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

The absolute instant center is the point in a link where the linear velocity is zero.

P

The absolute equimomental line is the line where the moment produced by a force is zero.

V

The linear velocities of points located at a distance r from the absolute instant center are the same.

r

r

M

The moments produced by a force located a distance r from the absolute equimomental line are the same.

P

X

Y

Z

Y

X

X

Y

Z

X

Y

P

C

D

C

D

C

D

D

C

B

E

B

E

B

E

B

E

A

F

A

F

A

F

A

F

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.

The Linear Velocities Field The Moment Field

I10 I20I12r1

r2

21 VV

122121 mFmF MM

r1r2

F1

F2

The relative instant center of two links is the point in both links where the angular velocities produce the same linear velocity.

The relative equimomental line of two forces is the line where the forces produce the same moment.

1

2

Relative instant center. Relative equimomental line.

I10

I20 I30

I12

I23

I13

m12

m13

m23

m10

m20

m30

1

2

3 1F

2F

3F

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.

Arnold-Kennedy Theorem Dual Kennedy Theorem

mA0mB0

mAB AOF

BOF

I122

I10 I20I12 I12 I12

I II

mAB

mAB

mAB

I12I12

mAB

mAB

1

I

II

III

III

Correlation between Instant Centers and Equimomental Lines.

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.

A

B D

C

O

I

II

III

IV

A face in the graph is a circuit without inner edges.

I

IV

III

II O

O

Reference face.

Each face in the original graph corresponds to a vertex in the dual graph.

Face I corresponds to the vertex I.Face II corresponds to the vertex II.Face III corresponds to the vertex III.Face IV corresponds to the vertex IV.Reference face O corresponds to the reference vertex O.Two faces are adjacent if they have at least one edge in common.

Original graph

Dual graph

Every two adjacent faces correspond to two adjacent vertices in the dual graph. The edge common to these two faces corresponds to the edge that connects the vertices in the dual graph.

12

3

4 5

67

8

1*

2*

3*

4*

5*

6*7*

8*

Faces O and I are adjacent. Vertices O and I are adjacent.Faces I and II are adjacent. Vertices I and II are adjacent.Faces II and O are adjacent. Vertices II and O are adjacent.Faces II and III are adjacent. Vertices II and III are adjacent.Faces O and III are adjacent. Vertices O and III are adjacent.Vertices I and IV are adjacent.Faces III and IV are adjacent. Faces I and IV are adjacent. Vertices III and IV are adjacent. Faces O and IV are adjacent. Vertices O and IV are adjacent.

Cutset is a set of edges so that if removed from the graph, the graph becomes disconnected.

A circuit is a closed path.Each cutset in the original graph corresponds to a circuit in the dual graph, and vice versa.

Edges 3, 4 and 5 constitute a cutset in the original graph.

Edges 3*, 4* and 5* form a circle in dual graph.Edges 1, 2 and 7 constitute a

circuit in the original graph.

Edges 1*, 2* and 7* form a cutset in dual graph.

Each circuit in the original graph corresponds to a cutset in the dual graph, and vice-versa. If two faces are adjacent in the original graph then their corresponding vertices are adjacent in the dual graph.

Constructing the dual graph from the original graph

1

2

3

4

1

A

B

2

2O/AV

90sin

90cos

3

3

O2 O4

3

4

O2 O4

A B

2

3

4

1

2 4

O2O/A

B/A

4O/B

O

I

2*

3*

4*3

The kinematic analysis yields the magnitudes and directions of the angular and relative linear velocities.

Link 2 is the driving link.

Kinematic system. Constructing the corresponding graph

Vertex O2 corresponds to joint O2.Vertex A corresponds to joint A.Vertices B and O4 correspond to joints B and O4, respectively. Edge 2 is the potential source that corresponds to the driving link 2.Edge 3 corresponds to link 3.

0V41 O/O

0

41 O/O

Edges 4 and 1 correspond to link 4 and the fixed link 1, respectively.

We can contract the edges with potential difference equal to zero.

The relative linear velocity corresponds to the potential difference.

B/AV

4O/BV

The relative velocity of link 2 corresponds to the potential difference of edge 2.The relative velocities of links 3 and 4 correspond to the potential differences of edges 3 and 4.

90sin

90cos

3

3

90sin

90cos

3

3

Constructing the dual graph.

The dual graph.

P

*4F

*3F

Potential differences in the original graph correspond to flows in the dual graph.Potential differences of edge 2 corresponds to the flow in edge 2*. Potential differences in edges 3 and 4 correspond to flows in edges 3* and 4*. Face I corresponds to vertex I.

Reference face O corresponds to reference vertex O.

Faces I and O are adjacent.Edge 4 is common to the two adjacent faces I and O thus the dual edge 4* is between the two adjacent vertices I and O.

Edge 3 is common to the two adjacent faces I and O thus the dual edge 3* is between the two adjacent vertices I and O.

For consistency, the direction of the edge in the dual graph is defined by rotating the edge in the original graph in CCW direction.

The potential source, edge 2, is between the two adjacent faces I and O in the original graph. Therefore, in the dual graph it corresponds to the flow source and it is between the two adjacent vertices I and O.

O

I

Constructing its topology.Augmenting the geometry to

the graph.

Adding the geometry.

Building the corresponding truss.

3**4**

I

O3

O4

P

(CW)

(CCW) (CW)

(CW)

(CCW)

(CW)

(compression)

(tension)

(compression)

Vertex I corresponds to joint I.The external force acts upon

joint I

3**

4**O4

O3

The meaning of a directed edge in the dual graph:

e=<t,h> is the flow (force) acting upon the head vertex (joint) by the edge (rod).

The force in rod 4** acts upon the ground in this orientation.

The topology arrow and the force arrow are in the same direction -> compression.Inverse directions - tension

The type is compression.

Two choices?The direction of the force in rod 4**.The force in rod 3** acts upon the ground in this orientation.The direction of the force in rod 3**.

The type is tension.

The corresponding truss.

The angular velocity in CW corresponds to compressing force.The angular velocity is CCW which corresponds to a tension.

Constructing the Dual of a Linkage

1

2

3

1

A

B

O2 O4

4

P

3**

I

O3

O4

4**

We obtain the dual systems.

4O/BV

**4F

B/AV

**3F

2O/AV

The relative linear velocity of the input link corresponds to the external force.The relative linear velocity of the link 3 corresponds to the force in the rod 3**.The relative linear velocity of the link 4 corresponds to the force in the rod 4**.

? ?

What is a counterpart to absolute linear velocity of the jointAt first, what is this absolute velocity?

1.The absolute linear velocity has a property of potential.

Why?Because, We can give any absolute velocities to the links, and they will satisfy the rule of velocities (vectors KVL).

2O/AV

4O/BV

Since linear velocity is associated with a joint in the linkage, its dual variable is associated with the face in the truss

On the other hand we know that velocity corresponds to force.

We have systematically developed a new variable in statics:

Absolute linear velocity corresponds to face force.

BF

AF

Joint Face

Velocity Force

Velocity of a joint Face Force

dual

dual

dual

+ +

1

2

3

4

5

PBB mFr

R

P

A

B

R

O

P

AB

R

O

mPO

mPA

mBO

mPB

mAO

P

??mPO

mPA

mAB

mAB

mBR

mBR

mRO

mRO

mAO

mPB

Pr

PAA mFr

PAAPA mFmP MM

PAAPA mFAmP rFrP

M

The moment produced by the forces P and FA upon the equimomental line mPA.

PFP

0FO

POOPOP mFmF MM

POOPOP mFOmFP rFrF

AF

Thus we obtain the face force FA.

Face force FP acts in the face P.

An arbitrary point on equimomental line mPA:

Face force FA acts in the face A.

Absolute equimomental line mBO has to be determined .

Pr

An arbitrary point on the equimomental line mPB:

PBBPB mFmP MM

PAAPA mFAmP rFrP

The moment produced by the forces P and FB on the equimomental line mPB.MThus we obtain the face force FB.

BF

Face force FB acts in face B.

In the same manner we can find the reaction R.Set arbitrary directions of the edges.Each force in the rod is the difference of the right and the left face forces (Right and left defined according to the direction of the arrow in the edge).

AF

BF

P

R

AOA1 FFFF

1A FF

0FO

Force in rod 1 is equal to the subtraction of face forces FA by FO.

1F

Force in rod 2 is equal to the subtraction of face forces P by FA.

A2 FPF

2F

Force in rod 3 is equal to the difference of face forces FA and FB.

BA3 FFF

3F

In the same manner, locate the forces in the other rods.

4F

5F

tension

compression

com

pres

sion

compression

compression

Same direction corresponds to compression.

Opposite direction corresponds to tension.

The equimomental lines that will locate mBO. The circuit corresponds to the vertex.

The circuit corresponds to the vertex.

A1 FF

5F

BF

R

3F

AF

BF

P

AF

2F

BF

P

4F

1

2

3

4

5

6

7

O

A

O

CO

AC

2,3

6

5,7

1

2

3

4

5

6

7

(i) links 2 and 3 are collinear, and (ii) links 5 and 7 are collinear.

In this case, the faces B and O (i.e., the reference vertex) of the truss have the same face force which indicates that:

These two conditions ensure that the mechanism is in a dead center position.

Finding and characterization of the dead center positions of the mechanism.

Another examples to find dead positions of the mechanism by Face Force.

Given mechanism topology

By means of the duality transformation, checking the stabiliy of trusses can be replaced by checking the

mobility of the dual linkage.

Duality relation between stability and mobility

Definitely locked !!!!!

Rigid ????

8

12’

2’

1’

11’

10’6’

7’

3’

5’

’

9’

R’

4’12’

9’

10’

R’11’

6 ’

7’

8’

2’

3’

5’1’

4’

8

5 9

2

4

7

10

11

1

12

6

3

11

7

3

4

122

1 5

8

9

106

Due to links 1 and 9 being located on the same line