1. 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.
Offer Shai
Faculty of Engineering, Tel-Aviv University.
And
Walter Whiteley
Department of Mathematics and Statistics ,York University.

2. 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

3. 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).

4. 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.

5. - 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).

6. 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

7. 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.

8. 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.

9. 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.

10. 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.