1. General view on the duality between statics and kinematics M.Sc student: Portnoy Svetlana Advisor: Dr. Offer Shai

2. 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.

3. 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 13 The two reported types of graph theory duality.

4. 2’ 3’ We obtain the dual systems. 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 2 A B O1O1 O3O3 3 1’ 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.

5. The relation between Maxwell reciprocal diagram and graph theory duality. C A B D F1F1 B C D V 1’ A B C D 1 2 3 4 5 6 7 8 9 O 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 C D B A 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 C The reciprocal diagram The isostatic framework O V1’V1’ The isostatic framework has a self-stress. B A O D C

6. 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. 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. Statics in 2DKinematics in 2D Theoretical results obtained from the duality between trusses and linkages.

7. 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.

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

9. An EQML m BD == 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. m DO corresponds to the intersection line between plane D and the projection plane - O. An EQML m CO == 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 m CD == 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 m BO == 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 1 2 3 4 5 6 7 8 9 B C D 1 2 3 4 5 6 7 8 9 O The projection plane. The isostatic framework The corresponding polyhedron O A B C D The dual Kennedy circle Constructing the dual Kennedy circle for finding all the equimomental lines. A 8 5 9 3

10. 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. 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. 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. 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 Every three planes must intersect at a point. Geometry in 3DStatics in 2DKinematics in 2D

11. 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 2 4 5 6 3 O I Stewart platform

12. 1 2 4 5 6 3 O I 6 1 23 4 5 I Every platform element corresponds to a vertex and every leg to an edge. The original graph O

13. 1 2 4 5 6 3 O I Stewart platform 6 1 23 4 5 I The original graph O The dual graph 1’ 2’ 3’4’ 5’ 6’ A BC D E F

14. 1 2 4 5 6 3 O I Stewart platform 6 1 23 4 5 I The original graph O The dual graph 1’ 2’ 3’4’ 5’ 6’ A BC D E F Serial robot Every joint corresponds to a link and an edge to a joint. A B C E D F 1’ 2’ 3’ 4’ 5’ 6’

15. 1 2 4 5 6 3 O I Stewart platform 6 1 23 4 5 I The original graph O The dual graph 1’ 2’ 3’4’ 5’ 6’ A BC D E F Serial robot Every joint corresponds to a link and edge to a joint. A B C E D F 1’ 2’ 3’ 4’ 5’ 6’

16. The force in the leg of Stewart platform is identical to the relative angular velocity of the corresponding joint in the dual serial robot. 4’ 5’ 4 The duality relation between Stewart platforms and serial robots.

17. 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 Z p Z=1 Y X p Z p The relation between kinematics and statics through projective geometry. KinematicsStatics 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.

18. 10 2 9 3 6 7 11 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 6 4 3 11 Which of the drawings is a projection of a polyhedron? O A B C D E Dual Kennedy circle 1 1 5 5 For example, checking the existence of the EQML m DO The EQML m DO should pass through three points The EQML m DO 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 m DO passes through the three points. Since all the EQML can be found, the drawing is a projection of a polyhedron. 1 3 11 7 10 8 Marking all known EQML. EQMLs needed for finding m DO. Example of a practical application based on the theoretical results obtained in this research.

19. 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.

20. Thank you!