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

Stewart Platform and Serial robotTruss and Mechanism

1’

2’

3’

4’

5’

6’

7’

8’

9’

10’

11’

12’

13’

1

2

3

4

5

6

7

8

9

10

11

12 1

3

The two reported types of graph theory duality .

2’

3’

We obtain the dual systems.

1O/AV

3O/BV

The relative linear velocity of the driving link is equal to the corresponding external force.

The absolute linear velocity corresponds to the new variable in statics – face force.

1

2A

B

O1 O3

3

B/AV

1’

P

'3F

'2F

AFF

BFF

PV10/A

'330/B FV

'2B/A FV

B30/B FFV

A

10/A FFV

What kind of a variable corresponds to the absolute linear velocity ?

Duality between trusses and linkages and the theoretical

results derived from it .

The relative linear velocity of each link is equivalent to the force acting in the corresponding rod.

The relation between Maxwell reciprocal diagram

and graph theory duality .

C

A B

D

F1

B

C

D

V1’

A B

C

D

12

3

45

6

7

8

9

O

1F

1F

BFFAFF

CFFDFF

The truss underlying the reciprocal diagram has infinitesimal motion.

Removing link 1 and turning its internal force (the blue arrow), into an external force acting upon a linkage in a locked position

The original and the dual graphs

The unstable truss dual to the linkage in a locked position.

A

Applying rotation to the reciprocal diagram.

The Relation between Static Systems, Mobile Systems and Reciprocity

AV

BV

CV

DV

CD

B

A

AV

BV

CV

DV

B

A

O D

C

Maxwell’s theorem 1864: The isostatic framework that satisfies E=2*V-3 has a self stress iff it has a reciprocal diagram.

B

1’

2’

3’ 4’

6’

9’5’

A

O

7’

8’

D

CThe reciprocal diagram

The isostatic framework

O

V1’

The isostatic framework has a self-stress.

B

A

O D

C

For every link there exists a point where its linear velocity is equal to zero.

For every force there exists a line where the moment it exerts is equal to zero.

joI0V

joI

j

jF jom

0Mjoj mF

For every two links there exists a point where their linear velocities are equal. .

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.

k

jkjk IkIj VV

jkkjkj mFmF MM

jF

kFjkm

j

Kinematics in 2DStatics in 2D

Theoretical results obtained from the duality between trusses

and linkages .

jkI

l

k

j

jlI

jkI

jlm

klI

klm

jkm

kF

jF

lF

The Kennedy Theorem. For any three links, the corresponding three relative instant centers

must lie on the same line .

The Dual Kennedy Theorem. For any three forces, the corresponding three relative equimomental lines

must intersect at the same point .

The isostatic framework that has a self-stress.

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 .

A B

C

D

1

2

3

45

6

78

9

O

A

B

D

C

O

12

3

4

5

67

89

The isostatic framework The corresponding polyhedron

a

b

c

d

e

f

a

b

c

d

e

f

Introducing a polyhedral interpretation for the theoretical results obtained from this duality .

An EQML mBD == An edge between vertices B and D in the dual Kennedy circle == An intersection line between

planes B and D in the polyhedron .

Triangle in the dual Kennedy circle == An intersection point of the corresponding three EQML.

An equimomental line between two adjacent faces== A known edge in the dual Kennedy circle between two corresponding vertices== An intersection line between two adjacent planes in the polyhedron.

An EQML between two nonadjacent faces == An unknown edge in the dual Kennedy circle between two corresponding vertices == An intersection line between two nonadjacent planes.

Two triangles that include the EQML in the circle == Points that this line passes through them.

mDO corresponds to the intersection line between plane D and the projection plane - O .

An EQML mCO == An edge between vertices C and O in the dual Kennedy circle == An intersection line

between planes C and O in the polyhedron .

An EQML mCD == An edge between vertices C and D in the dual Kennedy circle== An intersection line between

planes C and D in the polyhedron .

A face in the framework == A vertex in the Dual Kennedy circle == A plane in the polyhedron.

A reference face O== A reference vertex in the dual Kennedy circle==A projection plane in the

polyhedron .

An EQML mBO == An edge between vertices B and O in the dual Kennedy circle== An intersection line between

planes B and O in the polyhedron .

Geometric interpretation for the new variable – the equimomental line.

A

B

D

C

O

12

3

4

5

67

89

B

C

D

12

3

45

6

78

9

O

DOm

BOm

COm

The projection plane .

The isostatic framework

The corresponding

polyhedron BO

CO

O

A

BC

D

The dual Kennedy circle

BDm

CDm

Constructing the dual Kennedy circle for finding all the equimomental lines.

A

DOBD

CD

8

5

9

3

83 95

9583mDO

For every link there exists a point where its linear velocity is equal to zero.

For every force there exists a line where the moment it exerts is equal to zero.

0VjoI

j

jF jom

0Mjoj mF

For every two links there exists a point where their linear velocities are equal.

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.

jkjk IkIj VV

jkkjkj mFmF MM

kF

jkm

j

joI

k

jlI

jkI klI

The Kennedy Theorem. For any three links, the corresponding three relative instant centers must lie on the same line.

l

k

j

jF

jlm

klm

jkm

kF

jF

lF

The Dual Kennedy Theorem. For any three forces, the corresponding three relative equimomental lines must intersect at the same point.

For every plane there exists a line where it intersects the

projection plane .

J

O

For every two planes there exists a line where they intersect.

J

K

JO

JK

Every three planes must intersect at a point .

Kinematics in 2DStatics in 2DGeometry in 3D

jkI

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

1

24

56

3

O

I

P

Stewart platform

1

24

56

3

O

I

P

Stewart platform

61 2 3 4 5

I

Every platform element corresponds to a vertex and every leg to an edge .

The original graph

P

O

1

24

56

3

O

I

P

Stewart platform

61 2 3 4 5

I

The original graph

P

O

The dual graph

1’ 2’ 3’

4’ 5’ 6’

A

B C D E F

1

24

56

3

O

I

P

Stewart platform

61 2 3 4 5

I

The original graph

P

O

The dual graph

1’ 2’ 3’

4’ 5’ 6’

A

B C D E F

Serial robot

Every joint corresponds to a link and an edge to a joint.

A

BC

E

D

F

1’

2’3’

4’

5’6’

1

24

56

3

O

I

P

Stewart platform

61 2 3 4 5

I

The original graph

P

O

The dual graph

1’ 2’ 3’

4’ 5’ 6’

A

B C D E F

Serial robot

Every joint corresponds to a link and edge to a joint.

A

BC

E

D

F

1’

2’3’

4’

5’6’

The force in the leg of Stewart platform is identical to the relative angular velocity of the corresponding joint in the

dual serial robot . P

4F

5/4

4’

5’

4

The duality relation between Stewart platforms and serial robots.

5/44F

The magnitude and the unit direction of the force correspond to the second projective

point that is located at infinity .

"Force" applied at point ‘p’ F(p) is defined by: the force and the point .

Motion in the projective plane is a line that joints between two projective points .

The angular velocity and the instant center correspond to the first projective point.

The point p corresponds to the second projective point.

Adding a dimension and defining a projective plane on Z=1.

The point p corresponds to the first projective point.

Y

X

p

c

c

Z

p

Z=1 pc

Y

X

p

Z

p

Z=1pq

qf

q

pcM pqF

The relation between kinematics and statics through projective geometry.

Kinematics Statics

Motion of a point ‘p’ on a link - M(p) is defined by: the instant center, angular velocity and the

point on the link .

Adding a dimension and defining a projective plane on Z=1 .

Introducing the motion in the projective plane.

Introducing the “force” in the projective plane.

It follows that the magnitude of the angular velocity is equivalent to the magnitude of the force vector. The "force" in the projective plane is a line that

joints between two projective points, one is at the infinity .

10

29

3

6

711

4

A

B

C

D

E

O

8

DO

Testing whether a line drawing is a correct projection of a polyhedron.

A

B

C

E

O

D

10

8

7

9

2

64

3

11

Which of the drawings is a projection of a polyhedron?

O

A

B

C

D

E

Dual Kennedy circle

11

55 For example, checking the existence of the EQML mDO

The EQML mDO should pass through three points The EQML mDO cannot pass through the three points , thus

this drawing is not a projection of a polyhedron .

Finding all the EQML there exists a self stress in the configuration (Maxwell+Whiteley) it is a

projection of a polyhedron

The EQML mDO passes through the three points. Since all the EQML can be found, the drawing is a projection of a

polyhedron .

1

311

7

10

8

117

108 108

117

31 31

31108117mDO

Marking all known EQML.EQMLs needed for finding mDO.

Example of a practical application based on the theoretical results

obtained in this research .

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!