initial

and

= In order the

i = 1, ..., m

t is progressing.

_i

in which

are

ki e-'rt,

to make

critically

the

damped

solution

to the

(5.2.4.4)

requires

_i :

Substituting

(3.4.18)

and

equations

of motion,

the

yields

following

the

When

a and

/_ are

second

time

derivatives

solution

_3, the

numerical

nitude

and

(kli

(3.4.19)

- 92 constraint that

(5.2.4.5)

equation

a = 13 and

decay

optimally,

a > 0 so that

(5.2.4.G)

+ k2it)e -c*t

into

Lagrange

(5.2.4.1)

and

multiplier

for

replacing

/i from

holonomic

the

constraints

expression

BM-1BTA

merical

+

may

= BM-XF

equal

of the diverge

solution

frequency

to zero,

equation

constraint from

oscillates

of these

+ c + 2aBfl

the

(5.2.4.7)

exact

exact

depend

the

(5.2.4.3)

solution. the

(5.2.4.7)

recovers

equations

about

oscillations

+ _2_

on

For

where

tile

nonzero

solution. the

original

values

nu-

_ and

The

mag-

of o and

77 /3. Park and Chiou (1986,1987) have shown that for someMBD problems Baumgarte's technique suffersfrom the following drawbacks: (1) When BM-1B

T is ill-conditioned,

is considerably (2)

The

errors

constant (3)

the

committed

values

numerically

this a,

the

constraint

physical

and

time

the

are

truncated

after

a key

stepsize

h.

equations holonomic the

in which

t is the current

integration

process. (t + h),

a.s

decay

of dynamic

the

damping

of motion,

with

one

time

problems. terms

and

to dominate

make

the system

(5.2.4.8)

with

can

series

the

of

coefficients

approximate

expansion.

at the

analysis

the In such

(t + h) time

step

terms:

+ h (t) +

and

the

one

equations

derivative

an

between

Taylor's

constraint

time,

procedure,

relationships

by using

second

Since

in this

Mathematically,

h is the

4 8)

time stepsize

constraint

conditions

used should

in thc

numerical

be vanished

a_

becomes 2. + _

(5.2.4.5)

role

to obtain

• (t + h) =

Comparing

natural

"7 will cause

/3 play

constraint

expansion,

step

multipliers

stiff.

an

time

equations

of the equations

is undertaken

and

nonlinear

of the

solution

a

method fl,

in the

of a, fl, and

numerical

Since

of Lagrange

degraded.

regardless

Large

the accuracy

(5.2.4.1),

2 + _

= 0

o_ and/3

can

1 h

h

(5.2.4.9)

be expressed

in terms

of h

78

Replacementof

a and/3

in (5.2.4.5)

/0,1, 2

It

is noted

shown

that

in (5.2.4.6).

used,

But

this modified

constraint and

(5.2.4.10)

_.

Recently,

the

optimal

can

be damped

5.2.5

Null The

alternative cedures

null

space

methods

the

time

step

damped

technique

fl so that

damps

constant

is

out

value

a method

the of a

to determine

the constraint concerning

has

algorithm

indeed

assigned

difficulties

as one

integration

a and

T remain

(5.2.4.11)

critical

developed

But the

by the

thogonal

complement

span

subspace

method

to deal system

a matrix

spanned

the

reach

[56] have

constants

to

violations

the

appearance

unanswered.

Method

to eliminate

introduced

lead

i --- x/-Z-i -

an arbitrarily

Yang

BM-1B

Space

not

than

out efficiently.

of an ill-conditioned

5= i),

of Baumgarte's

and

stabilization

(5.2.4.10)

if a constant

faster

Bae

1 _(-1

=

does

version

violations

by

[30,54]

with

of the of B,

C.

DAEs

its variations

that

constraint

method

rows

and

forces.

by considering constraint and

adopt

special

Hemami the

Jacobian

A T be a (n -

[17,18,31,32!

m

and

Let

x n matrix

pro-

Weimer

dimensional

matrix. m)

numerical

are

i5-1

subspacc C be the whose

orrows

By definition

ATB

T = 0

(5.2.5.l)

Premultiplying (3.4.13)by A T and using the result given by (5.2.5.1),Lhc governing second-order differential equations become

Since

the

uniquely

choice

of A T is not

determined.

ATMIi

= ATF

unique,

the

reduced

de Jalon

et al.

Recently,

(5.2.5.2) system

equations

!17,181

have

are

utilized

not this

79 concept to obtain the null spaceof the constraint Jacobian matrix by using natural coordinates. Their procedurestarts with the assumption that independent expressed

velocities

fii can

be selected

as a projection

of fi which

E is the

(5.2.5.3)

with

matrix (3.4.18)

is not

which

defines

the

linear

singular,

= 0, we conclude

Combining

combination.

yields

fl = [B

}

(5.2.5.4)

then

-a

that

fl = A6 _ where

A is an n x (n - m)

null space

of B.

Replacement

matrix

u i ¢- 0, which

whose

(5.2.5.6) column

of fi in (3.4.18)

Bfl

But

be

(5.2.5.3)

,_

If Ot

can

as fii = Eft

where

n - m

= BAIl

' = 0

by (5.2.5.6)

a basis leads

of the

to

(5.2.5.:)

implies

BA =0 Transposing

constitute

of (5.2.5.8)

(5.2.5.s)

gives

ATB

T = 0

(5.2.5.9)

80 Equation (5.2.5.9) hasachievedthe goal of constructing the null spaceof the constraint Jacobianmatrix sothat constraint forcescan beeliminated. Augmenting (5.2.5.2) and (3.4.19), the final system of equationsof the present procedure can be rewritten in the following matrix form

= Note

that

violates

(5.2.5.10)

not

the constraint

To avoid

these

only

conditions

drawbacks,

stabilization

technique

independent

variables

destroys

the

when

one

can

or reduce by taking

-g i j symmetry

time

either

and the

(5.2.5.10)

integration adopt

express time

of the

matrix

but

algorithms

the

are used.

Baumgarte

(5.2.5.2)

differential

constraint

in terms

of system

of (5.2.5.6)

as

fi = Aft i + Afi_

Substituting

(5.2..5o11)

into

(5.2.5.2)

ATMAfi

This

expression

eliminates

of symmetrizing However, arising

this from

integrate time

approach the

fact

the

independent

derivative

A need

acceleration

can

derivative/it

are obtained

resolution iteration

of the becomes

i = ATF

equations suffers

that

without from

accelerations

Since

by solving

unavoidable.

equations The

i

forces

violating

two

major

system

dependent

equations. the

drawback

first

are

thc

matrix

one

used

to

A and

its

indcpcn(tcnt

A and

velocity

a costly

the goal

matrix

so that

null space

using

achieves

constraint

null space

the

second

and

algorithms

in advance

by

(5.2.5.12)

drawbacks,

integration ii _, the

to be evaluated

constraint

_ ATM/kfl

constraint

if numerical

be determined.

(5.2.5.11)

yields

the system

system

also

and

its time position,

Newton-Raphson is that

during

the

81 processof time integration somepositions can be reachedwhich can cause the matrix in (5.2.5.4) to becomesingular. The reason is that the chosen independent variables do not numerically represent uniquely the motion of all the possiblemechanismsduring the processof simulation. To be able to continue the integration process,a new matrix E must be chosen. Since these solution proceduressuffer to some degree from computational difficulties, we have motivated to procedures which

that

involve

technique)

overcome either

such

constraint

or constraint

developed

in the

and

efficiency.

5.3

Penalty

Constraint

in the

of trying

to eliminate

in constraint concept

(Lanczos,

motion,

e is the the

constraint

stabilized

their

scheme)

numerical

will be

robustness

Technique

the

objective

equations. constraint

is to minimize

In the

penalty

violations,

of the the

the

constraint

the

technique,

errors

In other

violation

being

words,

committed

the

constraint

ac-

instead

by applying

equations,

of the

error

the

Lagrange

equations

as

[51])

= --,

where

procedures

to emphasize

from

solution

solution

sections

control

are determined 1949

(penalty

will be controlled.

of proportional

new

partitioning

constraint the

alternative

(natural

method,

equations

multipliers

stabilization

Stabilization

In Baumgarte's

These

for

elimination

following

cumulated

difficulties°

look

penalty

final

coefficient.

equation

0 < e
. 0

-2o JOINT

2

I I

-40 0

20

40

60

80

100

X (in) Fig.

7.6.1

Crane

Tip Trajectory

of Rigid

and

Flexible

Members

6

.-j. _J

¢J O

I

0

I

I

I

I

0 Time

Fig.

7.6.2

Specified

I

I

I

5

Crane

Tip

Velocity

10 (sec)

of Rigid

and

Flexible

Members

149

1600

_•

8OO

._

JOINT s

/ L\\

,o,_,,---_

o -800

-1600

I

I

I

I

I

0

I

7.6.3

1600

I

I

5 Time

Fig.

.I

Crane

Joint

Torques

10 (sec)

(Rigid

Members)

.vs,

Time

!i

800

,o_,_/,

tt_

JOINT

S

,An,"_A",,,.,,.A,A.,,t,,,,,.A,.%:/)m f..,._"

0

¢ -800

o

JOINT 2 ----/

_

/

W -1600

I

I

I

I

I

I

I

I

I

5 Time

Fig.

7.6.4

Crane

Joint

Torques

TO (sec)

(Flexible

Members)

.vs.

Time

150 spacecrane from one position to another position in space (Fig. 7.6.5), a holonomic constraint at the y-direction on the tip of the third link is imposed as follows:

yt(t) =

to < t