CHECKING MOBILITY AND DECOMPOSITION OF LINKAGES Proceedings of the ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE 2011 August 29-31, 2011, Washington, DC, USA CHECKING MOBILITY AND DECOMPOSITION OF LINKAGES VIA PEBBLE GAME ALGORITHM Adnan Sljoka Department of Mathematics and Statistics, York University. Email: adnanslj@mathstat.yorku.ca Offer Shai Faculty of Engineering, Tel-Aviv University. Email: shai@eng.tau.ac.Il And Walter Whiteley Department of Mathematics and Statistics ,York University. Email: whiteley@mathstat.yorku.ca

Outline of the talk: 1. When and for what it was developed. 2. The main idea behind this algorithm. 3. Employing the pebble game for checking mobility of linkages with binary links. 4. Employing the pebble game for decomposing linkages into Assur Graphs. 5. Further research/work

1. When and for what it was developed. Pebble Game was developed by D. J. Jacobs and B. Hendrickson 1995, from the Department of Physics and Astronomy, Michigan State University. It was developed to study protein rigidity and flexibility (motion).

2. The main idea of this algorithm. - Each pebble ( ) represents one dof of a joint. - Each joint is assigned with two pebbles, two dof. - Main rule: you can assign a pebble to a link IFF its two end joints have two pebbles on each (guarantees there is no redundancy – topology SS). - When you assign a link by a pebble from a joint, that joint becomes the tail joint and the link becomes a directed link. The links with pebbles are directed and termed “covered”. - You can move pebbles from joints only through directed links an changing the directions of the links, respectively.

- The ground links are covered at the end without any restrictions. Results: After all the links are covered (except the driving link), only one free pebble should be left. When the run terminates, we can conclude: a. If there are links that were not covered -> there is a redundancy. b. The directed cut-sets define the Assur Graph decomposition. (directed cut-set: a set of links whose removal separates the linkage into at least two parts).

THE DECOMPOSITION INTO ASSUR GRAPH GROUNDING THE DRIVING LINK We want to cover link 7. Joint E has only one pebble, thus we transfer one pebble from joint C to E through directed links. Therefore, the direction of link 5 is reversed. A pebble is assigned to link 5 from joint E thus E is the tail joint. We want to cover link 4. Links 4,9,8 constitute a cut-set since removing them separates the linkage. E is mobile D C 7 6 5 4 9 8 E 5 C is mobile E C B is mobile 4 7 Covering the ground links. 6 B B A is mobile D A A 2 2 D is mobile 8 9 3 3 1 1

Pebble Game can identify regions with Redundancy If a link can’t be covered (it is impossible to bring two pebbles to each of its end joints), it indicates that there is a region with redundancy. It is also possible to find the redundantly region.

2 A B 1 3 5 6 D C 4 7 8 Link 6 is uncovered, no pebble was assigned to it, thus there is a redundancy.

Further Research/work 1. In 3d it was mathematically proved that this algorithm decomposes any 3d Linkage into 3d Assur Graphs. 2. In 3d there are still unsolved mobility problems, such a the - ‘double banana’. 3 In 2d and 3d to decompose the linkages into body-bar Assur Graphs.

Thank you !! More details can be found: Shai O., Sljoka A. and Whiteley W., "Directed Graphs, Decompositions, and Spatial Rigidity" submitted to Discrete Applied Mathematics, 2010.