DETC2012-71289

1. Proceedings of the ASME 2012 International Design Engineering Technical Conferences &Computers and Information in Engineering ConferenceIDETC/CIE 2012August 12-15, 2012, Chicago, IL, USA DETC2012-71289 Mobility Determination Of Mechanisms Based On Rigidity Theory Slavutin Michael School of Mechanical Engineering Faculty of Engineering Tel Aviv University Tel Aviv, Israel Offer Shai School of Mechanical Engineering Faculty of Engineering Tel Aviv University Tel Aviv, Israel Andreas Müller Institute of Mechatronics Technical University Chemnitz Germany

2. Rigidity theory Rigidity theory deals mostly with the topological computation. Mechanism theory is mainly concerned with the geometrical analysis and generic statements. In rigidity theory there are two main types of representations: 1.Bar-joint representation. 2. Body-bar representation.

3. Bar and joint representation Between any two joints (2d-revolute, 3d-spherical) there exists at most one bar/constraint. Binary Link/one constraint 2d – revolute joint 3d – spherical joint 2d – revolute joint 3d – spherical joint

4. Body-bar Representation Bodies connected by bars (constraints). 1 Lower kinematicpair 2 1 2 Higher kinematic pair 2 1 1 2 Hinge Spherical joint 1 2

5. Representing a mechanism by Bar-joint and Body-bar representations Body-bar: Two bodies and three bars. Bar-joint: Nine bars and seven joints.

6. Definition of a Body-Bar Assur Graph • Graph G is a 2d/3d Body-Bar AssurGraph IFF • G has 3/6 DOF • 2. G does not contain any sub-graph • (of more than one element) which also • has 3/6 DOF. • There are at most 2/5 constraints between • any two bodies.

7. 2d Body-bar Assur Graph 1 1 2 2 3 3 (a) (b) Body-bar Assur Graph Not a body-bar Assur Graph

8. Analysis mechanisms through body-bar graphs Rigidity theory + Projective Geometry

9. Analysis mechanisms through body-bar graphs Property 1: The length of the projection of the velocity of any point P, onto a line (not at infinity) with coordinates, denoted by , is equal to the scalar productof Z, with the line coordinate vector:

10. Body-bar rigidity matrix D 2 E C A 1 B F Each joint of a boy imposes constraints on its motion. xC-xB xByC - xCyB xB-xC yC-yB yb-yC xCyB - xByC 0 1 xD-xE xEyD - xDyE 0 xE-xD yE-yD yD-yE xDyE - xEyD 2 xBll yBll xBl yBl Bll Bl 0 Fx 0 0 0 0 1 -yF = -1 xF Fy 0 0 0 0 0 vAx -yA Ax 0 1 0 0 0 xA 0 0 0 vAy -1 Ay 0

11. The spatial body-bar rigidity matrix for mechanisms Property 2:For the constraint between the two bodies there will be at most six entries written according to the following equation: From the latter equation it is possible to derive the projection of each point of body A, let it be Ai, on the line of the bar/constraint connected to it, li, as follows:

12. Example: Body-Bar Rigidity Matrix of Stewart Platform The schematic description of the Stewart Platform.

13. Pebble game Algorithm • Determines the generic mobility of the system and each element. • Reveals redundancies. • Decomposes the system into Assur Graphs (body-bar or bar-joint).

14. The main rule/theorem underlying pebble game: The sum of the pebbles on both two end elements before assigning a pebble to a constraint should be at least: dof of a rigid body + 1.

15. Assigning pebbled to the elements in dimension d Element can be body or joint. To each element we assign pebbles according to its dof.

16. Moving pebbles to the constraints from the elements Each constraint reduces onedof thus we move one pebble from the end element to the constraint. Elementi Element j Elementi Element j Element i Element j The directed constraint is directed from the element from which the pebble was taken.

17. 6 8 5 4 2 2 1 1 3 0 3 1 3 4 8 5 7 6 0 2

18. 7 6 Mechanical system 8 5 4 2 2 1 1 3 0 3 1 3 4 8 5 Body-bar Graph 0 7 6 2