Download Constructing the dual graph from the original graph

The Duality between Planar
Kinematics and Statics
Dr. Shai, Tel-Aviv, Israel
Prof. Pennock, Purdue University
The Outline for the tutorial:
A Study of the Duality between Kinematics and Statics
• Two new concepts for statics that are derived from kinematics:
Equimomental line and face force.
• Transforming theorems and rules between kinematics and statics.
• Characterizing and finding dead center positions of mechanisms
and the stability of determinate trusses.
• Correlation between Instant Centers and Equimomental Lines.
• Graph theory duality principal and the dual of linkages – trusses.
• Detailed example of the face force and the procedure for deriving
the face force.
• Transforming Stewart platforms into serial robots and vice versa.
• Checking singularity through the duality transformation.
• Applying the duality transformation for systematic conceptual
design.
• Discussion and suggestion for future research in this area
Kinematics
Statics

P


The absolute instant center is the point
in a link where the linear velocity is zero.
The absolute equimomental line is the
line where the moment produced by a
force is zero.

V




M
r

P
The linear velocities of points located at
a distance r from the absolute instant
center are the same.

r
The moments produced by a force
located a distance r from the absolute
equimomental line are the same.
The Linear Velocities Field
The Moment Field
Y
Y
F
E
D
Z
Z
C

P
F
E
D


C
B
A
X
B
A
X
Y
Y
F
F
E
E
D
D
X
X
C
B
A
C
B
A
The field of the linear velocities
produced by the angular
velocity. The linear velocity is
zero at the absolute instant
center.
The field of the moments produced
by the force. The moment is zero
along the absolute equimomental
line.
Relative instant center.

2

1
I10
r1
I12
r2
Relative equimomental line.
F1
I20
 
V1  V2
The relative instant center of two
links is the point in both links
where the angular velocities
produce the same linear velocity.
r1
r2
F2


M F1 m 12  M F2 m 12
The relative equimomental line of
two forces is the line where the
forces produce the same
moment.
Arnold-Kennedy Theorem

2
I20
I23
Dual Kennedy Theorem
I30

3
I12

F1
m10
m30

F3
m13
m12

F2
m23
m20

1
I10
I13
The Arnold – Kennedy Theorem.
For any three links, their three
relative instant centers must lie
on the same line.
The Dual Kennedy Theorem. For any three
forces, their three relative equimomental lines
must intersect at the same point.
Correlation between Instant Centers and Equimomental Lines.
I
I12
I12
II

1
I10
I12

2
I20
I12
III
I12
II
mAB
I
mA0
mAB
mB0
mAB
mAB

FAO

FBO
III
mAB
mAB
I12
The idea behind the transformation of Kinematic systems
(Linkages) into Static systems (determinate trusses)
• Each engineering system can be represented into
mathematical model based on graph theory.
• There are mathematical relations between the graph
representations such as the graph theory duality.
• For example, the representations of linkages and trusses
were found to be dual. Thus, linkages and determinate
trusses are dual systems.
The following slides will show the process of constructing
dual engineering systems on the basis of the graph
theory duality principle.
Constructing the dual graph from the original graph
Original graph
1*
3
A
1
2
II
5
4
I
O
3*
C
2*
III
7
B
6
D
II
I
O
5*
III
7*
IV
O
4*
Dual graph
6*
IV
8
8*
Reference face.
Cutset is a set of edges so that
A circuit
is 1*,
a closed
Edges
3*,
4* andpath.
2*
5* form
7*
Each circuit in the original graph corresponds to a cutset in the dual graph, and
ifEdges
removed
the
graph,
the
Face
IV
I corresponds
II
III
corresponds
corresponds
to
to
to
the
the
vertex
vertex
vertex
II.
III.
IV.
a
circle
cutset
inin
dual
dual
graph.
graph.
Aand
face
in
the
graph
isgraph
aadjacent
circuit
without
inner
edges.
Each
Reference
infaces
the
face
Ograph
corresponds
corresponds
totoI.graph
the
to
athen
vertex
Each
3,
1, 4
2from
cutset
5
7face
in
constitute
Two
the
original
aadjacent
are
corresponds
ifthe
they
have
a
at
circuit
in their corresponding
vice-versa.
If two
faces
areoriginal
in the
original
graph
becomes
disconnected.
in original
the
dual
reference
graph.
vertex
O.
cutset
circuit
the
in dual
the
graph,
least
and
graph.
vice
versa.
in
common.
vertices
are
adjacent
inone
theedge
dual
graph.
Vertices
II
IOand
and
IV
are
are
adjacent.
adjacent.
Every
adjacent
faces
correspond to two
adjacent
vertices
in
the dual graph.
Facestwo
III
Oand
III
and
andIV
IIO
III
IIV
are
are
are
are
adjacent.
adjacent.
adjacent.
adjacent.
III
and
IIO
IIII
IV
are
are
adjacent.
adjacent.
The
edge
to adjacent.
these two faces corresponds
theIVedge
that connects the
Faces
O common
and IV are
Vertices O to
and
are adjacent.
vertices in the dual graph.
Constructing the Dual of a Linkage
Augmentingthe
thedual
geometry
Constructing
graph.to
The meaning of a directed edge in the
Constructing
its
topology.
the graph.
cos  90 O
dual
graph: system.

Kinematic
Constructing the
B


3
 cos3  90

is sinthe
3 flow
90  3(force)
corresponding graph
 sin   90 
3

 3
e=<t,h>
acting upon the
 A / B (CCW) B
2*

A

A


head vertex
(joint)
by
the
edge
(rod).
(CCW)
The topology
arrow and the
arrow are in

(CW)
 force
3*

VA / B
 direction -> compression.
the same
VB / O
2
4
The force in rod
acts upon the ground
2
VA / 4**
O
4
Inverse directions - tension 4
I
Two
in this orientation.
direction
of the
inacts
rodthe
4**.
Thechoices?
force
in external
rod
3** force
acts
upon
ground
The
force
upon


(CW) 2 2


B/ O
in thisThe
orientation.
Vertex
I corresponds
to joint
direction
ofjoint
theIforce
in rodI. 3**.
3
4
4
2
O2
The type is tension.
OThe type is compression.
1

VO1 / O4  0
3
4*
3
O4
1
(CW)
A / O2
4 (CW)
O
O
O
2
4
Adding
 1the geometry.
 O1 / O4  0
O source
Vertex
O
to
Faces
joint
. and
Othe
adjacent.
EdgeLink
2The
is 2the
is
potential
the
Edge
driving
corresponds
link.
that
corresponds
to
link
3.the
to
driving
link
2. the twofaces
Vertex
A
corresponds
The
potential
to
joint
source,
4O
A.
edge
2,
relative
The
relative
velocity
velocities
linear
of
oflink
velocity
links
2to
corresponds
3
corresponds
and
4
toOthe
to
the
Edge
3
is
common
to is
the
two
adjacent
243corresponds
2Icorrespond
For
consistency,
direction
ofbetween
the
edge
Vertices
B
and
O
correspond
joints
B
and
, are
respectively.
We
can
contract
the
edges
with
4
The kinematic
analysis
yields
the
Edges
4
and
1
correspond
to
link
4
and
the
fixed
link
1,
3**
adjacent
Iequal
faces
and
O
Iand
and
thus
Othe
in dual
the
original
edge
4*
graph.
is between
potential
difference
differences
potential
of
difference.
of
edge
edges
2.
3Reference
and
4.
4**to the
faces
Ito
O
thus
the
dual
edge
3* is
face
O
corresponds
in
the
dual
graph
is
defined
by
rotating
potential
difference
zero.
The
angular
velocity
in
CW
magnitudes
and
directions
of
the
angular
Potential
differences
in
edges
3
and
4
of the
edge
original
2 Therefore,
graph theintwo
respectively.
adjacent
vertices
Ivertices
and O. Itoand
the
dual
graph
it corresponds
theO.
angular velocity
is CCWthe
which
between
the two
adjacent
tothe
reference
vertex
O.
edge
inFace
original
graph
in
CCW
The
corresponds
to
compressing
force.
and
relative
linear
velocities.
correspond
to
flows
in
edges
3*
and
4*.
corresponds
to
the
flow
the
in
edge
dual
graph.
2*.
I
corresponds
to
vertex
I.
Pcorresponds
I
flow
source and it is between the costwo
to a tension.
Building
the
direction.
 adjacent
90




4**
vertices I and
O.
sin   90 
corresponding
truss.

F
3*
O
4
3
3
4
(tension)
3**
O3
The corresponding truss.
(compression)

P
The dual graph.

F4* (compression)
Since
velocity
associated
with ain statics:
We have systematically
developed
a new
variable
We linear
obtain
theisdual
systems.
dualits dual variable is
jointB in the linkage,
Face
At
first, what
isthe
thisface
absolute
velocity?
What Joint
is a
counterpart
to
absolute
linear
velocity of the joint
associated
with
in
the
truss
O
dual
3
The
relative 1.The
linear
velocity of
the link
input
3 link
4
corresponds
corresponds
to the
toofthe
force
external
in the rod
force.
3**.
4**.
absolute
linear
velocity
has
a property
potential.
Velocity
Force
A
+
2
O2
+
3
On the other
handdual
we know that velocity
Why?
4
3**
Absolute
linear
velocity
face force.
Velocitycorresponds
of a joint to force.corresponds
FacetoForce
Because, We can give any absolute velocities to the links, and
4**
they will satisfy the rule of velocities (vectors KVL).
O
I


FB 
P
1
4
1
?

FA

VA / B

VA / O2

VA / O2

F3**
?

F4**
VB / O4

VB / O4
O4
2
1 is equal to to
thecompression.
Same
direction
Force
in rod corresponds
3
difference of
subtraction
ofA face
forces
P
F by
byFFAO. .
face
forces F
and F
.
manner,
locate
theB forcesA to
in tension.
the other rods.
Opposite
direction
corresponds
 In the same
M
 M F m
An arbitrary point on the
 F m

equimomental
FP  r FPmPO  FO  r FO mPO
PB:

 line mPA
P
PO
O
PO
M
FO  0 
 M BA
m
P m
F m
Face force FABP


P


P  r PmPA  FA  r FA mPA
acts
the B.
F in face
F
face A.
P.
 
F F P
The moment produced by the

P rod isFthe difference of the
Each force
in the

forces
P and
FAB The
on the
upon
the line
Absolute
circuit
equimomental
corresponds
circuit
The equimomental
lines
that
Set
of the(Right
edges.
rightarbitrary
andF the directions
left face forces
and
left
m
In the same manner
we
can
find
the
reaction
R.
Thus
we
obtain
the
face
force
F
.
equimomental
line
m
.
m
to
the
has
vertex.
to
be
corresponds
determined
.
to
will
locate
m
.
m
B
PB
A
PA
BO
BO
defined according to the M
direction of the arrow
the vertex.
m
in the edge).
r
PA
PB
PA
PB
A
B
A
2
3
PB
PO
BO

P
rP

FB

P
P

F4
4

F4
      
F32FA F
P
F1 
A
FOFF
AB FA
compression
2
F2

A 
F3 FA
?
?

FB
FB  m PB
3
B
rP M
P
mPA
 rFA  m PA

F1A
1

FB

F5 compression
R
mRO
mPO
R
P
mAO
mPB
mBR
O
mPA
mBR
5

R
 
F1  FA
FO  0
O


FA  F1

mAB
R m
AO
mRO

FB

R

F5
B
mAB
A
Finding and characterization of the dead
center positions of the mechanism.
O
2
7
1
4
3
2
O
4
5
A
1
5
3
7
O
C
6
6
2,3
5,7
C
6
A
In this case, the faces B and O (i.e., the reference vertex) of the truss have the
same face force which indicates that:
(i) links 2 and 3 are collinear, and (ii) links 5 and 7 are collinear.
These two conditions ensure that the mechanism is in a dead center position.
Another examples to find dead positions of the
mechanism by Face Force.
Given mechanism topology
Duality relation between stability and mobility
Rigid
????
12
12
2
4
2
8
8
4
3
7
5
1
3
9
7
1
9
5
10
6
6
11
10
11
Due to links 1 and 9 being
located on the same line
2’
Definitely
locked !!!!!
2’
12’
4’
12’
8’
3’
1’
4’
7’
5’
6’
9’
10’
11’
1’
8’
3’
9’
7’
5’
6’
10’
11’
R’
R’
By means of the duality transformation, checking the
stabiliy of trusses can be replaced by checking the
mobility of the dual linkage.