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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 The two reported types of graph theory duality. Truss and Mechanism Stewart Platform and Serial robot 11’ 11 10 3’ 4 4’ 3 12 10’ 13 9 2 7’ 5 12’ 9’ 7 2’ 13’ 5’ 1 6’ 1’ 8’ 6 8 We obtain the dual systems. What kind of a variable corresponds to the absolute linear velocity? B velocity The relativelinear linearvelocity of the driving is equal tothe thein corresponding each link to islink equivalent to force acting in the The absolute corresponds the new variable statics – face force. external force.2 rod. corresponding 2’ A Duality between trusses and linkages and theV theoretical FF derived from it. results 3 1 1’ FF B O1 VA / 0B1 P FFF 2' A VB / 0033 FFF 3' B 1 O3 B / O3 A VA / B F2 ' VA / O1 P F3' 3’ The Relation between Static Systems, Mobile Systems and Reciprocity F1 O O FFA F1 B A FF B D C Removing link 1 and turning its internal force (the blue arrow), into an external force acting upon a linkage in a locked position D 6 9 5 V11’ ’ 3 8 C 4 B The isostatic The isostatic framework framework has a self-stress. F1 A The relation between Maxwell reciprocal diagram and graph theory duality. 3’ 2’ D B The original and the dual graphs Applying rotation to the reciprocal diagram. VC D D O 9’ 5’ 1’ The unstable truss dual to the linkage in a locked position. VB 7’ C 6’ B VD The truss underlying the reciprocal diagram has infinitesimal motion. O C VA 4’ The reciprocal diagram B VD A 8’ A C VB B A 7 FFC FFD Maxwell’s theorem 1864: The isostatic framework that satisfies E=2*V-3 has a self stress iff it has a reciprocal diagram. 2 1 VC D C A VA Kinematics in 2D Statics in 2D For every link there exists a point where its linear velocity is equal to zero. j VI jo 0 I jo For every force there exists a line where the moment it exerts is equal to zero. Fj m jo M Fj m jo 0 Theoretical results obtained For every two links there exists a point where their For every two forces there exists a line, such that linear velocities are equal. . from the duality between the moments exerted by the two forces on each point on the line are equal. Fj trusses and linkages. k j Vj I jk Fk I jk m jk Vk M Fj m jk M Fk m jk I jk The Kennedy Theorem. For any three links, the corresponding three relative instant centers must lie on the same line. l j k I jk I jl The Dual Kennedy Theorem. For any three forces, the corresponding three relative equimomental lines must intersect at the same point. Fj m jl m jk I kl Fk Fl mkl 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. Introducing a polyhedral interpretation for the theoretical results obtained from this duality. e A c a 2 D 1 O e c A 6 B 7 D a 1 3 d 8 7 6 5 C 5 b 8 C 9 d 2 B 3 9 b 4 4 f The isostatic framework that has a self-stress. The isostatic framework O f The corresponding polyhedron mBO Geometric interpretation for the new variable – mBD The isostatic framework the equimomental line. 2 1 O A 7 3 8 B D mCO 6 D 9 C mDO C B Constructing the dual Kennedy circle for finding all the equimomental lines. 38 59 CD A A 3 8 m CD mDO 3 8 5 9 5 9 4 5 The corresponding polyhedron O The dual Kennedy circle D BO 1 2 7 B 6 C 8 3 9 5 O Triangle the dual Kennedy == AninDual intersection Two triangles that include the EQML in the circle ==D m corresponds totwo the intersection line A reference face Abetween reference the AnDO face equimomental EQML inin the between m framework ==O== An line edge nonadjacent == between Acircle vertex twovertex adjacent vertices faces B C and and O D in in BD CO CD BO point ofunknown the corresponding three EQML. Points that this line through them. between plane D and the projection plane -the O.between dual Kennedy circle==A projection plane inline faces== == O Kennedy the inAn dual the A dual Kennedy circle known Kennedy == edge edge circle circle== Apasses plane in circle in== the the An in An dual == the dual intersection intersection An Kennedy polyhedron. Kennedy intersection line between line polyhedron. circle between between planes C and B planes Dtwo O inCthe corresponding andpolyhedron. O in the polyhedron. vertices== An vertices intersection == An intersection line between line between two adjacent two planes in the polyhedron. nonadjacent planes. 4 CO The projection plane. DO BD Kinematics in 2D For every link there exists a point where its linear velocity is equal to zero. j Geometry in 3D For every force there exists a line where the moment it exerts is equal to zero. Fj VI jo 0 I jo Statics in 2D For every plane there exists a line where it intersects the projection plane. m jo J M Fj m jo 0 O JO 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. F For every two links there exists a point where their linear velocities are equal. j k j I jk Vj I jk Vk I jl I jk K JK M Fj m jk M Fk m jk I jk The Kennedy Theorem. For any three links, the corresponding three relative instant centers must lie on the same line. j J m jk Fk k For every two planes there exists a line where they intersect. The Dual Kennedy Theorem. For any three forces, the corresponding three relative equimomental lines must intersect at the same point. Fj l I kl m jk Fk m jl Fl mkl Every three planes must intersect at a point. 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 P 3 2 I 4 1 6 O 5 Stewart platform I 1 2 3 4 5 6 O The original graph Every platform element corresponds to a vertex and every leg to an edge. P 3 2 I 4 1 6 O 5 Stewart platform P A I 1 1’ B 2’ C 3’ D 4’ E 5’ F 6’ 2 3 4 5 6 O The original graph The dual graph P 3 2 I 4 1 6 O 5 Stewart platform P A I 1 1’ B 2’ C 3’ D 4’ E 5’ 2 A 5 6 O The original graph Every joint corresponds to a link and an edge to a joint. F 5’ 6’ 4 F 6’ The dual graph 3 P 3 2 I 4 E 1’ 4’ B 3’ 2’ D 1 6 O 5 C Serial robot Stewart platform P A I 1 1’ B 2’ C 3’ D 4’ E 5’ 2 3 F 6’ P F A 6 The original graph Every joint corresponds to a link and edge to a joint. 5’ 6’ 5 O The dual graph 4 3 2 I 4 E 1’ 4’ B 3’ 2’ D 1 6 O 5 C Serial robot Stewart platform P The duality relation between Stewart platforms and serial robots. 4 / 5 5’ 4’ The force in the leg of Stewart platform is identical to the relative angular velocity of the corresponding joint in the dual serial robot. F4 4 / 5 P 4 F4 The relation between kinematics and statics through projective geometry. The p corresponds to the first Adding a dimension and Itto defining follows the magnitude "Force" of thepoint magnitude angular and the unit direction The Motion Introducing point angular of in p the a corresponds point the velocity projective motion ‘p’ on in aplane the link the the instant - is M(p) second aathat line is that defined Introducing thevelocity “force” Adding applied a dimension at point and‘p’ defining F(p) is defined a of the by: The "force" in the projective plane is a line that projective point. planetwo onto Z=1. is equivalent thethe magnitude force of the correspond force vector. to Z=1. the second projective projective center by: joints the between correspond instant point. plane. center, projective the angular first points. velocity to and in the projective plane. the projective force and plane the on point. joints between two projective points, one is at point that is located at infinity. projective point on the point. link. the infinity. Z M c p Z c p Z=1 F q p Z=1 p c p Y q p Y p p f q c X Kinematics X Statics q Testing whether a line drawing is a correct projection of a polyhedron. Which of the drawings is a projection of a polyhedron? Finding all the EQML there exists a self stress in 7 11 the configuration (Maxwell+Whiteley) it is a projection of a polyhedron mDO 7 11 8 10 1 3 8 10 8 10 O 10 A 2 1 9 1 8 D 3 10 11 B 9 10 E 3 A 2 Example of a practical application based on the theoretical results 7 obtained in this research. 7 E O O 8 A 1 8 D 11 E 11 7 D 3 B DO 4 C 6 C B 6 4 Dual Kennedy circle 5 Marking all known EQML. EQMLs needed for finding mDO. For example, checking the 1 3 1 3 existence of the EQML mDO The EQML mDO should pass through three points The EQML mDO cannot pass The EQML mDO passes through the three points , thus through the three points. Since all the EQML can be found, the this drawing is not a drawing is a projection of a projection of a polyhedron. polyhedron. C 5 7 11 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!