Computer modeling of spatial mechanisms for kinematic analysis and synthesis
by John Michael Lowell
A thesis submitted in partial fullfillment of the requirements for the degree of MASTER OF SCIENCE
in Mechanical Engineering
Montana State University
© Copyright by John Michael Lowell (1982)
Abstract:
A methodology is presented from which a general purpose computer code for determining the
kinematics (position, velocity, and acceleration) of spatial mechanisms may be generated. This
methodology permits the equations which describe the mechanism kinematics to be assembled and
solved by a program which only needs to read a concise data base describing the mechanism and type
of solution desired.
The inherent flexibility in this method permits modification of the mechanism configuration and
characteristics by altering the data base describing the mechanism. Changing relatively few parameters
in the data base makes possible the synthesis of motion. That is, for a given desired path or
configuration of the mechanism, the values of the mechanism parameters (coordinates) can be
determined without alterations in computer code. The method presented is well suited for
multi-positional kinematic analysis and synthesis of complex spatial mechanisms with multiple degrees
of freedom.
Two examples of kinematic analysis using this method are presented. The first is a three degree of
freedom closed loop mechanism. The second is the robot arm used on the space shuttle, which is a
six-degree-of freedom mechanism. Examples of position, velocity, and acceleration solutions are given
for both mechanisms. STATEMENT OF PERMISSION TO COPY
In presenting this thesis in partial fulfillment of the require­
ments for an advanced degree at Montana State University, I agree that
the Library shall make it freely available for inspection.
I further
agree that permission for extensive copying of this thesis for schol­
arly purposes may be granted by my major professor, or, in his absence,
by the Director of Libraries.
It is understood that any.copying or
publication of this thesis for financial gain shall not be allowed
without my written permission.
Signature
Date
T n y
Oufi,
/9 A l .
. COMPUTER MODELING OF SPATIAL MECHANISMS FOR
KINEMATIC ANALYSIS AND SYNTHESIS
by
JOHN MICHAEL LOWELL
A thesis submitted in partial fullfillment
of the requirements for the degree
of
MASTER OF SCIENCE
in
Mechanical Engineering
Approved:
Chairperson, Graduate Committee
tP.
Head, Major Department
Graduate Dean
MONTANA STATE UNIVERSITY
Bozeman, Montana
M a y , .1982
iii
ACKNOWLEDGEMENTS
The
Dr.
H.
author
W.
wishes
Townes
for
performing this work.
to t h a n k Dr. D. 0. B l a c k k e t t e r
their
advice
and
assistance
and
in
TABLE OF CONTENTS
Page
V I T A .......................
ACKNOWLEDGEMENTS
LIST OF TABLES
ii
....................
iii
.............................
v
LIST OF F I G U R E S ................
vi
N O M E N C L A T U R E .................. .......................
ABSTRACT
...........
CHAPTER I.
CHAPTER II.
CHAPTER III.
CHAPTER IV.
CHAPTER V.
vii
ix
INTRODUCTION ............................ *.
DEVELOPMENT OF V E C T O R EQUATIONS
I
. . . .
5
ASSEMBLY AND SOLUTION OF
KINEMATIC EQUATIONS
. . . ............
11
MODELING AND SYNTHESIS . . . . . . . .
CONCLUSIONS
LITERATURE CITED
........................
21
34
36
V
LIST OF TABLES
Table
Page
1
Tripod Position Solution .........................
25
2
Tripod V e l ocity solution
25
3
Tripod Acceleration Solution
4
LI Array Values For Shuttle Arm Model
5
IP Array Values For Shuttle Arm Example
6
Shuttle Arm Position Solution
7
Shuttle Arm Velocity Solution
8
Shuttle Arm Acceleration Solution
. . . . . . . . . . . .
..................
.25
. . . . .29
. . . .
29
31
..................
. ...........
32
33
vi
LIST OF FIGURES
ZiSJlTS
Page
1
Schematic of Vector I
2
Flow Chart of Equation
Assembly and Solution P r o c e d u r e . . . . . . . .
18
Flow Chart for Solution to
Geometry using Newton's Method
19
3
......................
................
.
7
4
Vector Model of T r i p o d ....................
23
5
Space Shuttle Mechanical A r m .................. ■
, 27
6
Vector Model of Shuttle A r m .................. 28
vii
NOMENCLATURE
Symbol
Description
{ AA}
column matrix containing
derivative of (PA)
(AA1 )
second time derivative of (PAt )
(Equation 2.12)
Ia1 ]
3 X 4 acceleration coefficient array for
vector I (Equation 2.11)
cI
4 X 1
array
c 'l
first time derivative of
c ni
second time derivative of
M 1]
[D] .
i
second
time
cosine squared constraint coefficient
(Equation 2.5)
*
Cj
c^
(Equation 2.9)
(Equation 2.13)
3X4
displacement coefficient
vector I (Equation 2.2)
array
n X m assembled displacement matrix for model
(Equation 3.1)
number of vectors in model
(IA)
array used to identify known and unknown
accelerations
(IP}
array used to identify known and unknown
geometry parameters
(IV)
array used to identify known and unknown
velocities
j
[LI]
m
for.
number of vector paths in the model
path information matrix, stores information
as to which vector is in which path and its
direction
number of kinematic equations, m is equal to
j times three plus i
viii
Symbol
np
Description
number of parameters defining model, np is
equal to i times four
{PA}
column matrix containing all known and unknown
geometry parameters
(PAl)
1 X 4
2.3)
(PD)
path displacement matrix, contains X, Y , and
Z displacement from beginning to end of each
vector path
(
(VA)
c o lumn matrix containing first time derivative
of (PA)
(VA1 )
time derivative of (PAj)
(V1)
3 X 4 velocity coefficient array
(Equation 2.7)
p a r a m e t e r a r r a y for v e c t o r I ( Equation
(Equation 2.8)
for vector
Subscripts
a
denotes parameter in direction angle
coordinate system
c
denotes parameter in. direction cosine
coordinate system
I
corresponding to vector I
I
ix
ABSTRACT
A m e t h o d o l o g y is p r e s e n t e d f r o m w h i c h a g e n e r a l
p u r p o s e c o m p u t e r c o d e for d e t e r m i n i n g the k i n e m a t i c s
(position, velocity, and acceleration) of spatial m e c h anisms
m a y be g e n e r a t e d . T h i s m e t h o d o l o g y p e r m i t s the e q u a t i o n s
wh i c h describe the m e c h a n i s m kinematics to be a ssembled and
solved by a program which only needs to read a concise data
base describing the m e c h a n i s m and type of solution desired.
Th e i n h e r e n t f l e x i b i l i t y in this m e t h o d p e r m i t s
modification of the m e c h a n i s m configuration and character­
istics by altering the data base describing the mechanism.
C h a n g i n g r e l a t i v e l y f e w p a r a m e t e r s in the d a t a b a s e m a k e s
p o s s i b l e the s y n t h e s i s of m o t i o n . T h a t is, f o r *a g i v e n
desired path or configuration of the mechanism, the values
of the m e c h a n i s m parameters (coordinates) can be determined
without alterations in computer code. The m e t h o d presented
is well suited for multi-positional kinematic analysis and
s y n t h e s i s of c o m p l e x s p a t i a l m e c h a n i s m s w i t h m u l t i p l e
degrees of freedom.
T w o e x a m p l e s of k i n e m a t i c a n a l y s i s u s i n g thi s m e t h o d
are presented. The first is a three degree of freedom closed
l o o p m e c h a n i s m . The s e c o n d is the robot a r m u s e d on the
space shuttle, w h i c h is a six-degree-of freedom mechanism.
Examples of position, velocity, and acceleration solutions
are given for both mechanisms.
CHAPTER I
s
INTRODUCTION
Rapid advances
in robotics and automated equipment for
industrial,
underwater,
demands
engineers
on
and
to
space
design
applications
and a n a l y z e
have
placed
increasingly
sophisticated three-dimensional mechanisms. While the use of
v e c t o r m e c h a n i c s to g e n e r a t e the e q u a t i o n s
n e c e s s a r y for
the kinematic analysis of three-dimensional mechanisms
well known procedure,
the d e r i v a t i o n and
is a
s o l u t i o n of the
resulting equations is a lengthy process which typically has
required a problem
This t h e s i s
specific computer code for solution.
d oes not e n l a r g e on the b o d y of k n o w l e d g e
related to m e c h a n i s m
which
a
theory,
but describes a methodology by
three-dimensional,
described with
rigid
a relatively concise
base is used as input to a computer
and
sol v e
the
velocity,
and
represent
problem
the
b ody
equations
which
acceleration
mechanism
of
model.
mechanism
data base.
can
be
Thi s data
code which can assemble
determine
each
The
the
parameter
computer
d e p e n d e n t a n d can be u s e d for e i t h e r
position
used
code
to
is not
analysis
or
s y n t h e s i s by c h a n g i n g the data base. S y n t h e s i s as used in
this thesis
mechanism
refers
to determining
the
configuration
required to obtain a given position,
of
the
velocity,
acceleration of some m e m b e r or point in the mechanism.
or
This
2
computer
code
developing
frees
the
programs
designer
which
are
from
only
the
useful
necessity
for
a
of
single
m e c h a n i s m and difficult to modify.
In
finite
application
element
modeled
and
equations
method
technique
the
for
the
[1]
governing
the
presented
in
the
way
equations
position,
resembles
the
mechanisms
are
assembled.
v e locity,
and
Matrix
acceleration
*
r e l a t i o n s of an a r b i t r a r y v e c t o r
mechanics.
are d e r i v e d u s i n g v e c t o r
The o v e r a l l m e c h a n i s m e q u a t i o n s are a s s e m b l e d
from these blocks of equations using information from a data
base which describes the m e c h a n i s m configuration and type of
solution
desired.
Kinematic problems are typically formulated as problems
in
vector
analysis.
However,
a
variety
of
solution
techniques can be applied to the resulting vector equations.
Some
of the most
graphical
methods.
c o m m o n l y used
solutions,
While
vector
these
solution
a l g ebra,
techniques
are
techniques
and
complex
methods
of
include
number
vector
a n a l y s i s , e a c h m e t h o d of s o l u t i o n has i n h e r e n t a d v a n t a g e s
and
disadvantages.
Historically,
graphical
techniques
[2,3]
have been the
p r e d o m i n a t i n g t e c h n i q u e for s o l u t i o n of the k i n e m a t i c s of
planar
mechanisms.
Since
solution
by
graphical
methods
3
requires
d r a w i n g .of the position,
polygons,
become
its
accuracy
is
velocity and acceleration
limited.
Also,
graphical
ver y t e d i o u s w h e n m u l t i p l e p o s i t i o n
methods
s o l u t i o n s are
desired. G r a p h i c a l m e t h o d s are d i f f i c u l t to use on spatial
mechanisms.
Solutions
complex numbers
advantages
by
algebraic
[2,3]
over
methods,
whether
or vector algebra
graphical
techniques.
[2,3],
based
have
Accuracy
on
several
is
limited
onl y by the a c c u r a c y of the p r o b l e m data and the n u m e r i c a l
evaluation of the solution. Once the problem has been put in
an a l g e b r a i c form, a k i n e m a t i c s o l u t i o n can be o b t a i n e d at
different positions
simultaneous
usually
e q u a tions.
requires
obtain a solution.
spatial
of the m e c h a n i s m
However,
tedious
an
by s o l v i n g
algebraic
mathematical
[3],
although
the
solution
manipulations
Algebraic methods are also
,mechanisms
sets of
applicable
algebra
is
to
to
more
complicated than for planar mechanisms.
Chace, at the U n i v e r s i t y of M i c h i g a n , w a s one of the
first to use vector notation to obtain closed form solutions
to
three-dimensional
approach
consists
tetrahedron
equation
of
in
vector
equations
[4],
defining
solutions
spherical
coordinates
to
The
a
Chace
vector
according
to
the u n k n o w n s in the e q u a tions. The r e s u l t i n g n i n e p o s s i b l e
4
solutions
are
reduced
to
explicit
closed-forms
and
can
be q u i c k l y e v a l u a t e d . C h a c e u s e d this t e c h n i q u e to d e v e l o p
the ADAMS (Automatic Dynamic Analysis of Mechanical Systems)
computer
code,
one
of
the
more
widely
use d
industrial
programs' for analysis of three-dimensional mechanisms.
Garcia de Jalon,
new
technique
coordinates
to
Serna,
using
solve
and Viadero
matrix
the
[5] have offered a
methods
kinematics
of
and
spatial
Lagrangian
mechanisms.
In this technique the m embers of a mechanism are represented
by three or more points which are the Lagrangian coordinates
for
the m e m b e r .
Geometric constraint equations
c o n s t a n t d i s t a n c e b e t w e e n points,
surfaces,
c o n s t a n t a r e a Of p l a n a r
and constant volume of solids are specified.
derivatives
analysis.
requiring
of
th e s e
However,
equations
this method
are
used
for
Time
kinematic
requires a previously known
position which limits it's use for multi-position analysis.
The p u r p o s e of this t h e s i s is to p r e s e n t a m e t h o d using
classical
vector
mechanics
from
which
a general
purpose
computer code may be developed to perform kinematic analysis
and synthesis of spatial mechanisms.
CHAPTER II
DEVELOPMENT OF VECTOR EQUATIONS
An
arbitrary
vector
in
coordinate
system,
parameters.
One p a r a m e t e r
the
other
can
be
The
three direction
one
three
be
defining
using
the
defined
define
an
in
Cartesian
terms
its
of
direction.
direction angles
parameters
given
position,
for
three-dimensional
four
is the l e n g t h of the v e c t o r and
either
calculated
relationships
be
parameters
defined
can
will
a
are
not
the
arbitrary
vector
or cosines.
independent
other
velocity,
Direction
two.
and
will
as
any
Equations
acceleration
be
developed
in
terms of these four parameters and their derivatives. One or
m o r e v e c t o r s can be used to d e f i n e a m e m b e r or l o c a t e po i n t s
of interest in a mechanism.
With
vectors
defined
in
this
o p t i o n of d e v e l o p i n g the k i n e m a t i c
either
two
path
the direction angles
coordinate
vector
equations
equations.
of
use
of
direction
Use
of
results
the
forms
cosine
in
direction
results
has
in t e r m s
the direction cosines.
direction
direction
while
define vector
Use
one
equations
systems yield different
eq u a t i o n s .
define
or
manner,
nonlinear
an g l e
the vector
to
algebraic
coordinates
angles
of
These
coordinates
in nonlinear
direction
of
the
to
transcendental
also
c o n v e r g e n c e p r o b l e m s w h e n an i t e r a t i v e m e t h o d
causes
is used to
6
solve
the
nonlinear
mechanism.
The
equations
time
derivatives
constraint
equations
coordinates
are
parallel
to one
matrices
for
for
the
of
expressed
the
in
identically
satisfied
of the
T h i s can
axes.
the v e l o c i t y
and
geometry
of
cosine
squared
direction
when
a
the
angle
vector
is
r e s u l t in s i n g u l a r
acceleration
relations
at
certain positions of a mechanism. Using direction cosines as
*
p a r a m e t e r s r e s u l t s in e q u a t i o n s w h i c h take less
time
to
construct,
show
better
convergence
computer
characteristics
w h e n u s i n g an i t e r a t i v e t e c h n i q u e to solve them, and their
derivatives do not yield a singular
set of equations when a
vector
is
axes.
cosine
variable
attempting
and
parallel
to convert
coordinates
variables,
one
of
parameters
accelerations
coordinate
to
from
systems.
known
i nto
more
the
can
and
and
result
in
unknown
out
of
commonly
However,
problems
angular
the
used
use
cosine
units
of
when
velocities
variable
in
other
Due to the a d v a n t a g e s of us i n g c o sine
all equations were developed using
the direction
cosines.
For
convenience
and.
ease
of
programming,
the
parameters defining the arbitrary vector I shown in Figure I
are stored in a parameter array {PA}, and subscripted in the
following manner:
7
PA(4I-3)
PA(4I-2)
PA(4I-1)
PA(4I)
F I G U R E
1
S c h e m a t i c
= L
= Cos(Ax )
= cos(Ay )
= Cos(Az )
of
V e c t o r
I
8
P A(1*4-3) = Length of
P A (I*4-2) = Cosine of
P A ( I M - I ) = Cosine of
P A (I *4)
= Cosine of
vector I
the angle between X axis and vector I
the angle between Y axis and vector I
the angle between Z axis and vector I
Two additional matrices, a velocity array
acceleration
array
derivatives
of
subscripted
the
in the
{A A } s t o r e
{PA}
same
the
matrix
man n e r .
first
{VA}, and an
and s e c o n d
respectively,
With vector
time
and
are
I d e f i n e d in
this manner the component relations can be written in matrix
form as:
[dj](PA1 ) =
x component
J y component
z component
(2 .1 )
I
where
fpA(n+l)
= X PA(n+2)
P A (n + 3 )
[dr]
(PA1 )
J
0
0
0
PA (n)
P A (n + 1 ) >
PA(n+2)
P A (n+3)
0
0
0
(2 . 2 )
(2.3)
I
n = 41 - 3
I = vector number
(2.4)
W h i l e e q u a t i o n (2.1) d e f i n e s the c o m p o n e n t r e l a t i o n ­
ships for a single vector,
as
the
three
direction
an additional
cosines
equation
is needed
are not i n d e p e n d e n t .
T his
equation states that the sum of the squares of the direction
9
cosines must equal one. In matrix form this is:
Lc1J (PAi) =i
(2.5)
where
L0
Lc iJ
PA (n + 1 )
PA (n+2)
PA(n+3)J
j
The v e l o c i t y r e l a t i o n s for v e c t o r I are g e n e r a t e d by
taking the time derivative of the component
in equation
(2.1) yielding the following
[ V 1 ] (VA1 )
relations given
equa t i o n s :
X component of velocity
= <Y component of v e l o c i t y >
Z component of velocity
(2 .6 )
where
P A (n+1)
P A (n+2)
P A (n + 3 )
[V 1 ]
V A (n)
< V A (n + 1 ) >
V A (n+2)
V A (n+3)
(VA1 ]
P A (n)
0
0
0
PA(n)
0
0
0
PA(n)
(2.7)
I
(2 . 8 )
I
The t i m e d e r i v a t i v e of the c o s i n e s q u a r e d c o n s t r a i n t
equation
(2.5) yields the additional
[C1ij
velocity
relationship:
(VA1 } = 0
(2.9)
where
(J'lj
=
L0
2*PA(n+l)
2*PA(n+2)
2*PA(n+3)J
j
10
The
by
acceleration
taking
the
relations
second
time
for
vector
derivative
of
I are generated
the
component
relations which y i e l d s :
X
<Y
Z
U
[v j ]{AA j } + [a j]{V A j }
component of acceleration
component of acceleration r
component of acceleration
I
(2710)
where
4
V A (n + 1 )
V A (n+2)
V A (n+3)
[a j ]
VA (n)
0
0
AA (n)
< AA(n+l) >
AA(n+2)
A A (n + 3 )
(AA1 )
The
second
equation
time
0
V A (n)
0
(2 .11)
I
(2 .12 )
I
derivative
of
(2.5) yields the additional
Lf"lj
0
0
VA (n)
the
cosine
acceleration
constraint
equation:
(VA1 ) = 0
(2.13)
where
#
[f"lj
= L0
Equations
relationships
relations
can
2*VA(n+l)
(2.1)
for
an
then
be
thru
2*VA(n+2)
(2.13)
arbitrary
use d
to
2*VA(n+3)j
define
vector.
define
the
kinematic
These
models
j
of
vector
large,
complicated three-dimensional m e c h anisms with a wide variety
of
linkages.
CHAPTER III
ASSEMBLY AND SOLUTION OF KINEMATIC EQUATIONS
Computer
assembly
of the k i n e m a t i c
vector model of a m e c h a n i s m
requires
equations
for a
information describing
the configuration of the model. The information necessary to
generate
the
equations
vectors
displacement
which
describe
vector
contained
in
from
to finish.
start
a given
paths
path
and
consists
the
More will
be
of
relative
said about
i
how
these paths
are
determined
in the following chapter
on
modeling.
A
set
of
matrices
will
be
used
to
store
the
information describing the vector model of the mechanism.
path information matrix,
[LI],
A
stores the information as to
w h i c h v e c t o r is c o n t a i n e d in w h i c h pat h and its direction.
The
[LI]
matrix,
which
is
number
of p a t h s
by
number
of
v e c t o r s in size, c o n t a i n s one r o w of i n f o r m a t i o n for each
pa t h in the m e c h a n i s m . E a c h row c o n t a i n s an i d e n t i f i e r for
each v e c t o r
vector
minus
in the m o d e l ,
this i d e n t i f i e r
is in the particular
one
if in the
path
negative
is a one if the
in the positive direction,
direction,
and
zero
if
the
v e c t o r is not in the path. The t e r m s in the [LI] m a t r i x are
used as m u l t i p l i e r s
on the
[dj],
[v%],
and
[aj]
matrices
when assembling the kinematic equations for the mechanism.
A second information matrix,
(PD], contains the X,
Y,
12
and Z displacements
This
{P D } m a t r i x
length.
paths
The
and
from beginning to end of a vector path.
is
values
three
of
non-zero
the
for
times
{PD}
ope n
the
array
paths.
number
of
are z e r o
The
paths
in
for c l o s e d
terms
PD(3*J-2) ,
P D (3* J - 1 ), and P D (3*J) r e p r e s e n t the d i f f e r e n c e s in the X,
Y , and Z c o o r d i n a t e s ,
respectively,
from
the b e g i n n i n g to
end of the J - th path.
Geometry
The generation of the geometry
equations for
the model
f r o m the i n f o r m a t i o n s t o r e d in the [LI] and [PD] a r r a y s is
done
in the
following
man n e r .
The
matrix
[D]
is used
to
store the a s s e m b l e d d i s p l a c e m e n t e q u a t i o n s for all v e c t o r
pa t h s plus the c o s i n e
s q u a r e d c o n s t r a i n t e q u a t i o n s and is
generated as follows:
(3.1)
[D]
[d i ]* L I (I ,I)
[d-J *LI(2,1)
[d 2 ] *LI (1,2)
[d2 ] * L I (2,2)
.........[d n p I * L I (I ,i)
......... [dnp ] * L I (2,i)
[d1 ]*LI(j,l)
Cl
0
0
[d 2 ]* L I (j ,2)
0
......... (dn p i*LI(j,i)
• • • • •
0
.........
0
•
c2
0
•
•
0
. 0
0 . . . ,
............. C i
w h e r e j is n u m b e r of p a t h s , i is the n u m b e r of vectors, and
13
np is the n u m b e r of p a r a m e t e r s in the model.
The geometry equations are then written as:
' D 1 {PA) '
This
is a set of m
nonlinear
i?j}
.
equations
,3.2,
in np p a r a m e t e r s ,
w h e r e m is equal to the n u m b e r of v e c t o r s plus three t i m e s
the n u m b e r of paths, and np is g r e a t e r than m. T h e r e f o r e np
minus
m
parameters
must
parameters is the number
be
known.
The
number
of
known
of degrees of freedom available in
the model.
The
{IP}
array
unknown parameters
one,
is
then PA(L)
known.
The
rearrange
is
used
to
identify
the
in the following manner:
is unknown;
if IP(L)
information
equation
equations by moving
(3.2)
in
to
the
a
if IP(L)
equals zero,
{IP}
m-by-m
known
array
set
equals
then PA(L)
is
of
and
u sed
to
nonlinear
known quantities to the right-hand side
of the equation.
On c e
equation
(3.2)
has
bee n
rearr a n g e d ,
used to solve for the geometry of the mechanism.
has
used
the
Newton-Raphson
method
it m a y
be
The author
or generalized
Newton's
m e t h o d [6] for solution. N e w t o n ' s m e t h o d s o l v e s the set of
nonlinear
linear
equations
equations
in
several
(3.2)
times
by
solving
a related
to converge
to
the
set
of
solution
I
14
of the nonlinear equations.
storage
of
the
convergence,
but
[D ]
Newton's method does not require
matrix
does
and
require
results
an
values of the unknown parameters.
initial
in
guess
As nonlinear
general have several possible solutions,
very
rapid
for
the
equations
in
the initial guesses
m u s t be r e l a t i v e l y c l o s e to the v a l u e s w h i c h r e s u l t in the
mechanism
being
in
the
desired
orientation,
otherwise
i
convergence to an alternate orientation may occur. For small
movements
of the mechanism,
an excellent guess
the previous position serves as
for the current position analysis.
Velocity
On c e
the
geometry
of
the
mechanism
is
known,
the
velocity relations for the mech a n i s m are generated by taking
the time derivative of equation
(3.2) which yields:
[ V ] {VA} = {0}
(3.3)
where
[V] =
(3.4)
[V1 ] *LI(1,1)
[V1 ]*LI(2,1)
[v2 ]*LI(1,2)
[v2 ] * L I (2,2)
[VjP L K j fI)
Iv2P L K j , 2)
S':
0
0
.........
.........
[vn p ]*LI(l,i)
[v n p ] * L I (2,i)
[vn p P L I ( j , i)
0
0
. 0
. c'i
15
The {IV} a r r a y is used to i d e n t i f y k n o w n and u n k n o w n
velocity parameters
used
to
identify
information
in
in the s a m e
the
the
manner
geometry
{ IV}
array
the
{IP}
parameters.
equation
a r r a y is
Using
(3.3)
the
can
be
r e a r r a n g e d so that all k n o w n q u a n t i t i e s m o v e to the right
h a n d side and are s u m m e d .
parameters
are
expressed
K n o w n t i m e d e r i v a t e s of a n g u l a r
in
terms
of the cosine
coordinate
s y s t e m in the f o l l o w i n g m anner.
Let P A a a n d V A a r e p r e s e n t
the
corresponding
direction
angle
and
it's
derivative
r e s p e c t i v e l y , the c o r r e s p o n d i n g t i m e d e r i v a t i v e in c o s i n e
coordinates,
V A c f is
then
calculated
with
the
following
coordinate transformation:
V A c = -VAa Sin(PAa )
Equation
velocities
(3.3)
is
a
linear
set
(3.5)
of
equations
in
and is s o l v e d u s i n g an exact t e c h n i q u e such as
Gauss elimination
[7].
Acceleration
Wifih the p o s i t i o n s
components
from
known,
the acceleration
the time derivates
[ A ] {VA}
I
and v e l o c i t i e s
,
of equation
+
of the
mechanism
relations can be obtained
(3.3) which yield:
[ V ] {AA} = {0}
(3.6).
16
where
(3.7)
I a 1 ] * L I ( 1 ,1)
Ia1 ] * L I (2 ,I )
[ a 2 ] * L I ( I ,2)
[a2 ] * L I (2,2)
[a 1] * L I ( j , l )
c"l
0
0
Since
equation
(a2 ] * L I (j ,2)
......................
. . . . .
[ anp ] * L I ( I , i)
[ aj - j p] * L I ( 2 , i )
.......................t a n p H L I ( j , i )
c "2
0
•
0
0
O •
•
•
.
.
.
.................................. C " i
[A] and {VA} are f u n c t i o n s of k n o w n quantities,
(3.6) can be
rewritten as:
[ V ] {AA}
= -{Y}
(3.8)
where
{Y} =
[ A ] {VA}.
(3.9)
The { I A } a r r a y is use d to i d e n t i f y k n o w n and u n k n o w n
acceleration parameters
in the same manner as the {IP} array
identifies geometry parameters.
{IA}
array
equation
(3.6)
can
Using the information in the
be
known quantities are moved to the
to
rearranged
{Y } . The known accelerations are expressed
second
time
derivative
of
that
all
right hand side and added
coordinate system in the following manner.
the
so
in the cosine
Let AA a represent
P A a , the
corresponding
17
acceleration
in c o s i n e
coordinates,
A A c , is g i v e n by the
following coordinate transformation:
AAc = -AAa Sin(PAa ) - V A a 2Cos(PAa )
Once rearranged equation
equations
in
(3.6) is a m - b y - m
accelerations
and
may
be
(3.10)
set of l i n e a r
solved
by
Gauss
elimination or some other exact technique.
A f t e r the k i n e m a t i c c a l c u l a t i o n s are c o m p l e t e d ,
the
calculated first and second time derivatives of the angular
parameters can be expressed in terms of the direction angle
coordinate
system
by
the
following
coordinate
transformations:
V A a = -VAc/ s i n ( P A a )
(3.11)
AA a = -AAc - V A a 2COt(PAa )
(3.12)
One should note that
[V] is the total derivative of the
[D] m a t r i x and is u s e d in N e w t o n ' s m e t h o d to c a l c u l a t e the
differential
changes
in
{PA}
for
each it e r a t i o n .
The
[V]
m a t r i x is a l s o the c o e f f i c i e n t m a t r i x in the v e l o c i t y a n d
acceleration
calculations.
Figures
2 and
3 show
flow
diagrams for a computational scheme which takes advantage of
the fact that only the [V] m a t r i x n e e d be s t o r e d and can be
rearranged
using the {IP},
{IV}, a n d { I A } a r r a y s to solve
18
START
INPUT
[LI], {IP}
INPUT KNOWN POSITIONS AND
GUESSES FOR UNKNOWN POSITIONS
INPUT KNOWN VELOCITIES
AND ACCELERATIONS
SOLVE FOR GEOMETRY}*
G ENERATE
TvTI
TRANSFORM KNOWN VELOCITIES
TO COSINE COORDINATES
REARRANGE
[VI USING
SOLVE FOR VELOCITIES
USING GAUSS ELIMINATION
TRANSFORM KNOWN ACCELERATIONS
TO COSINE COORDINATES
MULTIPLY OUT
REARRANGE
[V]
[A]{VA} |
USING
{IA}
SOLVE FOR ACCELERATIONS
USING GAUSS ELIMINATION
TRANSFORM TO DESIRED
COORDINATES FOR PU TPUI
FIGURE
2
Assembly
Flow
and
Chart
Solution
of
Equation
Procedure
19
GENERATE
[V] USING
APPROXIMATE {PA}
GENERATE RESIDUAL
BY
[D ] {PA}
REARRANGE
[V] USING
{IP}
SOLVE FOR DIFFERENTIAL CHANGE
IN {PA} WITH GAUSS ELIMINATION
UPDATE
(PA)I
/ T E S T FORs
CON V ERGANCE
RETURN
FIGURE
to
3
Geometry
Flow
Chart
using
for
Solution
Newton's
Method
-20
for u n k n o w n p o s i t i o n s ,
other
coefficient
velocities,
matrices
can
multiplication to a column array
and a c c e l e r a t i o n s . All
be
reduced
by
matrix
and m o v e d to the right-hand
side of the equations.
Th i s m e t h o d is q u i t e g e n e r a l and can be u s e d to solve
for
a variety
{IP},
{IV},
of
and
known
{IA}
and
unknown
arrays
need
parameters
not
be
since
identical.
the
This
*
p r o c e d u r e a l l o w s k n o w n g e o m e t r y p a r a m e t e r s to hav e e i t h e r
known or unknown derivatives.
CHAPTER IV
MODELING AND SYNTHESIS
Use of c o n c i s e input to d e f i n e a m e c h a n i s m m o d e l for
computer
generation
of
the
kinematic
equations
efficient way to perform kinematic analysis.
is
an
The method used
here to define a m e c h a n i s m model is a vector path method.
A
vector path is compo s e d of vectors which define the relative
p o s i t i o n of m e m b e r s ,
mechanism.
model
interest,
<$
It is important to realize that a vector need not
be on or define
the
or l o c a t e p o i n t s of i n t e r e s t in the
to
an actual physical link, but m a y be part of
constrain
other
vectors,
locate
points
of
or to be used for synthesis. A m e c h a n i s m model is
d e f i n e d by a c o m p l e t e set of v e c t o r paths. A l o o p is a pat h
w h i c h s t a r t s a n d e n d s at the s a m e point.
defining
arrays
paths
defined
and
in
l o op s
the
is
stored
previous
The i n f o r m a t i o n
in the
chapter.
[LI]
The
and
{PD}
vector
path
method allows relatively concise input for model definition,
yet
is
adaptable
even
complicated internal
Two
informal
defining
very
complex
mechanisms
with
structures.
"rules"
a complete
mechanism.
to
set
have
been
of v e c t o r
developed
paths
when
to
aid
in
modeling
a
These vector path rules are:
I. A v e c t o r p a t h m u s t be m a d e a r o u n d e a c h " i n t e r n a l "
lo o p
in
the
mechanism.
These
loops
h ave
zero
offset
and
22
usually define a solid body other than a rod.
2.
If
"grounded",
system,
one
or
i.e.
more
points
attached
to
a single ground point
in
the
the
mechanism
Newtonian
is selected,
are
coordinate
and vector paths
are g e n e r a t e d f r o m this p o i n t to e v e r y other g r o u n d point.
The d i f f e r e n c e s
two
points
in the three C a r t e s i a n c o o r d i n a t e s of the
yield
the
corresponding
terms
in
the
path
4
displacement array,
{PD}.
Rod s w i t h ball j o i n t s on e a c h end can be r e p r e s e n t e d
by
a
single
vec t o r ,
d e f i n e d by m o r e
while
more
complicated
t han one vector.
R ods w i t h
members
are
pinned joints
n e e d t w o a d d i t i o n a l v e c t o r s , one to r e p r e s e n t the pin, a n d
the oth e r to d e f i n e the a n g l e the p i n m a k e s w i t h the rod. A
plate
that
has
three
other
members
connected
to
it
is
r e p r e s e n t e d by the t h r e e v e c t o r s w h i c h f o r m a c l o s e d p a t h
around
the
members
connection
with
more
m o d e l e d as s e v e r a l
points.
than
three
plates,
Complex
three-dimensional
connection
each containing
points
can
be
three vectors
which form a loop thru three connection points.
Example I
A vector
mechanism,
model
of a tripod,
a three e l e m e n t
spatial
is s h o w n in F i g u r e 4. Each of the t h r e e v e c t o r s
s h o w n in the m o d e l r e p r e s e n t s a rod w i t h ball j o i n t s on e a c h
23
Z
FIGURE
4
Vector
Model
of
T r i p o d
24
end.
Th i s
f rom
the
o r i g i n to the p o i n t (10,0,0) a l o n g v e c t o r s one and two,
the
other
one
model
from
and
the
origin
three.
configuration
contains
The
two
vector
to the p o i n t
[LI]
paths,
(0,10,0)
information
one
along vectors
matrix
for
this
is then:
[LI]
The assembled
for this model
1 - 1 0
1 0 - 1
=
[V] coefficient matrix from
equation
(3.4)
is:
[V1 ]
-[v2 ]
[0]
[V1 ]
[0]
-[v3 ]
c 'l
0
0
0
c'2
0
0
0
w h i c h is a s y s t e m of nin e e q u a t i o n s in t w e l v e p a r a m e t e r s .
The
[V]
matrix
is the onl y c o e f f i c i e n t
must be generated and stored as rearranging
the information in the [IP],
the
necessary
velocities,
and
coefficient
and
matrix
which
[V] according to
{IV], and {IA] arrays generates
matrices
accelerations,
to
solve
respectively
for
(see
positions,
Figures
2
3).
Tables
1-3
give
an
example
of
the
solution
to
the
k i n e m a t i c s of the t r i p o d m e c h a n i s m w h e n all t h r e e l e n g t h s
25
TABLE I TRIPOD POSITION SOLUTION
===========================================================
Underlined values are known inputs
Vector
Number
I
2
3
Length
(inches)
10.00
8.00
12.00
Angle with Axis
X
Y
47.16
113.58
55.48
(Degrees)
Z
73.74
69.51
126.87
47.34
32.11
55.62
4
TABLE 2
TRIPOD VELOCITY SOLUTION
Underlined values are known inputs
Vector
Number
Length
Velocity
(in/sec)
I
2
3
I .00E+00
2.00E+00
- I .00E+00
Angular Velocities Relative
To Coordinate Axis
(rad/sec)
X
Y
Z
1.75E-01
-2.00E-01
-6.68E-02
-•2.73E-02
-2.00E-01
1.24E-02
3.37E-03
- I .67E-01
- I .75E-01
TABLE 3 TRIPOD ACCELERATION SOLUTION
=========================== =================================
Underlined values are known inputs
VECTOR
NUMBER
I
2
3
LENGTH
ACCELERATION
(in/sec2 )
5.00E-01
-5.00E-01
7.50 E-OI
ANGULAR ACCELERATION RELATIVE
TO COORDINATE AXIS
(rad/sec2 )
X
Y
Z
-9.86E-02
8.46E-02
1.26E-01
-4.06E-02
1.15E-01 -6.33E-02
-1.75E-02 -1.22E-02
4.52E-02
26
are
known.
Simply
altering
the
{IP},
{IV},
and
{IA }
information matrices would allow solution to the kinematics
problem with a different set of known parameters.
Example Z
The mechanical
Figure 5
[8],
arm used on the space shuttle shown in
is a s p a t i a l
mechanism
with
six d e g r e e s
of
f r e e d o m . The v e c t o r m o d e l of the a r m is s h o w n in F i g u r e 6,
with
the
[LI]
information
matrix
for
the m o d e l
given
in
Table 4. This model of the arm consists of seventeen vectors
a r r a n g e d in nine paths. A total of fo r t y four e q u a t i o n s in
sixty
eight
analysis.
either
parameters
This
analysis
depending
model
or
on which
of
must
the
synthesis
twenty
knowns in the {IP},
{IV},
be
arm
of
four
solved
can be
for
kinematic
used
to
robotic motions
parameters
are
perform
of the arm
specified
and {IA} arrays.
For k i n e m a t i c a n a l y s i s of the space s h u t t l e arm,
following
of
all
parameters
the
parameters),
vectors
would typically be known,
except
six
(a
total
the
the lengths
of
sixteen
two of the direction cosines of vector one and
the six degrees of freedom in the mechanism,
twenty
as
four known parameters.
The
for a total of
six degrees of freedom
in
the mechanism can be specified in the model in the following
manner:
no
FIGURE
5
Space
Shuttle
Mechanical
Ar m
F I G U R E
6
Vector
Model
of
S h u t t l e
A r m
29
TABLE 4
LI ARRAY VALUES FOR SHUTTLE ARM MODEL
=================================================
Path Number
kDCD'JCTiUl^UIMM
I
===========
I
0
0
0
0
0
0
0
I
2
3
4
5
0 0
I
0
0 I
0 0
0 0
0 0
0 0
0 0
I I
== = =
0
0
0
I
I
0
0
0
I
0
0
0
0
0
0
I
I
I
Vector Number
I 8 9 10 11
LI Array Values
0 I I
0 0 0
0 I
0 I
0 0
0 I
0 0 I 0
0 I
0 0 0 I
0 0 0 0 0 0
0 -I
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
I
0 0 0 0 0
======
6
12 13 14 15 16 17
0
0
0
0
0
0
0
I
0
0 0
0 0
0 0
0 0
0 I
0 I
0 -I
I
00
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
I
0
0
TABLE 5
IP ARRAY VALUES FOR SHUTTLE ARM EXAMPLE
============================================================
IP = I means unknown parameter value
IP = 0 means known parameter value
Vector Number
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
=============
Length
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
0
Direction Cosine Parameters
X
Y
Z
0
0
I
I
I
0
I
I
0
I
I
0
0
I
I
I
I
I
I
0
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
0
I
I
I
I
I
I
I
I
I
I
I
I
I
================================
=====
30
Shoulder Pitch
Shoulder Yaw
Elbow Pitch
Wrist Pitch
Wrist Yaw
Wrist Roll
-
Z angle
Y angle
Z angle
Z angle
X angle
Y angle
of
of
of
of
of
of
vector
vector
vector
vector
vector
vector
two
seven
three
four
five
thirteen
Th e s e six p a r a m e t e r s w o u l d t hen h ave k n o w n v e l o c i t i e s a n d
accelerations
parameters
associated
would
with
typically
t h e m . The
have
known
other
known
velocities
and
i
accelerations of zero.
The
{IP}
array
for
this
set
of
known
parameters
is
g i v e n in T a b l e 5. For this e x a m p l e the {IV} a n d {IA} a r r a y s
were
the
necessary.
same
as
the
{IP}
array
although
this
Tables 6-8 give an example of position,
was
not
velocity,
and acceleration solutions for the shuttle arm model.
Synthesis of mech a n i s m or servo coordinates of the arm
is
accomplished
by
specifying
the
length
and
two
of the
d i r e c t i o n c o s i n e s for v e c t o r six and a l l o w i n g any three of
the degrees of freedom
to be unknowns.
by appropriate changes in the
{IP},
This is accomplished
{IV},
with no need to alter any computer code.
and
{IA} matrices,
31
TABLE 6
SHUTTLE ARM POSITION SOLUTION
=========================================================
Known Parameters are Underlined
Vector
Number
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Length
(inches)
36.00
251.00
287.00
36.00
36.00
576.75
100.00
106.28
270.19
295.44
106.28
50.91
36.00
36.00
50.91
106.28
50.91
Angle with Axis
X
Y
90.00
8.07
11.40
40.09
40.00
177.38
87.00
92.82
159.91
160.07
107.96
121.65
91.37
50.07
174.17
99.68
95.04
90.00
92.97
92.95
92.30
85.87
87.40
3.00
159.98
108.75
106.84
157.82
139.06
5.00
91.93
87.01
18.01
94.29
(Degrees)
Z
180.00
82.50
79.00
130.00
129.70
90.37
90.00
70.20
96.97
100.34
77.43
66.90
85.19
40.00
95.00
105.04
6.62
32
TABLE 7
SHUTTLE ARM VELOCITY SOLUTION
Known Velocities are Underlined
Vector
Number
Length
Velocity
(in/sec)
I
0.00E+00
2
0.Q0E+00
3
0.00E+00
4
0. 00E+00
5
0.00E+00
6
-6.75E+00
7
0.OOE+OO
8
0.00E+00
0.00E+0n
9
10
o .ooE+on
o.ooE+nn
11
o .ooE+no
12
13
0.00E+00
14
0. 00E+00
15
0.00E+00
o .ooE+nn
16
17
0 .00E+00
==============
Angular Velocities Relative
To Coordinate Axis
♦
(rad/sec)
X
Y
Z
0.00E+00
0 .00E+00
0.0E+00
- I .13E-01
I .OOE-Ol
-4.89E-02
8.34E-02
- I .OOE-Ol
-5.01E-02
4.67E-02
5. 00E-02
-4.00E-02
I .OOE-Ol
-8.95E-01
-3.06E-02
I .02E-01
I .03E-01
3.32E-03
— 5.00E-O2 -1.43E-07
5.00E-02
-4.70E-02
7.19E-03
1.43E-07
-1.15E-02
4.89E-02
-9.28E-02
-9.51E-02
5.01E-O2
9.39E-02
-6.01E-02
4.24E-02
-1.33E-02
- I .42E+00
9.73E-01
-4.19E-01
-I.OOE-Ol
1.64E+00
5.70E-01
-5.21E-02
-3.01E-02
5.00E-02
6. 85E-02
4.96E-02
-5.00E-02
6.14E-02
2.50E-02
- 1 .13E-02
-7.40E-02
6.12E-01
3.41E-01
============ ======================
TABLE 8 SHUTTLE ARM ACCELERATION SOLUTION
=============================
===========
Known Acceleration Inputs are Underlined
Vector
Number
Length
Acceleration
(in/sec2 )
I
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
0 .00E+00
0.OOE+OO
0. 00E+00
0 .00E+00
0. 00E+00
2.50E+01
0 .OOE+OO
0.00E+00
0 -00E+00
0.00E+00
0 . OOE+OO
0. OOE+OO
0.OOE+OO
0.OOE+OO
0 .OOE+OO
0.OOE+OO
0.OOE+OO
Angular Accelerations Relative
To Coordinate Axis
(rad/sec2 ) .
X
Y
Z
0. OOE+OO
0.OOE+OO
0.OOE+OO
I.OOE-Ol
- I . IlE-Ol
-5.07E-02
-I -OOE-Ol
I.IlE-Ol
-4.87E-02
-5.00E-02
- 4 . 93E-02
-3.35E-02
5. 0 0E— 02
1.70E+01
9.22E-01
- I . OOE+OO
-1.01E+00
-3.93E-02
-5.00E-02
5.00E-02
-1.32E-07
- 4 . 7 OE-02
4.72E- 02
1.33E-07
- 4 . 4 6 E- 02
5.07E-02
-9.03E- 02
-1.04E-01
4.87E-02
9.37E-02
-3.76E-02
3.48E-02
I .38E-02
-7.33E+00
- 1 .73E+01
-3.03E+01
I -OOE-Ol
1 .03E+01
3.87E+01
4.48E-02
-5.20E-02
-5.00E-02
-1.49E-02
6.04E-O2
5.00E-02
3.48E-02
3.94E-02
1.06E-02
-2.66E-03
-1.21E+01
-5.57E+00
CHAPTER V
CONCLUSIONS
The m e t h o d of m e c h a n i s m m o d e l i n g p r e s e n t e d a l l o w s a
designer
to
kinematic
write
modeling
mechanisms.
kinematic
a
general
of
purpose
multi-degree
'program
of
freedom
for
the
spatial
O n c e w r i t t e n t his c o d e can be u s e d to p e r f o r m
analysis
or
synthesis
of
a
wide
variety
of
m e c h a n i s m s w i t h no a l t e r a t i o n s of the c o m p u t e r code. O n l y
the da t a
base
which
describes
the m e c h a n i s m
and
type
of
solution desired are problem specific.
The m e t h o d p r e s e n t e d
their
time
derivatives
uses
to
the
define
direction
the
cosines
kinematics
and
of
a
mechanism. A pre- and post-processor program can be written
to
allow
inputs
and
outputs
to
be
in
any
convenient
coordinate system or units.
The method presented has several advantages. It is well
suited
degrees
for
of
multi-positional
freedom,
can
analysis,
model
very
permits
multiple
complex
spatial
mechanisms, permits a n a l y s i s of d i f f e r e n t c o n f i g u r a t i o n s ,
and can perform limited synthesis of multi-degree of freedom
spatial mechanisms.
LITERATURE CITED
LITERATURE CITED
Zienkiewicz, 0. C . , The Finite Element Method in
Engineering S c i e n c e . London: McGraw-Hill
Publishing C o m p a n y , 1971.
S h i g l e y f J. E . f and J. J. D i c k e r , Theory of Machines
and M e c h a n i s m s , New Y o r k : McGraw-Hill Publishing
Company, 1980.
Paul, B., Kinematics and Dynamics of Planar M a c h i n e r y .
Englewood Cliffs: Prentice-Hall, Inc., 1979.
C h a c e , M. A., "Vector Analysis of Linkages", J. Eng.
Ind., ASME Trans., ser. B, v o l . 85, no. 3, pp.
289-297, 1963.
4
Garcia de Jalon, J., Serna, M. A., Viadero, F., and
F l a q u e r , J., "A Simple Numerical Method for the
Kinematic Analysis of Spatial Mechanisms",
Journal of Mechanical Design, ASME Trans., vol 104,
no. I, pp 78-82, 1982.
O r t e g a , J. M., and Rheinbolt, W. C., Iterative Solution
of Nonlinear Equations in Several V a r i a b l e s .
New Y o r k : Academic Press, Inc., 1970
Hornbeck, R. W., Numerical M e t h o d s . New York: Quantum
Publishers, I n c . , 1975
T o w n e s , H. W., B l a c k k e t t e r , D. O., and L o w e l l , J . M.,
"Robot Kinematics Using a Microcomputer", presented
at Society for Computer Simulation Conference on
Modeling and Simulation on Microcomputers, Jan.
28-30, 1982, San Diego, CA. Published in the
proceedings.
MONTANA STATE UNIVERSITY LIBRARIES
stks N378.L9515@Theses
RL
Computer modeling o f spatial mechanisms
3 1762 00118770 5
UR
N378
L9515
cop.2
Lowell, J. M.
Computer modeling of
spatial mechanisms for
kinematic analysis...
DATE
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