The Duality between Planar Kinematics and Statics Dr. Shai, Tel-Aviv, Israel Prof. Pennock, Purdue University

1. The Duality between Planar Kinematics and Statics Dr. Shai, Tel-Aviv, Israel Prof. Pennock, Purdue University

3. 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

4. The absolute equimomental line is the line where the moment produced by a force is zero. The absolute instant center is the point in a link where the linear velocity is zero. The linear velocities of points located at a distance r from the absolute instant center are the same. The moments produced by a force located a distance r from the absolute equimomental line are the same.

5. Y Y F E F D E D C Z B A C X Z B A X Y Y F F E E D D X X C B C A B A The Linear Velocities Field The Moment Field 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.

6. I10 I20 r2 r1 F1 F2 Relative instant center. Relative equimomental line. I12 r1 r2 The relative equimomental line of two forces is the line where the forces produce the same moment. The relative instant center of two links is the point in both links where the angular velocities produce the same linear velocity.

7. Arnold-Kennedy Theorem Dual Kennedy Theorem I23 I20 I30 m13 m30 m10 m12 m23 I12 m20 I10 I13 The Dual Kennedy Theorem. For any three forces, their three relative equimomental lines must intersect at the same point. The Arnold – Kennedy Theorem. For any three links, their three relative instant centers must lie on the same line.

8. Correlation between Instant Centers and Equimomental Lines. I II III I10 I20 I12 I12 I12 I12 I12 I12 II mAB mA0 mAB I mB0 mAB III mAB mAB mAB

9. 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.

10. Constructing the dual graph from the original graph Original graph 1* 3 3* A C 1 II 5 2 4 II 2* 4* Dual graph O I III I 5* III O B D O 6 7 7* 6* IV IV 8 8* Reference face. Cutset is a set of edges so that if removed from the graph, the graph becomes disconnected. A circuit isa closed path. Edges 1*, 2* and 7* form a cutset in dual graph. Edges 3*, 4* and 5* form a circle in dual graph. Each circuit in the original graph corresponds to a cutset in the dual graph, and vice-versa. If two faces are adjacent in the original graph then their corresponding vertices are adjacent in the dual graph. A face in the graph is a circuit without inner edges. Face IV corresponds to the vertex IV. Face III corresponds to the vertex III. Face II corresponds to the vertex II. Face I corresponds to the vertex I. Each face in the original graph corresponds to a vertex in the dual graph. Edges 3, 4 and 5 constitute a cutset in the original graph. Edges 1, 2 and 7 constitute a circuit in the original graph. Each cutset in the original graph corresponds to a circuit in the dual graph, and vice versa. Two faces are adjacent if they have at least one edge in common. Reference face O corresponds to the reference vertex O. Vertices II and O are adjacent. Vertices O and III are adjacent. Vertices I and IV are adjacent. Every two adjacent faces correspond to two adjacent vertices in the dual graph. The edge common to these two faces corresponds to the edge that connects the vertices in the dual graph. Faces I and IV are adjacent. Faces II and O are adjacent. Faces III and IV are adjacent. Faces O and I are adjacent. Faces I and II are adjacent. Faces II and III are adjacent. Faces O and III are adjacent. Vertices O and I are adjacent. Vertices III and IV are adjacent. Vertices II and III are adjacent. Vertices I and II are adjacent. Faces O and IV are adjacent. Vertices O and IV are adjacent.

11. Constructing the Dual of a Linkage Augmenting the geometry to the graph. Constructing the dual graph. The meaning of a directed edge in the dual graph: e=<t,h> is the flow (force) acting upon the head vertex (joint) by the edge (rod). The force in rod 4** acts upon the ground in this orientation. Constructing its topology. Kinematic system. O O Constructing the corresponding graph B 3 3 2* 4* A (CCW) B The topology arrow and the force arrow are in the same direction -> compression.Inverse directions - tension The type is compression. A (CCW) 3* (CW) 2 4 2 4 I I Two choices? The direction of the force in rod 4**. The force in rod 3** acts upon the ground in this orientation. The external force acts upon joint I 4 (CW) The direction of the force in rod 3**. Vertex I corresponds to joint I. 2 (CW) (CW) O2 O4 O The type is tension. O2 O4 O3 1 Adding the geometry. 1 1 Edge 2 is the potential source that corresponds to the driving link 2. Link 2 is the driving link. Vertex O2 corresponds to joint O2. Edge 3 corresponds to link 3. Faces I and O are adjacent. Vertex A corresponds to joint A. The potential source, edge 2, is between the two adjacent faces I and O in the original graph. Therefore, in the dual graph it corresponds to the flow source and it is between the two adjacent vertices I and O. Edge 4 is common to the two adjacent faces I and O thus the dual edge 4* is between the two adjacent vertices I and O. The relative velocity of link 2 corresponds to the potential difference of edge 2. The relative velocities of links 3 and 4 correspond to the potential differences of edges 3 and 4. The relative linear velocity corresponds to the potential difference. Edge 3 is common to the two adjacent faces I and O thus the dual edge 3* is between the two adjacent vertices I and O. O4 For consistency, the direction of the edge in the dual graph is defined by rotating the edge in the original graph in CCW direction. Vertices B and O4 correspond to joints B and O4, respectively. We can contract the edges with potential difference equal to zero. The kinematic analysis yields the magnitudes and directions of the angular and relative linear velocities. Edges 4 and 1 correspond to link 4 and the fixed link 1, respectively. 3** Reference face O corresponds to reference vertex O. 4** The angular velocity in CW corresponds to compressing force. Potential differences in edges 3 and 4 correspond to flows in edges 3* and 4*. Potential differences in the original graph correspond to flows in the dual graph. Potential differences of edge 2 corresponds to the flow in edge 2*. The angular velocity is CCW which corresponds to a tension. Face I corresponds to vertex I. Building the corresponding truss. I 4** O4 (tension) (compression) 3** (compression) O3 The dual graph. The corresponding truss.

12. dual dual dual We obtain the dual systems. Since linear velocity is associated with a joint in the linkage, its dual variable is associated with the face in the truss We have systematically developed a new variable in statics: Joint Face + At first, what is this absolute velocity? + What is a counterpart to absolute linear velocity of the joint B O3 3 The relative linear velocity of the input link corresponds to the external force. The relative linear velocity of the link 3 corresponds to the force in the rod 3**. The relative linear velocity of the link 4 corresponds to the force in the rod 4**. 1.The absolute linear velocity has a property of potential. Velocity Force A Why? On the other hand we know that velocity corresponds to force. O4 4 Absolute linear velocity corresponds to face force. Velocity of a joint Face Force 2 3** Because, We can give any absolute velocities to the links, and they will satisfy the rule of velocities (vectors KVL). 4** O2 O4 I 1 1

13. Force in rod 2 is equal to the subtraction of face forces P by FA. Force in rod 1 is equal to the subtraction of face forces FA by FO. Same direction corresponds to compression. Opposite direction corresponds to tension. Force in rod 3 is equal to the difference of face forces FA and FB. In the same manner, locate the forces in the other rods. An arbitrary point on equimomental line mPA: An arbitrary point on the equimomental line mPB: Face force FP acts in the face P. Face force FA acts in the face A. Face force FB acts in face B. The moment produced by the forces P and FB on the equimomental line mPB. The moment produced by the forces P and FA upon the equimomental line mPA. Each force in the rod is the difference of the right and the left face forces (Right and left defined according to the direction of the arrow in the edge). Absolute equimomental line mBO has to be determined. The circuit corresponds to the vertex. The circuit corresponds to the vertex. The equimomental lines that will locate mBO. Set arbitrary directions of the edges. In the same manner we can find the reaction R. mPB Thus we obtain the face force FB. Thus we obtain the face force FA. mPO mBO P compression O 2 mPA mRO mPO A P R mAO 4 tension 1 compression compression 3 B mPB mBR mPA O compression mBR 5 mAB A B R mAB mAO mRO

14. Finding and characterization of the dead center positions of the mechanism. O 2 7 1 4 4 5 3 O 2 5 3 7 O A C 1 6 6 2,3 5,7 C A 6 In this case, the faces B and O (i.e., the reference vertex) of the truss have the same face force which indicates that: (ii) links 5 and 7 are collinear. (i) links 2 and 3 are collinear, and These two conditions ensure that the mechanism is in a dead center position.

15. Another examples to find dead positions of the mechanism by Face Force. Given mechanism topology

16. 12 12 2 2 4 8 4 3 3 7 5 9 1 7 1 9 5 6 6 10 10 11 11 8 Rigid ???? 2 ’ 12 ’ 4 ’ 8 ’ 3 ’ 2 ’ 1 ’ 7 ’ 5 ’ 9 ’ 12 ’ 8 ’ 6 ’ 4 ’ 10 ’ 3 ’ 9 ’ 7 ’ 1 ’ 1 1 ’ 5 ’ 6 ’ 10 ’ R’ 11 ’ R’ Definitely locked !!!!! Duality relation between stability and mobility Due to links 1 and 9 being located on the same line By means of the duality transformation, checking the stabiliy of trusses can be replaced by checking the mobility of the dual linkage.