Download The two reported types of graph theory duality.

General view on the duality between
statics and kinematics
M.Sc student: Portnoy Svetlana
Advisor: Dr. Offer Shai
The outline of the talk
1. The two reported types of graph theory duality.
2. Duality between trusses and linkages and the
theoretical results derived from it.
3. The relation between Maxwell reciprocal
diagram and graph theory duality.
4. Introducing polyhedral interpretation for the
theoretical results obtained from this duality.
5. The second type of duality – duality between
Stewart Platforms and serial robots and its
projective geometry interpretation.
6. Example of a practical application
based on the theoretical results
obtained in this research.
7. Further
research.
The graph theory
duality
The two reported types of graph
theory duality.
Truss and Mechanism
Stewart Platform and Serial robot
11’
11
10
3’
4
4’
3
12
10’
13
9
2
7’
5
12’
9’
7
2’
13’
5’
1
6’
1’
8’
6
8
We obtain the dual systems.
What kind of a variable corresponds to the absolute linear velocity?
B velocity
The
relativelinear
linearvelocity
of the
driving
is
equal
tothe
thein
corresponding
each
link to
islink
equivalent
to
force
acting
in the
The
absolute
corresponds
the
new
variable
statics
– face
force.
external
force.2 rod.
corresponding
2’
A
Duality between trusses and
linkages and theV theoretical
FF derived from it.
results

3
1
1’
FF B
O1

VA / 0B1  P
FFF
2' A


VB / 0033  FFF
3' B
1
O3
B / O3
A
VA / B

F2 '

VA / O1

P

F3'
3’
The Relation between Static Systems, Mobile Systems and
Reciprocity

F1
O
O
FFA

F1
B
A
FF B
D
C
Removing link 1 and
turning its internal force
(the blue arrow), into an
external force acting upon a
linkage in a locked position
D
6
9
5
V11’
’
3
8
C
4
B
The isostatic
The isostatic
framework
framework
has a self-stress.
F1
A
The relation between
Maxwell reciprocal diagram
and graph theory duality.
3’
2’
D
B
The original and the
dual graphs
Applying rotation to the
reciprocal diagram.

VC
D
D
O
9’
5’
1’
The unstable truss dual to
the linkage in a locked
position.

VB
7’
C
6’
B

VD
The truss underlying the
reciprocal diagram has
infinitesimal motion.
O
C

VA
4’
The reciprocal diagram
B

VD
A
8’
A
C

VB
B
A 7
FFC
FFD
Maxwell’s theorem 1864: The
isostatic framework that satisfies
E=2*V-3 has a self stress iff it has
a reciprocal diagram.
2
1

VC
D
C
A

VA
Kinematics in 2D
Statics in 2D
For every link there exists a point where its linear
velocity is equal to zero.

j

VI jo  0
I jo
For every force there exists a line where the
moment it exerts is equal to zero.

Fj
m jo

M Fj  m jo  0
Theoretical results obtained
For every two links there exists a point where their For every two forces there exists a line, such that
linear velocities are equal.
.
from
the duality
between
the moments
exerted by the two
forces on each

point on the line are equal.
Fj
trusses
and
linkages.


k
j

Vj
I jk

Fk
I jk
m jk

 Vk


M Fj  m jk  M Fk  m jk
I jk
The Kennedy Theorem. For any three links, the
corresponding three relative instant centers
must lie on the same line.

l

j

k
I jk
I jl
The Dual Kennedy Theorem. For any three forces,
the corresponding three relative equimomental lines
must intersect at the same point.

Fj
m jl
m jk
I kl

Fk

Fl
mkl
Maxwell’s theorem (1864): Isostatic framework that satisfies E=2*V-3
has a self stress IFF it is a projection of a polyhedron.
The sufficient part was proved only in 1982 by Walter Whitely.
Introducing a polyhedral
interpretation for the theoretical
results obtained from this duality.
e
A
c
a
2
D
1
O
e
c
A
6
B
7
D
a
1
3
d
8
7
6
5
C
5
b
8
C
9
d
2
B
3
9
b
4
4
f
The isostatic
framework
that has a self-stress.
The isostatic
framework
O
f
The corresponding polyhedron
mBO Geometric interpretation for the new variable –
mBD
The isostatic
framework
the equimomental line.
2
1
O
A
7
3
8 B
D
mCO
6
D
9
C
mDO
C
B
Constructing the dual Kennedy circle
for finding all the equimomental lines.
38
59
CD
A
A
3
8
m CD
mDO  3  8  5  9
5
9
4
5
The
corresponding
polyhedron
O
The dual
Kennedy circle
D
BO
1
2
7
B
6
C
8
3
9
5
O
Triangle
the
dual
Kennedy
==
AninDual
intersection
Two
triangles
that
include
the
EQML
in
the
circle
==D
m
corresponds
totwo
the
intersection
line
A
reference
face
Abetween
reference
the
AnDO
face
equimomental
EQML
inin
the
between
m
framework
==O==
An
line
edge
nonadjacent
==
between
Acircle
vertex
twovertex
adjacent
vertices
faces
B
C and
and
O
D in
in
BD
CO
CD
BO
point
ofunknown
the
corresponding
three
EQML.
Points
that
this
line
through
them.
between
plane
D
and
the
projection
plane
-the
O.between
dual
Kennedy
circle==A
projection
plane
inline
faces==
==
O
Kennedy
the
inAn
dual
the
A
dual
Kennedy
circle
known
Kennedy
==
edge
edge
circle
circle==
Apasses
plane
in
circle
in==
the
the
An
in
An
dual
==
the
dual
intersection
intersection
An
Kennedy
polyhedron.
Kennedy
intersection
line
between
line
polyhedron.
circle between
between
planes
C and
B
planes
Dtwo
O
inCthe
corresponding
andpolyhedron.
O in the polyhedron.
vertices== An
vertices
intersection
==
An intersection
line between
line between
two adjacent
two
planes
in the polyhedron.
nonadjacent
planes.
4
CO
The projection plane.
DO
BD
Kinematics in 2D
For every link there exists
a point where its linear
velocity is equal to zero.

j
Geometry in 3D
For every force there exists
a line where the moment it
exerts is equal to zero.

Fj

VI jo  0
I jo
Statics in 2D
For every plane there exists
a line where it intersects the
projection plane.
m jo
J

M Fj  m jo  0
O
JO
For every two forces there
exists a line, such that the
moments exerted by the two
forces on each point
 on the
line are equal. F
For every two links there
exists a point where their
linear velocities are equal.

j

k
j
I jk

Vj
I jk

 Vk
I jl
I jk
K
JK


M Fj  m jk  M Fk  m jk
I jk
The Kennedy Theorem. For any
three links, the corresponding
three relative instant centers must
lie on the same line.

j
J
m jk

Fk

k
For every two planes there exists
a line where they intersect.
The Dual Kennedy Theorem. For
any three forces, the corresponding
three relative equimomental lines
must intersect at the same point.

Fj

l
I kl
m jk

Fk
m jl

Fl
mkl
Every three planes must
intersect at a point.
Consider a Stewart platform system.
The projective geometry interpretation
(4D) of the second type of graph theory
duality yielding the duality between

Stewart platforms and serial robots
P
3
2
I
4
1
6
O
5
Stewart platform
I
1
2
3
4
5
6
O
The original graph
Every platform element corresponds to
a vertex and every leg to an edge.

P
3
2
I
4
1
6
O
5
Stewart platform

P


A
I
1
1’ B 2’ C 3’ D 4’ E 5’ F 6’
2
3
4
5
6
O
The original graph
The dual graph

P
3
2
I
4
1
6
O
5
Stewart platform

P


A
I
1
1’ B 2’ C 3’ D 4’ E 5’
2
A
5
6
O
The original graph
Every joint corresponds to
a link and an edge to a
joint.
F
5’
6’
4
F 6’
The dual graph


3

P
3
2
I
4
E
1’
4’
B
3’
2’
D
1
6
O
5
C
Serial robot
Stewart platform

P


A
I
1
1’ B 2’ C 3’ D 4’ E 5’
2
3
F 6’

P
F
A
6
The original graph
Every joint corresponds to
a link and edge to a joint.
5’
6’
5
O
The dual graph


4
3
2
I
4
E
1’
4’
B
3’
2’
D
1
6
O
5
C
Serial robot
Stewart platform

P
The duality relation between Stewart platforms and serial robots.

4 / 5
5’
4’
The force in the leg of Stewart platform
is identical to the relative angular
velocity of the corresponding joint in the
dual serial robot.


F4  4 / 5

P
4

F4
The relation between kinematics and statics
through projective geometry.
The
p corresponds
to
the
first
Adding
a dimension
and
Itto
defining
follows
the magnitude "Force"
of
thepoint
magnitude
angular
and
the
unit
direction
The
Motion
Introducing
point
angular
of
in
p
the
a corresponds
point
the
velocity
projective
motion
‘p’ on
in
aplane
the
link
the
the
instant
- is
M(p)
second
aathat
line
is that
defined
Introducing
thevelocity
“force”
Adding
applied
a dimension
at
point
and‘p’
defining
F(p)
is
defined
a of the
by:
The
"force"
in
the
projective
plane
is
a
line
that
projective
point.
planetwo
onto
Z=1.
is equivalent
thethe
magnitude
force
of
the
correspond
force
vector.
to Z=1.
the second projective
projective
center
by:
joints
the
between
correspond
instant
point.
plane.
center,
projective
the
angular
first
points.
velocity to
and
in
the
projective
plane.
the
projective
force
and
plane
the
on
point.
joints
between
two
projective
points, one is at
point that is located at infinity.
projective
point on the
point.
link.
the infinity.
Z
M  c  p
Z
c  p
Z=1
F  q  p
Z=1
p
c
p
Y
q  p
Y


p
p

f  q
c
X
Kinematics
X
Statics
q

Testing whether a line drawing is a correct projection of a polyhedron.
Which of the drawings is a projection of a polyhedron?
Finding all the EQML  there exists a self stress in
7  11
the configuration  (Maxwell+Whiteley) it is a
projection of a polyhedron
mDO  7  11  8  10  1  3
8  10
8  10
O
10
A
2
1
9
1
8
D
3
10
11
B
9
10
E
3
A
2
Example of a practical application
based on the theoretical
results
7
obtained in this research.
7
E
O
O
8
A
1
8
D
11
E
11
7
D
3
B
DO
4
C
6
C
B
6
4
Dual Kennedy circle
5
Marking
all known
EQML.
EQMLs
needed
for finding
mDO.
For example,
checking
the
1 3
1 3
existence of the EQML mDO
The EQML mDO should pass through three points
The EQML mDO cannot pass
The EQML mDO passes
through the three points , thus
through the three points. Since
all the EQML can be found, the
this drawing is not a
drawing is a projection of a
projection of a polyhedron.
polyhedron.
C
5
7  11
Further research:
- Employing the theoretical results for additional practical
applications, such as:
combinatorial rigidity – found to be important in biology, material science, CAD
and more.
- Developing new synthesis methods.
- Applying the methods for static-kinematic
systems, such as: deployable structures,
Tensegrity Systems and more.
Thank you!