1. The Equivalence between Static (rigid) and Kinematic (flexible, mobile) Systems through the Graph Theoretic Duality Dr. Offer Shai Tel-Aviv University shai@eng.tau.ac.il

2. 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?

3. 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. 1 2 3 4 5 6 7 8 9 A B C D E O O O O

4. 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. 1 2 3 4 5 6 78 9

5. Example – truss (isostatic frameworks) External forces can be considered as bars inducing force. Static Structure 1 2 3 4 5 6 78 9

6. Duality between trusses (isostatic frameworks) and linkages (kinematic/mobile system, mechanism) REPRESENTING TRUSSES Graph G F - 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 G F – bar, external force, reaction in T  edge. pin-joint  vertex. forces  flows in G F.

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

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

9. Duality between trusses (isostatic frameworks) and linkages (kinematic/mobile system, mechanism) PROOF Let T be a truss (isostatic framework) and G F 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.

10. Kinematic Linkage Static Structure Example 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 78 9 1 2 3 4 5 6 7 8 9 3 5 7 8 2 5 4 1 9 A B C D E O A B C D E O O O O

11. Kinematic Linkage Static Structure 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 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 78 9 A B C D E O A B C D E O O O O

12. Kinematic Linkage Static Structure The equilibrium of forces in the structure is thus equivalent to compatibility of relative velocities in the linkage Example 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 78 9

13. 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 ? 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 78 9 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. A B C D E O O O O

14. Face force – a multidimensional generalization of mesh currents

15. 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. II VI III V I IV O

16. Discrete-continuous description of a linkage 1 2 3

17. Discrete-continuous description of a linkage Absolute instant center of a link defines a velocity field in each point of the plane 1 I1I1

18. 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. 1 2 3 I1I1 I3I3 I2I2 I 12 I 23 Relative instant centers Absolute instant centers

19. 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

20. Dual discrete-continuous description of truss - constraints 1 2 3 I1I1 I3I3 I2I2 I 12 I 23 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

21. Kennedy theorem in Linkages and Isostatic frameworks

22. A B D C O O A B C D AB C D 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1’ 3’ 5’ 6’ 2’ 4’ 7’ 8’ 9’ 2 3 4 5 6 7 8 9 F1F1 2 7 8 9 4 5 6 3 3’ V 1’ 2’ 4’ 5’ 6’ 7’ 8’ 9’ O C D B A 2’ 3’ 1’ 6’ 4’ 5’ 9’ 8’ 7’ B 1’ 2’ 3’ 4’ 6’ 9’ 5’ A O 7’ 8’ D 1’ 3’ 5’ C 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 7 3’ V 1’ 2’ 4’ 5’ 6’ 8’ 9’ 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

23. 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 124 234 - Face  vertex. - Edge (x,y)  intersection between the two faces (x,y) 0 1 2 3 4 -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 1 2 3 4 01 03 02 04 23 34 12 14 24 13 234 124 024 013 0

24. 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 Laman Graph Assur Graph Laman Graph

25. Assur Graphs (Minimally Rigid Graphs with respect to vertices) Assur graph Laman Graph + Laman Graph

26. 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

27. 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

28. Conjecture 1 Every Assur graph can be constructed through merging several Fundamental Assur Graphs (FAG). I1I1 I2I2 I3I3 I4I4 I5I5 I1I1 I2I2 I3I3 I5I5 Merging two Fundamental Assur Graphs yielding new Assur Graph:

29. Combining the latter Assur Graph with Laman Graph yielding Laman Graph: I1I1 I2I2 I3I3 I5I5 Assur Graph Laman Graph Resulted Laman Graph

30. The possible hierarchy of Laman graphs by means of the Assur graphs. Fundamental Assur graph Laman Graphs Ak =  Fi merging of several fundamental Assur graphs. M

31. 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.

32. (a) Serial Robot (b) The Dual Stewart Platform