The Equivalence between Static (rigid) and Kinematic (flexible, mobile) Systems through the Graph...

The Equivalence between Static (rigid) and Kinematic (flexible, mobile) Systems

through the Graph Theoretic Duality

Dr. Offer ShaiTel-Aviv University

[email protected]

Outline of the talk• The graphs representing static systems (trusses, isostatic frameworks)

and kinematic systems (linkages, flexible, mobile systems, mechanisms).

• Derivation of the duality between trusses (isostatic frameworks) and linkages (mechanisms, flexible systems).

• Revealing concepts in statics through the duality relation : equimomental line dual to instant center; face force dual to absolute linear velocity.

• Discrete-continuous description of linkages and trusses (isostatic frameworks).

• The relation between the cross-section test and valid equimomental lines.

• The relation between the duality introduced in this talk and reciprocity.• Conjectures – Assur graphs (a type of Laman graph that is minimal

rigid with respect to vertices). Another decomposition (construction) of Laman Graphs. Realization: strict self-stress and ALL the inner joints have infinitesimal flex. Deterministic algorithm for constructing it?

Kinematic Linkage

Kinematic linkages

Kinematic linkage is a planar system built of rigid links interconnected through revolute joints. The variables of interest in linkage analysis are: joint absolute linear velocity, link relative linear velocity, link angular velocity (the rate of the angle position change)The linkage has one degree of freedom, namely rotation of one of the links (referred as a driving link) uniquely determines the velocities of all the links in the mechanism.

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Truss

Trusses (isostatic frameworks, bar and joint frameworks, bridges)

Truss is built of rigid bars and supports (fixed and mobile) connected through pinned joints. External forces can applied upon the truss through some of the pinned joints thus inducing internal forces and reactions in the truss bars and supports. The trusses considered here are statically determinate, namely the values of the externally applied forces uniquely determine the internal forces and the reactions of the truss.

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Example – truss (isostatic frameworks)

External forces can be considered as bars inducing force.

Static Structure

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Duality between trusses (isostatic frameworks) and linkages (kinematic/mobile system, mechanism)

REPRESENTING TRUSSES

Graph GF - directed graph such that each edge is associated with a vector called flow and in each cutset the sum of the flows is equal to zero, namely,

Representing Trusses through GF – bar, external force, reaction in T edge. pin-joint vertex. forces flows in GF .

0)G()G( FF FQ


REPRESENTING LINKAGES

Graph G - directed graph. each vertex we associate – potential. each edge we associate – potential difference (difference between the potentials of its end vertices) . The sum of the potential differences in each circuit is equal to zero,

G - link edge joint vertex. relative linear velocity potential difference.

0)G()G(

B


• Proposition For every an isostatic framework with external forces there exist a dual linkage, L, with given driving links, satisfying:

• Link in L corresponds a truss element (rod, external force, reaction) of T, and vice versa.

Links and bars are perpendicular. • Relative velocity of each link in L == force acting in the corresponding element in T.

e

e*

V(e*)

F(e)


PROOF• Let T be a truss (isostatic framework) and GF its graph, thus: • Suppose that its dual graph is G - namely • Let us construct a mechanism such that the dual link is

perpendicular to the corresponding bar in the truss, thus:

• The magnitude of the forces in the truss and the magnitudes of the relative velocities in the dual linkage are identical.

0)G()G( FF FQ

0)G()G( FF FQ

)e(F̂*)e(ˆ

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*

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( ) ( ) 0( ) ( )

( ) ( )

ˆ ˆ( ) ( )

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G Ge F e

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Q F

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Kinematic Linkage Static Structure

Example

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The links and the corresponding bars are perpendicular, therefore the relative velocity

of each link of the linkage is equal to the internal force in the corresponding rod of the

structure

Example

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The equilibrium of forces in the structure is thus equivalent to compatibility of relative

velocities in the linkage

Example

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Translating the concept of absolute linear velocity into statics

Absolute linear velocity in linkage• Associated with each joint of the

linkage• Relative linear velocity of each link

is determined through the subtraction between absolute velocities of two adjacent joints.

Kinematical Linkage Static Structure

?

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Face force in truss• Associated with each face of the truss• Internal force of each truss bar is

determined through the subtraction between face forces of two adjacent faces.

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Face force – a multidimensional generalization of mesh currents

Maxwell diagram describes both the forces in the structure and the velocities in the mechanism

In the linakage coordinates of the points in the diagram correspond to the absolute velocities of mechanism while in the

structure they correspond to the face forces of the corresponding

faces.

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Discrete-continuous description of a linkage

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Discrete-continuous description of a linkage Absolute instant center of a link defines a velocity field in

each point of the plane

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Discrete-continuous description of a linkage - constraints

The velocity fields are subject to the constraints at relative instant centers (joints), where the velocities of the two links

are obliged to be equal.

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Relative instant centers

Absolute instant centers

Dual discrete-continuous description of truss

Face force and its line of action (absolute equimomental line) defines the moment

at every point on the plane.

Absolute instant center of a link defines a velocity field in each point of the plane

Dual discrete-continuous description of truss - constraints

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I1I3

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Relative instant centers

Absolute instant centers

The fields of moments are subject to the constraints at relative equimomental lines

(bars), where the moments exerted by the two face forces are obliged to be

equal

The velocity fields are subject to the constraints at relative instant centers

(joints), where the velocities of the two links are obliged to be equal.

Relative equimomental

line

Kennedy theorem in Linkages and Isostatic frameworks

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Projection of polyhedron

The Polyhedron

The reciprocal diagram

The truss underlying the reciprocal diagram is non stable

truss (the joints can have infinitesimal flexes)

Removing link 1 and turning its internal force, F1, into an external force acting upon a linkage in a locked position

The graph and its dual

The dual of the linkage is the non rigid truss (isostatic framework) identical to the truss underlying the reciprocal diagram

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Applying infinitesimal motion (pure rotation) to the reciprocal diagram

around the axis passing through point O

The Relation between Static Systems, Mobile Systems and Reciprocity

The relation between realizability of polyhedra and equimomental lines

The method suggested by Whitely (1979), implemented (Ros, 2001) for checking the realizability

of a drawing as a projection of polyhedron is by checking that every three planes intersect at one

point

From Maxwell Theorem it follows that since it is a projection if polyhedron that it has self-stress, that

can be checked by equimomental lines

124234

- Face vertex.

- Edge (x,y) intersection between the two

faces (x,y)

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-Each circuit of size 3 – a point due to dual

Kennedy theorem.

234=23 34

124=12 14

24=124v234

013=01 03 13

024=02 04 24

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234124

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Assur Graphs (Minimally Rigid Graphs with respect to vertices)

Definition: Assur graph is a special graph satisfying the following properties:

A set of the graph vertices are defined to be "interface vertices".

Connecting Assur graph to another graph through the interface vertices maintains the rigidity property of the latter.

Assur graphs are minimal with respect to vertices. Assur graphs cannot be decomposed into other Assur graphs.

Interface Vertex

Interface Vertex

Laman Graph

AssurGraph

Laman Graph

Assur Graphs (Minimally Rigid Graphs with respect to vertices)

Assur graph

Laman Graph

+

Laman Graph

Fundamental Assur Graphs• Fundamental Assur Graphs is a

subsets that that are generated by ONE DETERMINISTIC recursive operation.

• Special property of fundamental Assur Graphs: When all the interface vertices of the fundamental Assur graphs are contracted to one, the graph becomes self-dual.

Recursive operation

Merging of Fundamental Assur Graphs

• Property: Merging of two fundamental Assur graphs yields an Assur graph:

Assur Graph Ak after merging Fundemtals Fi and Fj. M

Ak = Fi Fj M

Fi Fj

Fi Fj

Interface vertex

Interface vertex

Conjecture 1

Every Assur graph can be constructed through merging several Fundamental Assur Graphs (FAG).

I1I2

I3

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Merging two Fundamental Assur Graphs yielding new Assur Graph:

Combining the latter Assur Graph with Laman Graph yielding Laman Graph:

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Assur Graph

Laman Graph

Resulted Laman Graph

The possible hierarchy of Laman graphs by means of the Assur graphs.

Fundamental Assur graph

Assur graph

Laman Graphs

Ak = Fi merging of several fundamental Assur graphs. M

Conjecture 2:

• Li = Aj + Lk Every Laman graph is a combination of Assur graphs interconnected through the interface vertices together with one basic Laman graph.

Conjecture 3:• Laman Graph corresponds to a static rigid framework (truss, isostatic framework) that has

a special geometry, in which the framework possesses self-stress and all the bars corresponding to the Assur graph are mobile IFF the Laman graph consists of one and only one Assur graph.

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)a (Serial Robot

)b (The Dual Stewart Platform

The Equivalence between Static (rigid) and Kinematic (flexible, mobile) Systems through the Graph...

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