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
