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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 P The absolute instant center is the point in a link where the linear velocity is zero. The absolute equimomental line is the line where the moment produced by a force is zero. V M r P The linear velocities of points located at a distance r from the absolute instant center are the same. r The moments produced by a force located a distance r from the absolute equimomental line are the same. The Linear Velocities Field The Moment Field Y Y F E D Z Z C P F E D C B A X B A X Y Y F F E E D D X X C B A C B A 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. Relative instant center. 2 1 I10 r1 I12 r2 Relative equimomental line. F1 I20 V1 V2 The relative instant center of two links is the point in both links where the angular velocities produce the same linear velocity. r1 r2 F2 M F1 m 12 M F2 m 12 The relative equimomental line of two forces is the line where the forces produce the same moment. Arnold-Kennedy Theorem 2 I20 I23 Dual Kennedy Theorem I30 3 I12 F1 m10 m30 F3 m13 m12 F2 m23 m20 1 I10 I13 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. Correlation between Instant Centers and Equimomental Lines. I I12 I12 II 1 I10 I12 2 I20 I12 III I12 II mAB I mA0 mAB mB0 mAB mAB FAO FBO III mAB mAB I12 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. Constructing the dual graph from the original graph Original graph 1* 3 A 1 2 II 5 4 I O 3* C 2* III 7 B 6 D II I O 5* III 7* IV O 4* Dual graph 6* IV 8 8* Reference face. Cutset is a set of edges so that A circuit is 1*, a closed Edges 3*, 4* andpath. 2* 5* form 7* Each circuit in the original graph corresponds to a cutset in the dual graph, and ifEdges removed the graph, the Face IV I corresponds II III corresponds corresponds to to to the the vertex vertex vertex II. III. IV. a circle cutset inin dual dual graph. graph. Aand face in the graph isgraph aadjacent circuit without inner edges. Each Reference infaces the face Ograph corresponds corresponds totoI.graph the to athen vertex Each 3, 1, 4 2from cutset 5 7face in constitute Two the original aadjacent are corresponds ifthe they have a at circuit in their corresponding vice-versa. If two faces areoriginal in the original graph becomes disconnected. in original the dual reference graph. vertex O. cutset circuit the in dual the graph, least and graph. vice versa. in common. vertices are adjacent inone theedge dual graph. Vertices II IOand and IV are are adjacent. adjacent. Every adjacent faces correspond to two adjacent vertices in the dual graph. Facestwo III Oand III and andIV IIO III IIV are are are are adjacent. adjacent. adjacent. adjacent. III and IIO IIII IV are are adjacent. adjacent. The edge to adjacent. these two faces corresponds theIVedge that connects the Faces O common and IV are Vertices O to and are adjacent. vertices in the dual graph. Constructing the Dual of a Linkage Augmentingthe thedual geometry Constructing graph.to The meaning of a directed edge in the Constructing its topology. the graph. cos 90 O dual graph: system. Kinematic Constructing the B 3 cos3 90 is sinthe 3 flow 90 3(force) corresponding graph sin 90 3 3 e=<t,h> acting upon the A / B (CCW) B 2* A A head vertex (joint) by the edge (rod). (CCW) The topology arrow and the arrow are in (CW) force 3* VA / B direction -> compression. the same VB / O 2 4 The force in rod acts upon the ground 2 VA / 4** O 4 Inverse directions - tension 4 I Two in this orientation. direction of the inacts rodthe 4**. Thechoices? force in external rod 3** force acts upon ground The force upon (CW) 2 2 B/ O in thisThe orientation. Vertex I corresponds to joint direction ofjoint theIforce in rodI. 3**. 3 4 4 2 O2 The type is tension. OThe type is compression. 1 VO1 / O4 0 3 4* 3 O4 1 (CW) A / O2 4 (CW) O O O 2 4 Adding 1the geometry. O1 / O4 0 O source Vertex O to Faces joint . and Othe adjacent. EdgeLink 2The is 2the is potential the Edge driving corresponds link. that corresponds to link 3.the to driving link 2. the twofaces Vertex A corresponds The potential to joint source, 4O A. edge 2, relative The relative velocity velocities linear of oflink velocity links 2to corresponds 3 corresponds and 4 toOthe to the Edge 3 is common to is the two adjacent 243corresponds 2Icorrespond For consistency, direction ofbetween the edge Vertices B and O correspond joints B and , are respectively. We can contract the edges with 4 The kinematic analysis yields the Edges 4 and 1 correspond to link 4 and the fixed link 1, 3** adjacent Iequal faces and O Iand and thus Othe in dual the original edge 4* graph. is between potential difference differences potential of difference. of edge edges 2. 3Reference and 4. 4**to the faces Ito O thus the dual edge 3* is face O corresponds in the dual graph is defined by rotating potential difference zero. The angular velocity in CW magnitudes and directions of the angular Potential differences in edges 3 and 4 of the edge original 2 Therefore, graph theintwo respectively. adjacent vertices Ivertices and O. Itoand the dual graph it corresponds theO. angular velocity is CCWthe which between the two adjacent tothe reference vertex O. edge inFace original graph in CCW The corresponds to compressing force. and relative linear velocities. correspond to flows in edges 3* and 4*. corresponds to the flow the in edge dual graph. 2*. I corresponds to vertex I. Pcorresponds I flow source and it is between the costwo to a tension. Building the direction. adjacent 90 4** vertices I and O. sin 90 corresponding truss. F 3* O 4 3 3 4 (tension) 3** O3 The corresponding truss. (compression) P The dual graph. F4* (compression) Since velocity associated with ain statics: We have systematically developed a new variable We linear obtain theisdual systems. dualits dual variable is jointB in the linkage, Face At first, what isthe thisface absolute velocity? What Joint is a counterpart to absolute linear velocity of the joint associated with in the truss O dual 3 The relative 1.The linear velocity of the link input 3 link 4 corresponds corresponds to the toofthe force external in the rod force. 3**. 4**. absolute linear velocity has a property potential. Velocity Force A + 2 O2 + 3 On the other handdual we know that velocity Why? 4 3** Absolute linear velocity face force. Velocitycorresponds of a joint to force.corresponds FacetoForce Because, We can give any absolute velocities to the links, and 4** they will satisfy the rule of velocities (vectors KVL). O I FB P 1 4 1 ? FA VA / B VA / O2 VA / O2 F3** ? F4** VB / O4 VB / O4 O4 2 1 is equal to to thecompression. Same direction Force in rod corresponds 3 difference of subtraction ofA face forces P F by byFFAO. . face forces F and F . manner, locate theB forcesA to in tension. the other rods. Opposite direction corresponds In the same M M F m An arbitrary point on the F m equimomental FP r FPmPO FO r FO mPO PB: line mPA P PO O PO M FO 0 M BA m P m F m Face force FABP P P r PmPA FA r FA mPA acts the B. F in face F face A. P. F F P The moment produced by the P rod isFthe difference of the Each force in the forces P and FAB The on the upon the line Absolute circuit equimomental corresponds circuit The equimomental lines that Set of the(Right edges. rightarbitrary andF the directions left face forces and left m In the same manner we can find the reaction R. Thus we obtain the face force F . equimomental line m . m to the has vertex. to be corresponds determined . to will locate m . m B PB A PA BO BO defined according to the M direction of the arrow the vertex. m in the edge). r PA PB PA PB A B A 2 3 PB PO BO P rP FB P P F4 4 F4 F32FA F P F1 A FOFF AB FA compression 2 F2 A F3 FA ? ? FB FB m PB 3 B rP M P mPA rFA m PA F1A 1 FB F5 compression R mRO mPO R P mAO mPB mBR O mPA mBR 5 R F1 FA FO 0 O FA F1 mAB R m AO mRO FB R F5 B mAB A Finding and characterization of the dead center positions of the mechanism. O 2 7 1 4 3 2 O 4 5 A 1 5 3 7 O C 6 6 2,3 5,7 C 6 A In this case, the faces B and O (i.e., the reference vertex) of the truss have the same face force which indicates that: (i) links 2 and 3 are collinear, and (ii) links 5 and 7 are collinear. These two conditions ensure that the mechanism is in a dead center position. Another examples to find dead positions of the mechanism by Face Force. Given mechanism topology Duality relation between stability and mobility Rigid ???? 12 12 2 4 2 8 8 4 3 7 5 1 3 9 7 1 9 5 10 6 6 11 10 11 Due to links 1 and 9 being located on the same line 2’ Definitely locked !!!!! 2’ 12’ 4’ 12’ 8’ 3’ 1’ 4’ 7’ 5’ 6’ 9’ 10’ 11’ 1’ 8’ 3’ 9’ 7’ 5’ 6’ 10’ 11’ R’ R’ By means of the duality transformation, checking the stabiliy of trusses can be replaced by checking the mobility of the dual linkage.