1. Modeling lattice modular systems with space groups
Nicolas Brener, Faiz Ben Amar, Philippe Bidaud
Laboratoire de Robotique de Paris
Université de Paris 6

2. Lattice Robot vs Crystal
Connectors have discrete positions in a lattice
Mobilities act on the position of the connectors
Mobility are discrete motions
Crystal
Atoms have discrete positions in a lattice
Symmetries act on the atoms positions
Symmetries are discrete isometries
Use of crystallographic knowledge
Discrete Motion  Chiral space groups
Connectors position  Wyckoff sets

3. Outline Discrete motion spaces Wyckoff positions Group hierarchy
Point groups
Lattice groups
Space groups
Wyckoff positions
Group hierarchy
Classification of lattice robots
Design of lattice robots
Next steps

4. Transformation Groups taxonomy
Isometry (length preserved)
Motion (length and orientation preserved)
Continuous
3D Isometry group
SE3
SO3
Translations group
Subgroup
point groups
230 Crystallographic Space groups types
Discrete rotation groups
Subset
65 Chiral Space Groups types (Sohncke Groups)
Discrete
32 Crystallographic point groups
11 Crystallographic rotation groups
14 Lattice groups types

5. Discrete motion spaces
Discrete rotation  chiral point group
Discrete translation  lattice group
Discrete motion  chiral space group

6. Discrete motion spaces
11 chiral point groups (rotation groups)
Finite sets
Point group 622
12 rotations
1 6-fold axes
6 2-fold axes
Generator:
{ 180° rotation along x, 60° rotation along z }
Geometric elements
6-fold rotation
2-fold rotation

7. Discrete motion spaces
Point group 432
24 rotations
3 4-fold axes
4 3-fold axes
6 2-fold axes
Generator:
{ 90° rotation along x, 90° rotation along y }
Geometric elements
4-fold rotation
3-fold rotation
2-fold rotation

8. Discrete motion spaces
Lattice groups
Infinite sets
Generators: 3 translations
14 types
Cubic lattice
{(a,0,0),(0,a,0),(0,0,a)}
Cubic centered lattice
{(-a,a,a), (a,-a,a), (a,a,-a)}
Cubic face centered lattice
{(a,a,0),(a,0,a), (0,a,a)}
Hexagonal lattice
{(a,0,0),(a/2, a√3/2, h)}
Triclinic lattice
{A,B,C}

9. Discrete motion spaces
Chiral Space Group
Infinite sets
Have rotations and translations (and screw)
65 types
Described in the « International Tables for Crystallography » edition th. Hahn

10. Discrete motion spaces
Geometric elements of P622, except screw axes
6-fold
3-fold
2-fold

11. Geometric elements of P432, except screw axes
Discret motion spaces
Geometric elements of P432, except screw axes
4-fold
3-fold
2-fold

12. Geometric elements of F432, except screw axes
Discret motion spaces
Geometric elements of F432, except screw axes
4-fold
3-fold
2-fold

13. Wyckoff positions and Connectors
identity
2-fold
3-fold
4-fold
6-fold
Orthogonal rotation symmetry
Tangential rotation symmetry

14. Space Group Hierarchy

15. Space Group Hierarchy Classification Minimum space group
60 other chiral space groups
Maximum chiral space groups
Minimum chiral space group
Classification
Minimum space group
Design in Maximum groups

16. Classification M-tran Mobilities match 4-fold axes of F432
Connectors match Wyckoff position e
The connectors have 4-fold symmetry

17. Classification Lattice System Space Group Connector M-tran F432 e
Wyckof position
Symmetry
orientation
M-tran
F432 e
4-fold
(1,0,0)
I-Cube
P432
Telecube P1
a,a,a
None NA
Atron
j,j
Molecule
3D universal j

18. Designing
Choose a space, and corresponding wyckoff position(s) and orientation(s).
A module is constructed by selecting some mobilities and connectors and gluing them together
Example: in F432 with 1 position in (a)

19. Designing
Choose a space, and corresponding wyckoff position(s) and orientation(s).
A module is constructed by selecting some mobilities and connectors and gluing them together
Example: in F432 with 1 position in (a)

20. Designing
Choose a space, and corresponding wyckoff position(s) and orientation(s).
A module is constructed by selecting some mobilities and connectors and gluing them together
Example: in F432 with 1 position in (a)

21. Designing
Choose a space, and corresponding wyckoff position(s) and orientation(s).
A module is constructed by selecting some mobilities and connectors and gluing them together
Example: in F432 with 1 position in (a)

22. Designing
Choose a space, and corresponding wyckoff position(s) and orientation(s).
A module is constructed by selecting some mobilities and connectors and gluing them together
Example: in F432 with 1 position in (a)
4-fold
2-fold
3-fold
Node
Connexion plate
Rotation axis

23. Conclusion Chiral Space groups give:
All possible mobility
All possible orientation, position and symmetries for connectors
Lattice design but chain type product (ex: M-tran)
Provides a framework to design several compatible systems, or extend existing systems with new modules.
But: it does not provide automatically an operational modular system…
Collision
Joint stops …

24. Further steps Formalize and automate classification.
Find rules on connector positions and mobilities that provide systems that efficiently reconfigure.
Develop a design interface.